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math.RA 2026-05-19 2 theorems

2-generated Monster type axial algebras fully classified

by Clara Franchi, Mario Mainardis +2 more

The Classification of the 2-generated Primitive Axial Algebras of Monster Type

Case analysis on parameters, subalgebras, axets and dimensions completes the list with explicit bases and products

abstract click to expand

Axial algebras of Monster type are a class of commutative algebras generated by special idempotents called axes. Some motivating examples of these algebras are the Griess algebra and the Norton-Sakuma algebras, relating to the Monster simple group. A long standing open problem is to classify the 2-generated axial algebras of Monster type. A huge milestone was accomplished by Yabe leading, with additional cases completed by Franchi, Mainardis, and McInroy, to the classification in the symmetric case. In this paper, we complete the classification. To do so, we split the proof into multiple cases: dealing with certain parameters, subalgebras, axets, and axial dimensions. Furthermore, we provide a basis, multiplication and information of the algebras in the classification; consolidating existing results on these algebras into one place.

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math.RA 2026-05-15 3 theorems

Octonion order from icosian doubling carries 240-element H4 shell

by Daniele Corradetti

Non-crystallographic systems of integers over composition algebras

The resulting self-dual rank-8 Zφ-order has genuinely octonionic multiplication and no norm-integral overorder or isotropic gluings.

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In this work we revisit classical systems of integers inside the real normed division algebras from the point of view of finite norm shells and root systems. Building on the icosian framework of Moody--Patera and on the integral root-system viewpoint of Chen--Moody--Patera and of Johnson, we isolate the precise axiomatic ingredients of the non-crystallographic analogue: an order over the golden ring $\Zphi$ together with a distinguished finite root shell whose Cartan coefficients lie in $\Zphi$. We show that the usual Gaussian, Eisenstein, Hamilton, Hurwitz and Coxeter--Dickson examples are recovered by separating the order, its units, and its distinguished finite shells; once the lattice requirement is replaced by a finite root-shell requirement, the golden integer ring becomes the natural coefficient ring for the non-crystallographic cases $H_2$ and $H_4$. We then construct a weak golden octonion order by Cayley--Dickson doubling of the icosian ring; the resulting free rank-$8$ $\Zphi$-order has a $240$-element finite shell of type $H_4\oplus H_4$ and its multiplication is genuinely octonionic. Finally, we prove (i) that this weak double is self-dual with respect to the polar norm pairing, hence has no strict norm-integral overorder, and (ii) that the first trace-integral discriminant tower over it contains no octonion-stable nonzero isotropic gluing.

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math.RA 2026-05-13 2 theorems

Evolution algebras from expanders meet Alon-Boppana bound

by Piero Giacomelli

Expander Evolution Algebras

The Cheeger constant of the graph sets connectivity and forces the optimal eigenvalue gap for the evolution operator over C.

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We introduce \emph{expander evolution algebras} (EEAs), a class of nonassociative algebras defined over an arbitrary field $\K$ in which the underlying undirected loopless graph of the algebra -- in the sense of Kowalski -- is an expander graph in the classical sense of Cheeger. Starting from the formal graph definition of Kowalski and the algebraic framework of Tian, we establish a dictionary between combinatorial expansion and algebraic structure: the Cheeger constant of the associated graph governs connectivity, the subalgebra lattice, the growth of the evolution sequence, and -- over $\R$ and $\C$ -- the spectral gap of the evolution operator. Over a general field $\K$ we prove that EEAs are always connected and simple (as evolution algebras), carry no proper large evolution subalgebras, and that every generator of a \emph{symmetric} EEA is algebraically persistent. Over $\C$ we obtain the sharp Alon--Boppana lower bound for the second eigenvalue of the evolution operator, leading to the definition of \emph{Ramanujan evolution algebras} as optimal expanders. We also construct families of EEAs from Cayley graphs of finite groups. We close with open problems.

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math.RA 2026-07-03

R[x;δ] strongly simple iff R simple

by Johan Öinert

Bimodules in differential polynomial rings

This gives a complete description of the R-sub-bimodules as only truncations or the full ring under those conditions.

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We study the $R$-sub-bimodule structure of differential polynomial rings $R[x;\delta]$ by introducing the notion of strong simplicity, requiring each nonzero $R$-sub-bimodule of $R[x;\delta]$ to be either $R[x;\delta]$ or the truncation $\sum_{i=0}^n R x^i$ for some $n \in \mathbb{Z}_{\geq 0}$. Our main result gives a complete characterization: $R[x;\delta]$ is strongly simple if and only if $R$ is simple, ${\rm char}(R)=0$, and the derivation $\delta$ is outer. We provide examples illustrating both when strong simplicity fails and when it holds.

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math.FA 2026-07-02

No indecomposable Banach space has the primary factorisation property

by Antonio Acuaviva, Bence Horváth +1 more

Pure infiniteness and primary factorisation

Absence for real and complex cases follows from relating the property to infiniteness of the quotient B(E)/M_E under the unique-maximal-idea

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We show that there is no real or complex indecomposable Banach space with the primary factorisation property (PFP). We relate the PFP of a Banach space $E$ to ring-theoretic infiniteness of $\mathcal{B}(E)$ and of $\mathcal{B}(E)/\mathcal{M}_E$, where $\mathcal{M}_E$ denotes the set of operators not factoring the identity on $E$, in the case it is the unique maximal ideal of $\mathcal{B}(E)$. For complex $E$ with the PFP, this quotient is purely infinite exactly when it is not scalar. We isolate the quantitative gap relevant to ultrapowers, identify classical sequence spaces as positive non-scalar cases, and show that Read's space $E_{\operatorname{R}}$ does not have the uniform PFP.

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math.RA 2026-07-02

Abe-Kanno subgroup correspondence extends to Hopf algebras

by Serge Skryabin

Hopf algebraic homogeneous spaces interpreted rationally: the Abe-Kanno theorem

The link between subgroups and invariant subfields of rational functions now holds for residually finite-dimensional Hopf algebras with arti

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We present a Hopf algebraic generalization of the Abe-Kanno theorem on a correspondence between subgroups of an algebraic group and invariant subfields of the field of rational functions. It applies to residually finite-dimensional Hopf algebras admitting an artinian classical quotient ring and is used in the paper to derive some general properties of such Hopf algebras.

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math.RT 2026-07-02

One derivation governs every double transposed Poisson structure on an algebra

by Maxime Fairon, Nikita Safonkin

Double Transposed Poisson Algebras

It yields GL_N-equivariant transposed Poisson brackets on representation algebras and on their invariant rings via the trace map.

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We introduce double transposed Poisson algebras, a noncommutative analogue of the transposed Poisson algebras of Bai, Bai, Guo and Wu that is compatible with the Kontsevich--Rosenberg principle. We first consider a simplified version which we call id-adapted double transposed Poisson algebras and then explore the general definition. We prove that every such structure on a unital associative algebra $\mathbb{A}$ is governed by a single derivation $\mathbb{A}\to\mathbb{A}\otimes\operatorname{S}(\mathbb{A}/[\mathbb{A},\mathbb{A}])$. Furthermore, this induces a $\operatorname{GL}_N$-equivariant transposed Poisson structure on each representation algebra $\mathbb{A}_N=\Bbbk[\operatorname{Rep}_N(\mathbb{A})]$. We also define $H_0$-transposed Poisson structures, the transposed counterpart of Crawley-Boevey's $H_0$-Poisson structures, and use the trace map to obtain a transposed Poisson structure on the ring of $\operatorname{GL}_N$-invariants $\mathbb{A}_N^{\operatorname{GL}_N}$.

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math.RT 2026-07-02

Silting mutation extended to infinite dimensions

by Diego Alberto Barceló Nieves

Large silting mutation in extriangulated categories

Theory defined for n-cosilting complexes over any ring and infinite n-tilting modules over finite-global-dimension rings.

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Silting mutation in triangulated categories, both at the level of objects and of subcategories, was introduced in arXiv:1009.3370, and later generalized to extriangulated categories in arXiv:2303.08125. It simultaneously encompasses the mutation theories of cluster-tilting objects in cluster theory and of compact 2-term silting complexes and support $\tau$-tilting modules in $\tau$-tilting theory. In this article, we develop an infinite-dimensional analog of silting mutation in extriangulated categories with set-indexed (co)products, which we then apply to obtain a theory of mutation for $n$-cosilting complexes over an arbitrary ring, as well as for infinite-dimensional $n$-(co)tilting modules over a ring of finite global dimension. The former theory is also shown to reinterpret the cosilting mutation introduced in arXiv:2201.02147.

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math.OA 2026-07-02

Graph conditions make ultragraph algebras residual finite-dimensional

by Daniel Gonçalves, Danilo Royer

Residual finite-dimensionality of ultragraph algebras via branching systems

For RFUM2 ultragraphs the combinatorial conditions become equivalent to RFD of both Leavitt path algebras and C*-algebras.

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We study residual finite-dimensionality for ultragraph algebras, both in the algebraic and in the C-star-algebraic settings. We introduce graph-theoretic RFD conditions for ultragraphs, extending the conditions that characterize RFD graph C-star-algebras. Using the boundary ultrapath branching system, we construct finite-dimensional branching-system representations associated to terminal boundary sets and no-exit cycles. These representations are used to prove that, whenever an ultragraph satisfies the graph-theoretic RFD conditions, its ultragraph Leavitt path algebra LK(G) is RFD, for every field K, and its ultragraph C-star-algebra RFD. For ultragraphs satisfying Condition (RFUM2), we prove converses in both settings. The analytic converse uses the groupoid model and the density of periodic points, while the algebraic converse is proved directly by finite-dimensional linear algebra. Thus, for RFUM2 ultragraphs, RFD of LK(G), RFD of C(G), and the graph-theoretic RFD conditions are equivalent. This gives, in particular, a common combinatorial description linking the algebraic and analytic theories, recovers the graph C-start-algebra characterization, and yields an algebraic characterization for Leavitt path algebras of graphs. We also construct an RFD ultragraph algebra which is genuinely outside the graph-algebra class in both settings.

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math.RA 2026-07-02

Block doubling on graphs creates high-corank Kac-Moody algebras

by Simon Beaudoin, Quentin Bonnefoy +3 more

On a new class of high-corank Kac-Moody algebras

Recursive families show exponential corank growth and link it to the multiplicity of adjacency eigenvalue 2.

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We present recursive constructions of several families of generalized Cartan matrices associated with Kac-Moody algebras, whose sizes and coranks grow exponentially. The constructions are encoded by connected multigraphs and by block-doubling operations on their associated symmetric generalized Cartan matrices. Equivalently, the corank problem is translated into a spectral graph-theoretic problem: the corank of $2\mathrm{Id}-\operatorname{Adj}(G)$ is the multiplicity of the adjacency eigenvalue $2$. We give two explicit recursive families, compute their spectra and coranks, and emphasize the difference between absolute exponential growth and relative asymptotic density. The resulting algebras are typically indefinite and singular of corank larger than one, and therefore contain several independent central directions and several isotropic radical directions in the root lattice. We also discuss alternative constructions and possible applications to the algebraic structures appearing in gravity, supergravity, string/M-theory and related generalized symmetry problems.

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math.RA 2026-07-02

Units generate finite rings when gcd-graphs connect

by Ján Mináč, Tung T. Nguyen +1 more

Sums of units in finite rings and applications to Cayley graphs

The link ties algebraic generation to graph connectivity and field equation solvability.

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The question of whether a ring is additively generated by its units has been studied from several perspectives in ring theory and algebraic graph theory. In this paper, we investigate this problem for finite rings, not necessarily commutative, and relate it to the connectedness of gcd-graphs, the existence of perfect state transfer, and the solvability of certain equations over finite fields. Additionally, we discuss a generalization of this question in which only certain normalized units are allowed in the generating set. Our work intersects algebra, number theory, and graph theory, and may be of interest to a broad audience.

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math.RA 2026-07-02

2-step nilpotent Lie superalgebras admit pure local superderivations

by Xiaohui Chi, Huiyi Zhang +2 more

Local (Anti-)Superderivations on Nilpotent Lie Superalgebras

The maps satisfy the derivation rule on every single element yet fail globally when the algebra is finite-dimensional and the field has char

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In this paper, we study local (anti-)superderivations on finite-dimensional nilpotent Lie superalgebras. Firstly, we prove that every finite-dimensional 2-step nilpotent Lie superalgebra over a field $\mathbb{F}$ with $\operatorname{char}\mathbb{F}\neq2$ admits pure local (anti-)superderivations (namely, local (anti-)superderivations that are not (anti-)superderivations). Then for $n$-step nilpotent Lie superalgebras over arbitrary fields with n greater than 2, we provide a sufficient criterion to guarantee the existence of pure local (anti-)superderivations. Furthermore, we show that 3-step nilpotent Lie superalgebras admit pure localsuperderivations.

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math.RA 2026-07-02

The paper establishes degree obstructions to the equivalence of generalized Airy…

by Yichuan Cao, Ruyong Feng +2 more

The Equivalence Problem for Generalized Airy Operators

Proves degree obstructions for equivalence of generalized Airy operators of the same type and answers Katz's 1987 question.

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In this paper, we establish degree obstructions to the equivalence of generalized Airy operators of the same type. As an application, we answer a question posed by Nicholas M. Katz in Inventiones Mathematicae (87, pp. 13-61,1987). The main results of Sections 3 and 4 were obtained through a close interactive collaboration between the authors and the artificial intelligence agent system MechMath Agent Team (MMAT).

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math.RA 2026-07-02

Solvability equals right nilpotency for transposed Novikov-Poisson algebras

by Jiarou Jin, Yanyong Hong

Solvability and nilpotency of transposed Novikov-Poisson algebras

The algebra is solvable exactly when it is right nilpotent and when its square is nilpotent; the same holds for the two underlying algebras.

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In this paper, we develop the theory of nilpotency and solvability for transposed Novikov-Poisson algebras. We first establish several equivalent conditions for a dialgebra to be nilpotent, and show that the lower central series of a transposed Novikov-Poisson algebra $P$ admits a simplified form. We then prove that $P$ is solvable if and only if it is right nilpotent, and also if and only if $P^2$ is nilpotent. Moreover, we show that nilpotency (respectively, solvability) of a transposed Novikov-Poisson algebra is equivalent to nilpotency (respectively, solvability) of both its underlying commutative associative algebra and its underlying Novikov algebra. Finally, we prove that It\^{o}'s theorem holds for transposed Novikov-Poisson algebras.

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math.RT 2026-07-01

Fukaya category uniquely deforms NilHecke algebra under Z2 grading

by Jasper van de Kreeke

Fukaya categories of Coulomb branches as unique deformations

Removing the matter divisor from horizontal Hilbert schemes reduces their Fukaya category to the single Z^2-graded deformation of the NilHec

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The symplectic geometry of Coulomb branches is complicated and it is particularly difficult to determine their Fukaya categories. Relative Fukaya categories present an approach to circumvent these difficulties by first computing the Fukaya category of the complement of a divisor and then solving a deformation problem. In this paper, we apply this approach to the specific case of horizontal Hilbert schemes by removing their matter divisor and narrowing down the set of possible deformations through an additional $ \mathbb{Z}^2 $-grading. We utilize an existing description of the Fukaya category after removal of the matter divisor, in particular we use a specific generating Lagrangian and the identification between its endomorphism algebra and the NilHecke algebra. The core of this paper consists of solving the deformation problem, after which we recover the result of Aganagic et al.

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math.AC 2026-07-01

Weaker hypotheses extend intersection theorems to DG-rings

by Luigi Ferraro, Zachary Nason

Intersection theorems over DG-rings revisited

The generalizations improve prior bounds and characterize Cohen-Macaulay DG-rings by the existence of finite length finite projective dimens

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In this work we generalize two recently proved intersection theorems for DG-rings. The Derived Improved New Intersection Theorem concerns the length of semi-free DG-modules over DG-rings and it was recently proved by the second author. We show that it holds under weaker hypotheses. Foxby's Intersection Theorem was generalized to DG-rings by Yang and we improve the inequality that they provided. As an application we prove a DG version of the classic result that finite length modules of finite projective dimension only exist over Cohen-Macaulay rings, generalizing another result of Yang.

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math.RA 2026-07-01

Jordan plane liftings yield Hopf algebras with completely prime ideals

by Tao Lu

On a family of liftings of the Jordan plane

Every nonzero ideal meets the center and the Dixmier-Moeglin equivalence holds for the family over the infinite cyclic group.

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We study a family of Hopf algebras arising as liftings of the Jordan plane over the infinite cyclic group. We determine their centres, prime and primitive spectra, and automorphism groups. We show that every prime ideal is completely prime and that every nonzero ideal intersects the centre nontrivially. We construct explicit simple modules corresponding to all primitive ideals and classify the finite-dimensional simple modules. Finally, we prove that these Hopf algebras satisfy the Dixmier--Moeglin equivalence.

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math.RA 2026-07-01

Subalgebras of graded matrix rings embed into elementary graded ones

by Pavel Sokolov

Embedding a graded matrix algebra into an elementary graded

Any subalgebra of a full matrix ring over a graded division ring fits inside an elementary graded matrix ring.

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M.V. Zaicev and S.K. Segal, as well as S. D\u{a}sc\u{a}lescu, B. Ion, C. N\u{a}st\u{a}sescu, and D. Raios Montes studied certain gradings on matrix rings and algebras - 'elementary' gradings. However, examples of gradings on a matrix ring that are not elementary are known. In the present article, we show that any subalgebra of a full matrix ring over an arbitrary graded division ring can be embedded in an elementary graded matrix ring.

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math.RA 2026-07-01

Two random elements generate special linear Lie algebras

by Urban Jezernik, Andoni Zozaya

Random generation of the special linear Lie algebra over finite fields

Probability tends to 1 as cardinality grows, except for the pairs (3,3) and (4,2).

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We prove that the special linear Lie algebra $\mathfrak{sl}_n(\textbf{F}_q)$ over a finite field of characteristic $p$ is generated by two random elements with high probability as $|\mathfrak{sl}_n(\textbf{F}_q)|$ tends to infinity, provided that $(n,p) \neq (3,3), (4,2)$.

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math.RA 2026-07-01

Zinbiel-dendriform bialgebras affinize exactly to completed pre-Lie bialgebras

by Qinxiu Sun

Infinite-dimensional pre-Lie bialgebras induced from Leibniz-dendriform bialgebras and Zinbiel-dendriform bialgebras

Tensor products with quadratic Z-graded algebras of the opposite type produce the structures and lift YBE solutions to the S-equation.

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In this paper, we establish a completed pre-Lie bialgebra structure on the tensor product of a Leibniz-dendriform bialgebra and a quadratic $\mathbb{Z}$-graded Zinbiel algebra. We also obtain such a structure on the tensor product of a Zinbiel-dendriform bialgebra and a quadratic $\mathbb{Z}$-graded Leibniz algebra. Moreover, a Zinbiel-dendriform bialgebra is precisely one whose affinization by a special quadratic $\mathbb{Z}$-graded Leibniz algebra is a completed pre-Lie bialgebra. Finally, using solutions of the ZD-YBE (resp.~LD-YBE) with invariant skew-symmetric parts in a Zinbiel-dendriform (resp.~ Leibniz-dendriform) algebra, we construct completed solutions possessing invariant symmetric parts of the $S$-equation in the induced pre-Lie algebra.

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math.RA 2026-07-01

Super Jordan plane localizes to matrix algebra over Weyl algebra

by Tao Lu

Prime spectrum and representations of the super Jordan plane

The identification proves the ring is prime and classifies its ideals and modules.

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We study the ring-theoretic structure and representation theory of the super Jordan plane $\mathcal{J}$ over fields of characteristic different from $2$. We prove that $\mathcal{J}$ is prime and classify its prime, primitive, and maximal ideals. We determine its classical ring of quotients and classify the finite-dimensional simple modules, while relating infinite-dimensional simple modules to those of the first Weyl algebra. Our approach is based on showing that a localization of $\mathcal{J}$ is a matrix algebra over a localization of the first Weyl algebra.

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math.GR 2026-06-30

Isomorphic even Artin groups of FC type share p-parts of defining graphs

by Marcos Escartín Ferrer, Giorgio Leoni +1 more

Cohomology rings and p-local behavior of even Artin groups

For every prime p the p-parts match, extending right-angled rigidity via cohomology and pro-p computations.

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We generalize to certain families of even Artin groups several classical results on right-angled Artin groups. In particular, we compute the cohomology ring, describe the pro-$p$ completion, and determine the $p$-Zassenhaus restricted Lie algebra in the FC case. As a by-product, we prove a rigidity result that implies that if two even Artin groups of FC type are isomorphic, then for every prime $p$, the $p$-parts of their defining graphs are isomorphic.

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math.RA 2026-06-30

Cyclic associative cohomology classifies extensions

by Hassan AlHussein

On the Cohomology of Cyclic Associative Algebras

H^2_cyc classifies extensions of cyclic associative algebras and sits between HC and HH via inclusions.

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We introduce a cohomology theory for cyclic associative algebras, a subclass of shift associative algebras defined by the identity $(xy)z = x(yz) = y(zx)$. This cohomology, denoted $H^\bullet_{\mathrm{cyc}}(A, M)$, is a subtheory of Hochschild cohomology obtained by restricting to cochains that satisfy a cyclic compatibility condition derived from the defining identity. We prove that $H^2_{\mathrm{cyc}}(A, M)$ classifies cyclic associative extensions of $A$ by a cyclic bimodule $M$. The universal derivation and the module of differential forms $\Omega^\bullet_{\mathbb{F}}(A)$ are constructed, and $(\Omega^\bullet_{\mathbb{F}}(A), d)$ is shown to be the universal cyclic differential graded algebra over $A$. For trivial coefficients, we establish natural inclusions $HC^n(A) \hookrightarrow H^n_{\mathrm{cyc}}(A, \mathbb{F}) \hookrightarrow HH^n(A, \mathbb{F})$, placing our theory intermediate between Connes' cyclic cohomology and Hochschild cohomology.

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math.RA 2026-06-30

New s-notions recover matrix irreducibility for all tensor orders

by Jianhong Xu

An Alternative Framework for Irreducibility and Primitivity of Nonnegative Tensors

Definitions of s-irreducibility and s-primitivity contain every matrix result as a special case while adding statements needed for higher-or

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Motivated by some recent studies on higher order Markov chains and well-known characterizations for irreducibility and primitivity of nonnegative matrices, we propose in this paper an alternative framework for irreducibility and primitivity of nonnegative tensors, giving rise to the concepts of s-irreducibility and s-primitivity. This framework includes the relevant results on matrices as its special cases, yet it expands existing results regarding irreducibility and primitivity for tensors. In addition to its tensor theoretic significance, such a framework has important implications for applied fields, especially when it comes to higher order Markov chains.

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math.GR 2026-06-30

Skew braces satisfy Schur-Zassenhaus for Hall ideals

by M. Ferrara, M. Trombetti

The Schur--Zassenhaus Theorem and Sylow's Third Theorem for Finite Skew Braces

Hall ideals admit sub-skew brace complements and Sylow p-subbraces number 1 mod p

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In this short note we establish the Schur--Zassenhaus Theorem and Sylow's Third Theorem for finite skew braces. More precisely, we prove that every Hall ideal of a finite skew brace admits a sub-skew brace complement, and more generally that every left ideal whose order is coprime to that of the Hall ideal can be embedded in such a complement. Using similar ideas we show that every left ideal of prime-power order is contained in a Sylow sub-skew brace. Finally, we prove that the number of Sylow $p$-sub-skew braces is congruent to $1$ modulo $p$, and provide examples showing that the corresponding containment property fails for arbitrary sub-skew braces.

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math.GR 2026-06-30

Submonoids of FIM(1) satisfy FP₂ exactly when finitely presented

by Carl-Fredrik Nyberg-Brodda

On homological finiteness properties and free inverse monoids

A criterion based on idempotent lattice actions proves the equivalence for finitely generated cases and shows free inverse monoids lack FP₂.

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We construct a simple and useful sufficient condition, based on actions on a lattice of idempotents, for monoids admitting homomorphisms to the monogenic free inverse monoid $\mathrm{FIM}(1)$ to not be of type $\mathrm{FP}_2$. This recovers a result of Gray and Steinberg that free inverse monoids are not of type $\mathrm{FP}_2$. The same technique is then used to show that a finitely generated submonoid of $\mathrm{FIM}(1)$ is of type $\mathrm{FP}_2$ if and only if it is finitely presented, answering a question of Cho & Ru\v{s}kuc.

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math.RA 2026-06-30

Cartan subsemigroup yields Steinberg algebra via ultrafilter groupoid

by Tristan Bice, Malcolm Jones +1 more

Steinberg Algebras of Ample Semicategories and their Boolean-Cartan Restriction Semigroups

The reconstruction characterizes algebras with suitable restriction subsemigroups as Steinberg algebras of their ultrafilter groupoids, exte

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We extend the construction of Steinberg algebras of ample groupoids to \'etale semicategories. We also relate ample semicategories to Boolean restriction semigroups via a representation result extending previously known results for categories. Furthermore, we prove a reconstruction result which characterises an abstract algebra $A$ with a certain Cartan-like restriction subsemigroup $B$ (subject to conditions resembling those defining quasi-Cartan pairs) as the Steinberg algebra of the ultrafilter groupoid of $B$. In this way we obtain a twist-free extension of previous Steinberg algebra reconstruction results.

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math.CT 2026-06-30

Braided cogroupoids create monoidal equivalences on comodules

by Thi Hoa Nguyen (LMBP)

Braided cogroupoids

The structures generalize transmutation and bosonization from Hopf algebras and relate representation categories while preserving tensor pro

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We introduce and develop the theory of braided cogroupoids, a class of algebraic structures generalizing cogroupoids in a braided setting. We show that braided cogroupoids induce monoidal equivalences between the associated comodule categories, and we generalize Majid's transmutation and bosonization of braided Hopf algebras to the cogroupoid setting. Several examples are studied in detail, including the braided $SL_{n}$ cogroupoid and the braided bilinear cogroupoid.

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math.AG 2026-06-30

Springer's odd degree theorem holds over LG rings

by Philippe Gille (ICJ, AGL +2 more

Quadratic Spaces and Orthogonal Groups over semilocal Rings

Norm principles of Scharlau and Knebusch also extend to quadratic forms over semilocal rings, yielding results on spin group cohomology.

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We prove Springer's Odd Degree Theorem for quadratic forms over LG rings, and Scharlau's and Knebusch's norm principles for quadratic forms over semilocal rings. We present applications to the flat cohomology of spin groups and {\'e}tale norm groups.

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math.RA 2026-06-30

Axial algebras survey adds many new open problems

by I. Gorshkov, S. Shpectorov

Axial Algebras: Questions and Conjectures

Paper complements prior review with additional questions on algebras generated by axes and their group connections.

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Axial algebras are non-associative algebras generated by idempotents, called axes, whose adjoint action satisfies a fusion law. When this fusion law is graded, axes naturally lead to automorphisms of the algebra, and so such axial algebras are inextricably linked with groups. This article is meant to complement the recent survey \cite{ms} by significantly expanding the list of interesting open problems suggested by the specialists in the field, and providing a further discussion of the related concepts and available results.

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math.RA 2026-06-29

Pure-tilting hereditary rings equal hereditary noetherian rings over von Neumann regular b

by Umamaheswaran Arunachalam

Pure projective tilting modules associated with a special ring and Goresntein properties

The equivalence also shows that associated module-category hearts are Grothendieck categories and ties Gorenstein tilting modules to a stati

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In this paper, we study pure-projective tilting modules and related classes of rings. We introduce the notion of a pure-tilting hereditary ring, namely, a ring over which every ideal is pure-projective tilting, and investigate its structural properties. We prove that a ring R is a pure-tilting hereditary ring if and only if R is hereditary noetherian over a von Neumann regular ring R. In the commutative case, we show that R is a pure one-tilting hereditary ring precisely when R is hereditary noetherian. Using Kaplansky conjecture, we establish a connection between pure-tilting hereditary rings and the hereditary noetherian property of prime factor rings. In category theory, for the torsion pair consisting of Gen of I and the orthogonal class of I in the category of R-modules, we establish that the associated Happel-Reiten-Smalo heart H sub I is a Grothendieck category. We also examine the characterization of Ext-orthogonal classes determined by pure projective tilting modules. In addition, we show that every Gorenstein pure projective tilting module is Gorenstein flat if and only if every Gorenstein pure projective tilting module is strict T-stationary, where T denotes the class of all finitely presented tilting modules. These results establish new links between tilting theory, hereditary ring conditions, and Gorenstein homological structures.

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math.RA 2026-06-29

Bijective skew PBW extensions over prime PI-algebras have nontrivial centers

by James Gómez, Claudia Gallego

The PI property of skew PBW extensions

The result lets researchers check if these generalized polynomial rings satisfy polynomial identities by inspecting their centers.

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In this article we study the polynomial identity (PI) property of skew PBW extensions. We show that every bijective skew PBW extension over a prime PI-algebra has nontrivial center. This fact allows us to determine, from the known description of the center in several classes of examples, whether such extensions satisfy a polynomial identity. Furthermore, building on results of Brown and Zhang \cite{BrownZhang2022}, we investigate the PI property of certain $\K$-algebras over fields of positive characteristic.

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math.RA 2026-06-29

Deformation maps unify operators on Lie conformal algebras

by Taoufik Chtioui, Sami Mabrouk +1 more

Deformation maps on quasi-twilled Lie conformal algebras

Right and left maps subsume r-matrices, Rota-Baxter operators and more as Maurer-Cartan elements in derived L∞-algebras.

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In this paper, we develop a unified approach for various operators on Lie conformal algebras. Given a quasi-twilled Lie conformal algebra $(\Ep,\Vs,\Ws)$, we introduce two dual families of operators: \emph{right deformation maps} $D:\Vs\to\Ws$ and \emph{left deformation maps} $B:\Ws\to\Vs$. Each family simultaneously subsumes several classical structures: modified $r$-matrices, crossed homomorphisms, derivations, and Lie conformal algebra homomorphisms in the right case, relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators, and deformation maps of matched pairs in the left case. Using Voronov's derived bracket method, we construct the controlling homotopy algebras: a curved $L_\infty$-algebra governing right deformation maps and an $L_\infty$-algebra governing left deformation maps, with Maurer-Cartan elements precisely characterizing each type. We further develop the associated deformation theories via twisted $L_\infty$-algebras and define cohomology complexes for both types of deformation maps, recovering and extending the cohomologies of all classical and conformal operators already developed in the literature.

0

math.GR 2026-06-29

Ideal chain conditions imply finite generation for soluble skew braces

by Massimiliano Di Matteo, Ramón Esteban-Romero +2 more

Chain conditions on skew braces and solutions of the Yang-Baxter Equation

This extends classical group theorems to the skew braces that encode non-degenerate solutions of the Yang-Baxter equation.

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Classical works of Hall and McLain show that solubility and local nilpotency play a key role in deriving finite generation in groups from maximal or minimal conditions on normal subgroups. In this work, brace-theoretical analogues of Hall's and McLain's results are analysed for skew braces satisfying the maximal or minimal condition on ideals. We also introduce finiteness and chain conditions on non-degenerate set-theoretic solutions of the Yang-Baxter equation, and their impact on associated structure and permutation skew braces of solutions is also described.

0

math.RA 2026-06-29

All extending involutions on Cayley-Dickson doubles classified

by Masood Aryapoor, Per Bäck +1 more

Involutions in the Cayley-Dickson construction

The classification also identifies isomorphisms between doubles and proves the classical involution is the only scalar one.

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We determine all involutions in the Cayley-Dickson construction that extend the involution of the original $*$-algebra. We also find all algebra isomorphisms between the resulting Cayley doubles that extend the identity automorphism of the original $*$-algebra, and consequently classify the resulting $*$-algebras up to $*$-algebra isomorphism. As applications, we show that Cayley doubles without zero divisors admit exactly one additional involution, prove that the classical Cayley-Dickson involution is the unique scalar involution, and obtain a classification of the $*$-algebras arising from $\mathbb{R}$ up to dimension 4.

0

math.CO 2026-06-26

C-vector projections fill bands except on Ã_n source-sink diagonals

by Sarah B. Brodsky

Coordinate projections of c-vectors of cluster algebras from the annulus

The Auslander-Reiten defect blocks the source-sink pair in annulus type while all other coefficient-one bands fill completely.

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For an acyclic cluster algebra, the $c$-vectors are, up to sign, the real Schur roots of the associated root system. We study the two-coordinate projections $(c_v, c_w)$ of this configuration: when the difference $c_v - c_w$ is bounded the image lies in a band of lattice lines, and we ask when the projection fills that band. A band-existence dichotomy, valid in every acyclic type, shows the difference is bounded if and only if the null root satisfies $\delta_v = \delta_w$. For affine type $\widetilde{A}_n$ (the annulus), in the source-sink orientation, we resolve the filling question completely: every coordinate projection fills its band except along the source-sink diagonal, which carries only the finite regular part. The obstruction is the Auslander--Reiten defect, which a projection sees on its diagonal exactly when the defect is a coordinate difference; the only such pair is the source-sink pair of $\widetilde{A}_n$, so the pattern depends on the chosen seed. More generally, every banded pair of null-root coefficient one fills, except these diagonals. Off the diagonal a banded pair in $\widetilde{E}_7$ fails to fill, so non-filling is not confined to type $\widetilde{A}_n$; a computation classifies the pairs of coefficient at least two over a range of affine types, where this $\widetilde{E}_7$ pair is the only further failure, and the general classification remains open.

0

math.RA 2026-06-26

Algebras with (x^3)^2 = (x^2)^3 keep explicit idempotent fusion rules

by Daniel J. F. Fox, Vladimir G. Tkachev

Commutative algebras satisfying univariate identities with vanishing Peirce polynomial

The identity forces a vanishing Peirce polynomial yet still yields multiplication rules between Peirce spaces for every lambda except 1/2.

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We introduce and study $(2,3)$-palintropic algebras, a class of commutative algebras defined by the identity $(x^{3})^2 - (x^{2})^3 = 0$. This specific relation is the simplest generator of the $2$-dimensional space of minimal-degree evanescent identities in degree $6$, and encompasses several well-studied structures, including Jordan and medial algebras. The primary motivation for investigating these algebras lies in their trivial Peirce polynomials, which removes a priori restrictions on the spectrum of the multiplication operator associated with an idempotent. In this paper, we review and further develop the theory of Peirce operators, Peirce polynomials, and second-order linearizations. We demonstrate that despite the triviality of the Peirce polynomial, any idempotent $c$ admits well-behaved, explicit fusion rules for multiplication between its $\lambda$-Peirce spaces for $\lambda \neq \tfrac{1}{2}$. Furthermore, we prove that multiplication by such an idempotent always constitutes an algebra homomorphism. Finally, we present concrete examples of $(2,3)$-palintropic algebras and provide applications of these algebraic structures to commutative polynomial maps.

0

math.GR 2026-06-26

Free right nilpotent skew braces have free groups

by Eric Jespers, Thomas Letourmy +3 more

Free Skew Braces and Free Solutions of the Yang--Baxter Equation

An explicit construction shows they are residually finite and Hopfian, with free solutions having solvable word problem.

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We offer a workable construction of the free right nilpotent skew braces of arbitrary class which allows us to prove (among many other things) that this free object has free additive/multiplicative groups, and that it must also be residually finite and Hopfian. We introduce the class of right nilpotent solutions, which correspond to right nilpotent skew braces. As a consequence of our construction, the free solutions in this class have a solvable Word Problem, and every law holding for finite solutions of the previous type also holds for every solution of the same type. In the remainder of the paper, we present further explicit realizations of free objects and explore their consequences. Among these are free two-sided skew braces of abelian type (with an abelian multiplicative group) and free centrally nilpotent skew braces of class 2.

0

math.RA 2026-06-26

E8 Lie algebras over rings include non-graded isomorphic forms

by Seidon Alsaody, Jari Desmet

Groups of type E₈ over rings via TKK-algebras and their extremal elements

An E7-torsor over the scheme of extremal pairs parametrizes graded classes and is non-trivial, unlike the field case.

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Over any commutative ring containing $\tfrac16$, we study Lie algebras $L$ of type $\mathrm{E}_8$ that arise from the Tits--Kantor--Koecher (TKK) construction on a Brown algebra, and their twisted forms. We construct a smooth scheme $\mathbf{Y}$ of pairs of extremal elements in $L$. When $L$ arises from the TKK-construction, we express the automorphism group, of type $\mathrm{E}_8$, as an $\mathrm{E}_7$-torsor over $\mathbf{Y}$. We show that twisting by this torsor produces the graded isomorphism classes of those algebras isomorphic to $L$, and parametrize these classes by using $\mathbf{Y}$. We show that this torsor is non-trivial, yielding isomorphic Lie algebras of type $\mathrm{E}_8$ that are not graded isomorphic, as opposed to the behaviour over fields.

0

math.KT 2026-06-25

Matrix stability guarantees Morita invariance in K-theory

by Eugenia Ellis, Emanuel Rodríguez Cirone

Matrix stability and Morita invariance

Bivariant algebraic K-theory respects Morita equivalence for G-algebras and G-graded algebras, including crossed-product equivalences under

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Let $G$ be a group. We prove that matrix stability for either $G$-algebras or $G$-graded algebras guarantees Morita invariance. As a consequence, bivariant algebraic K-theory (either $G$-equivariant or $G$-graded) is Morita invariant. In particular, we show that if $G$ is a finite group acting freely on a finite simplicial set $X$, then $\ell^X\rtimes G$ and $\ell^{X/G}$ are kk-equivalent. Here, $\ell^Y$ denotes the $\ell$-algebra of piecewise polynomial functions on $Y$ with coefficients in the ground ring $\ell$.

0

math.GR 2026-06-25

Skew left braces obey Grün's lemma and Baer's theorem

by A. Ballester-Bolinches, R. Esteban-Romero +2 more

Analogues of Gr\"un's lemma and Baer's theorem for skew left braces

Classical results on centres and central series extend to infinite skew left braces through their trifactorised groups.

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We prove in this paper some analogues of the well-known group-theoretical Gr\"un's lemma, stating that in a perfect group the first and the second centre coincide, and Baer's theorem, stating that if the quotient by the nth centre of a group is finite, then so is the $(n + 1)$th term of the lower central series, in the scope of nfinite slew left braces. These results represent significant improvements over previous work. The trifactorised group associated with a skew left brace will be crucial for our proofs.

0

math.DG 2026-06-25

Condition for trivializing deformations of Lie subalgebras

by Ilias Ermeidis

On the Moser trick for Lie subalgebras and foliations

Necessary and sufficient criterion given via direct Moser proof for foliations, extending to subalgebroids

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Given a smooth deformation of a Lie subalgebra, we establish a necessary and sufficient condition for its smooth triviality and derive an analogous criterion for Lie ideals. We then give a direct proof of the Moser trick for foliations, which forms the basis for extending this result to general Lie subalgebroids.

0

math.HO 2026-06-25

Ring C[D] turns ODE solution rules into algebraic facts

by Hussain Al-Rasheed

An Algebraic Viewpoint on Linear Differential Equations

Kernels and cosets replace heuristics once differential operators are polynomials acting on smooth functions.

abstract click to expand

Classical methods for solving linear ordinary differential equations, such as superposition, the method of undetermined coefficients, and the annihilator technique, are often presented as heuristic, procedural rules. In this article, we show that these methods admit a coherent algebraic interpretation when constant-coefficient linear differential operators are viewed as elements of the polynomial ring $\mathbb{C}[D]$, acting on spaces of smooth functions. Without invoking the formalism of $D$-modules or non-commutative operator algebras, we explain how homogeneous solution spaces arise as kernels of linear operators, how particular solutions form affine cosets, and how the search for solutions is an infinite-dimensional eigenvalue problem. Furthermore, we extend this algebraic framework to variable-coefficient equations, resolving the Euler equation through ring isomorphisms and framing d'Alembert's reduction of order as non-commutative operator factorization. We also explore the boundary of this linear theory, demonstrating how diffeomorphic linearization allows certain non-linear equations -- such as those of Bernoulli and Riccati -- to be mapped directly into the $\mathbb{C}[D]$-module framework. Finally, we contrast this framework with the multivariable ring $\mathbb{C}[D_x, D_y]$, using the loss of the principal ideal domain property to explain the intrinsic structural divergence of partial differential equations, and indicate further universal extensions to discrete difference equations and the Weyl algebra.

0

math.RT 2026-06-25

Cleft extension restriction functors match on singular equivalences under equivariance

by Miltiadis Karakikes

Equivariant Cleft Extensions and Singular Equivalences

The ordinary and equivariant versions induce singular equivalences together once the cleft extension lifts to the group action setting.

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We study the equivariant lifting of cleft extensions of abelian categories and its impact on singularity categories. Specifically, we establish the necessary framework for lifting a cleft extension to a G-equivariant cleft extension. Furthermore, we prove that a restriction functor associated to a cleft extension induces a singular equivalence if and only if its equivariant counterpart does. As a concrete application, we demonstrate that the skew group ring of a $G$-equivariant $\theta$-extension is isomorphic to a $\widehat{\theta}$-extension of the base skew group ring, allowing us to lift singular equivalences for these structures.

0

cs.LO 2026-06-24

Datalog solves CSPs exactly for structures from n+2 tournaments

by Sebastian Meyer, Florian Starke

Almost Symmetric Linear Arc Monadic Datalog and Transitive Tournaments

n-almost symmetric linear arc monadic Datalog works on those built by primitive positive constructions from transitive tournaments on n+2 ve

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We introduce $n$-almost symmetric Datalog and study $n$-almost symmetric linear arc monadic Datalog. We characterize the finite relational structures whose constraint satisfaction problem is solved by this Datalog fragment as those that can be primitive positively constructed from the transitive tournament on $n+2$ vertices. We also give characterizations in terms of a certain homomorphism duality (which we call $n$-fixed unfolded caterpillar duality) and in universal-algebraic terms (the existence of $k$-absorptive operations and of operations forming an elevator chain of length $n+1$). This article generalizes the results from Bodirsky and Starke about symmetric linear arc monadic Datalog.

0

math.RA 2026-06-24

All smooth central Sklyanin quadrics are standard

by Izuru Mori, Kenta Ueyama +1 more

Irreducible noncommutative quadrics

Classifying every singular case shows that smoothness forces the standard form.

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In this paper, we study irreducible noncommutative quadrics $S/(f)$ via noncommutative graded matrix factorizations. We show that the line modules over $S/(f)$ are described by the rulings arising from indecomposable noncommutative linear matrix factorizations of $f$ of rank $2$. We study when Zhang twists of a standard smooth irreducible noncommutative quadric are standard. Finally, by identifying all singular central Sklyanin quadrics, we prove that every smooth central Sklyanin quadric is standard.

0

math.GR 2026-06-24

Nilpotency blocks simple Lie skew braces above one dimension

by Marco Damele, Andrea Loi

On Simply Connected Simple Lie Skew Braces with Nilpotent Multiplicative Group

Simply connected examples with nilpotent multiplicative groups must reduce to one-dimensional abelian structures.

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We prove that a simply connected simple Lie skew brace with nilpotent multiplicative Lie group must be one-dimensional and abelian. Equivalently, if $(G,\cdot,\circ)$ is a simply connected Lie skew brace with nilpotent multiplicative Lie group and $\dim G>1$, then $(G,\cdot,\circ)$ is not simple. Thus, in the simply connected Lie setting, nilpotency of the multiplicative group is incompatible with simplicity in every dimension greater than one. The proof is carried out at the post-Lie algebra level. First, if the additive Lie algebra is solvable, then its nilradical is automatically an ideal of the associated post-Lie algebra. Second, when both Lie algebras underlying an integrable post-Lie structure are nilpotent, one always obtains a proper post-Lie ideal with trivial quotient. To pass from infinitesimal ideals to global ideals of the Lie skew brace, we show that trivial post-Lie quotients give rise to homomorphisms onto abelian trivial Lie skew braces, whose kernels yield connected closed ideals.

0

math.GR 2026-06-24

S_n keeps same units through three endomorphism levels

by Victoria Gould, Ambroise Grau +2 more

The endomorphism tower of a finite symmetric group

For n≥7, the groups of units in End_0(S_n), End_1(S_n), End_2(S_n) and End_3(S_n) are all isomorphic to S_n.

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We consider the endomorphism tower of a monoid $M$, that is, the sequence of monoids End$_i(M)$ where End$_0(M)=M$ and for all $i\geq 1$, End$_i(M)$ is the monoid of all endomorphisms of End$_{i-1}(M)$. We show that for a finite monoid $M$ this sequence does not stabilise in a finite number of steps. Our focus is then on the case where $M=\mathcal{S}_n$, the symmetric group on a finite number $n$ of points. It is well known that other than in exceptional cases (which are avoided by taking $n \geq 7$), the corresponding automorphism tower of $\mathcal{S}_n$ stabilises at the first step. In spite of the natural nature of this question, nothing was known of the endomorphism tower above the level $i=1$. We determine (for each $n \geq 7)$ the elements of End$_2(\mathcal{S}_n)$ and their multiplication and thus verify that the monoids End$_i(\mathcal{S}_n)$ for $i=0,1,2$ all have group of units isomorphic to $\mathcal{S}_n$. We show that the same is true of End$_3(\mathcal{S}_n)$.

0

math.AC 2026-06-23

Cluster structures on partial flag varieties classified by finite type

by Fayadh Kadhem

On Two Approaches to Cluster Structures on Partial Flag Varieties

Relating them to Schubert cell structures yields the classification and flags open questions from earlier work.

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Continuing our previous work, this paper closely studies the relationship between the cluster algebra structures on the coordinate ring of Schubert cells and those on the coordinate ring of partial flag varieties. We give a finite-type classification for these cluster structures and point out several results that were left open in our previous work.

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math.RA 2026-06-23

Quaternions extend to commutative unital rings

by Antony Telveenus

A Note on Quaternions over Commutative Rings

The definition preserves key properties and gives explicit results for the unit group over halidon rings.

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The ring of quaternions is defined over the field of real numbers. This article extends that framework by defining quaternions over a class of commutative unital rings and generalising several of their properties. It also studies the properties of the unit group of quaternions over commutative rings with special attention to halidon rings.

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math.RA 2026-06-23

Differential polynomial rings gain compatible grading only for γ-derivations

by Yassine Ait Mohamed

Graded differential polynomial rings

The condition makes the new grading explicit and lets classical simplicity and primeness results lift to the graded setting.

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Let $R$ be a $\Gamma$-graded ring and $\delta$ a derivation of $R$. We determine exactly when the differential polynomial ring $R[t;\delta]$ admits a grading compatible with that of $R$: this happens if and only if $\delta$ is a $\gamma$-derivation for some $\gamma$ in the centralizer of the support, in which case the grading is explicit and unique once $\deg(t)$ is fixed. Over an arbitrary group, we establish graded analogues of the classical simplicity, primeness, and Noetherianity theorems; in characteristic zero, $R[t;\delta]$ is gr-simple if and only if $R$ is $\delta$-gr-simple and $\delta$ is $\gamma$-outer, and in arbitrary characteristic we obtain a graded \"{O}inert--Silvestrov criterion when $\Gamma$ is orderable and the nonzero homogeneous elements of $R[t;\delta]$ are regular. Finally, we show that the differential polynomial structure is invariant under homogeneous graded equivalence.

0

math.RA 2026-06-23

Explicit inverse and determinant formulas derived for 7D geometric algebras

by K. S. Abdulkhaev, D. S. Shirokov

Explicit Formula for Inverse and Determinant in Geometric Algebras over Odd-dimensional Vector Spaces

Basis conjugation operations produce closed-form expressions for all odd-dimensional cases by reduction to the even case.

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In this paper, we present explicit formulas for the inverse and determinant in geometric (Clifford) algebras over vector spaces of dimension $n=7$. The derivation of these formulas is made possible by generalizing the concept of conjugation to basis conjugation operations. We further develop a general method for constructing such formulas over odd-dimensional spaces from the known even-dimensional case. To validate computational utility of the results, we provide a numerical implementation of the formulas. The code implementation is available at the repository github.com/kamranuz/clifford_7d. These formulas extend previous results for lower dimensions and offer new insights for applications in mathematical physics and computational geometry.

0

math.RA 2026-06-23

Standard polynomials characterize centers of path algebra subalgebras

by Yihao Zheng, Shenglin Zhu

Standard Polynomials for Principal Subalgebras mathbb{K}Q_(geq 1) of Path Algebras

Describing St2 and St3 elements gives explicit centers and 3-centers for principal subalgebras of path algebras.

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We investigate standard polynomials for principal subalgebras of path algebras. First, we use standard polynomials to study the $PI$-theory of principal subalgebras. Then we describe the $St_2$-elements and $St_3$-elements of principal subalgebras, giving a characterization of their centers and 3-centers. In addition, we apply these results to combinatorics on words of formal languages, obtaining some explanations from a combinatorial perspective.

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math.RA 2026-06-22

Local Rota-Baxter operators have rigid diagonals and free last columns

by Izzat Qaralleh, Farrukh Mukhamedov +1 more

Local Rota--Baxter operators of weight zero on nilpotent evolution algebras with maximal nilindex

On maximal-nilindex evolution algebras the diagonal coefficients form a one-scalar geometric string after the obstruction index, making the

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Let $E$ be an $n$-dimensional nilpotent evolution algebra of maximal nilindex over a field of characteristic zero. The Rota--Baxter operators of weights zero and one on such algebras were recently classified. In this paper we investigate the local analogue of the weight-zero case. We prove that every local Rota--Baxter operator of weight zero has a rigid diagonal part: the nonzero diagonal coefficients, when they occur, begin after the obstruction index determined by the structural matrix of $E$ and form a final geometric string governed by one scalar. In contrast, the last-column coefficients are arbitrary. This gives an explicit description of the class $\LRB_0(E)$ and shows that it is generally strictly larger than $\RB_0(\E)$. We also prove that the corresponding $s$-local notion produces no new operators. The classification is further interpreted as a finite union of quasi-affine strata, yielding a dimension comparison between $\LRB_0(E)$ and $\RB_0(E)$. Finally, we study ordinary weight-zero Rota--Baxter operators commuting with derivations and automorphisms and describe the resulting conditions in terms of the directed graph associated with $E$.

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math.RA 2026-06-22

Steinberg algebras arise via pseudofunctor from groupoid bicategory

by Ralf Meyer, Fabian Rodatz +1 more

A bicategorical perspective on Steinberg algebras

The construction extends to a map that turns correspondences into bimodules and identifies covariance rings with the resulting algebras.

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We show that the Steinberg algebra construction for ample groupoids is part of a pseudofunctor from the bicategory of ample groupoids and groupoid correspondences to the bicategory of rings with local units and nondegenerate bimodules. We define a covariance ring for diagrams in this bicategory of rings and show that it is a bicategorical limit. We compute the covariance ring for a diagram of ``proper'' bimodules over an Ore monoid. For diagrams coming from groupoid correspondences, we identify the covariance ring with the Steinberg algebra of its groupoid model.

0

math.RT 2026-06-22

Condition turns orthogonal systems into simple-minded ones

by Zhen Zhang

A note on simple-minded systems and weakly simple-minded systems over self-injective algebras

The necessary and sufficient test applies to domestic Brauer graph algebras and yields examples for the 2-domestic case.

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Let A be a self-injective algebra over an algebraically closed field. We study the relationship between simple-minded systems and weakly simple-minded systems in A-stmod. We present a necessary and sufficient condition for an orthogonal system to be a simple-minded system over domestic Brauer graph algebras. As a byproduct, we construct a class of simple-minded systems over 2-domestic Brauer graph algebras.

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math.RA 2026-06-22

Infinite subalgebras of curve vector fields reduce to finite-codim

by Lucas Buzaglo, Colin Ingalls

Lie subalgebras of vector fields on curves

The isomorphism implies non-Noetherian enveloping algebras and yields a classification for subalgebras of the Witt algebra.

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We study the subalgebra structure of Krichever-Novikov algebras, which are Lie algebras of vector fields on smooth affine curves. Our main result is that every infinite-dimensional subalgebra of a Krichever-Novikov algebra is isomorphic to a finite-codimensional subalgebra of another Krichever-Novikov algebra. We then present some applications of our main result. First, we show that the universal enveloping algebra of any such infinite-dimensional subalgebra is not noetherian. We then prove that all Krichever-Novikov algebras satisfy the Dixmier property that all their nonzero endomorphisms are automorphisms, except for the Witt algebra of vector fields on the once-punctured affine line. Finally, we provide an explicit classification of the infinite-dimensional subalgebras of the Witt algebra.

0

math.RA 2026-06-22

Every auto of infinite-rank nilpotent Lie algebra is induced by free one

by C. E. Kofinas

Tame Automorphisms of Free Nilpotent Lie Algebras of Countably Infinite Rank

Proof covers L∞,c of class c over char-0 fields, showing all automorphisms lift from L∞.

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Let $L_{\infty}$ be a free Lie algebra of countably infinite rank over a field of characteristic $0$ and let $L_{\infty, c}$ be the free nilpotent Lie algebra of countably infinite rank and class $c$. We prove that every automorphism of $L_{\infty, c}$ is induced by an automorphism of $L_{\infty}$.

0

math.CO 2026-06-22

Hyperplane arrangements recover ask zeta functions from Igusa zeta of cones

by Alec Schmutz

Ask zeta functions of central hyperplane arrangements

Matrices of linear forms for central arrangements have local ask zeta functions obtained by substitution into a truncated flag series derive

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Given a central hyperplane arrangement $\mathcal{A}$ defined over a field of characteristic zero, we construct matrices of linear forms whose local ask zeta functions are recovered by the Igusa local zeta function of the cone over $\mathcal{A}$. Our construction extends a previously established connection between ask zeta function of hypergraphs and the Igusa local zeta function of Boolean arrangements. From a combinatorial standpoint, we introduce the truncated flag Hilbert-Poincar\'e series of $\mathcal{A}$, obtained as a rank specialisation of the flag Hilbert-Poincar\'e series of the cone over $\mathcal{A}$. Whenever $\mathcal{A}$ admits good reduction over the finite field $\mathbb{F}_q$, suitable substitutions of the variables of its truncated flag Hilbert-Poincar\'e series recover the local ask zeta functions associated with $\mathcal{A}$. Such formulae provide a means to study the analytic properties of the ask zeta functions considered, as well as to derive their reduced and topological relatives.

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math.RT 2026-06-22

Minimal modules parabolically induced from Levi subalgebras

by Simon Goodwin, Lewis Topley +1 more

Parabolic induction for modular finite W-algebras

In classical and most exceptional types, this holds whenever the p-character lies in a unique sheet or the module is invariant under twistin

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We study the modules of minimal dimension for reduced enveloping algebras of Lie algebras of reductive algebraic groups using the theory of modular finite $W$-algebras. First of all we consider the case where the $p$-character lies in a unique sheet, and demonstrate that in classical cases and in most exceptional cases all minimal modules are parabolically induced from a Levi subalgebra and a rigid $p$-character. Secondly we consider the minimal modules which are invariant under twisting by the component group, showing that in classical cases and in most exceptional cases these are also parabolically induced from a Levi subalgebra and a rigid $p$-character.

0

cs.LO 2026-06-22

Closure atlases realize globally exactly when obstruction-free

by Jaehwan Kim

Closure Atlases and Local-to-Global Obstructions in Finite Closure Systems

Finite local closures extend conservatively to one global operator iff repeated propagation creates no local inconsistency.

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This paper studies finite closure operators on overlapping finite universes and gives an exact local-to-global obstruction criterion for conservative globalization. Given a finite family of local closure systems, its atlas-generated closure is obtained by repeatedly applying the local closure operators to the parts visible in each chart. This closure is the least global closure operator extending all chart closures. A chart-visible obstruction is a consequence produced by this global propagation that lies inside a chart but is not validated by that chart's own closure operator. The main theorem proves that a finite closure atlas has a global conservative realization exactly when no such obstruction occurs; in that case the atlas-generated closure itself is the conservative realization. The obstruction condition is finite and directly computable. The paper also records the indexed representation layer motivating the terminology. For a finite closure system, an indexed truth space selects closed theories as contexts and represents each element by the region of selected closed theories containing it. Closure consequence is always sound for region inclusion, and the full indexed space of all closed theories recovers the original closure consequence exactly; reduced indexed spaces can therefore create spurious region consequences by deleting separating closed theories. A formal opposite gives a four-region membership decomposition - only one, only the other, both, and neither - unless additional separation assumptions are imposed. Finally, overlap-compatible local closed theories glue by canonical union under the atlas-generated closure. The framework is finite, structural, and closure-theoretic; the logical terminology is used only as an interpretation of the underlying closure data.

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math.AG 2026-06-19

Linear spaces of matrices define eigenvector varieties

by Sandra Di Rocco, Bernd Sturmfels +1 more

Eigenvector Varieties

Points of the variety are the eigenvectors of the matrices, supplying a geometric object for Lie algebras and quantum systems.

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Any linear space of square matrices has an associated eigenvector variety. Its points are eigenvectors of matrices from that linear space. We present a systematic study of eigenvector varieties, with focus on Lie algebras and Hamiltonians of quantum systems.

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math.RA 2026-06-19

Unit group abelian iff weighted Leavitt algebra is domain

by Huynh Viet Khanh

Free subgroups in weighted Leavitt Path Algebras

For finite connected graphs over char-0 fields, the units form an abelian group precisely when the algebra has no zero-divisors.

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We study unit groups of weighted Leavitt path algebras. Let $K$ be a field of characteristic $0$ and let $(E,\omega)$ be a finite connected weighted graph. We prove that $L_K(E,\omega)^\times$ is abelian if and only if $L_K(E,\omega)$ is a domain. Equivalently, $L_K(E,\omega)^\times$ contains no non-cyclic free subgroup if and only if $L_K(E,\omega)$ is a domain.

0

math.RA 2026-06-19

Order embeddings on matrix domains take explicit form

by Peter Semrl

Order embeddings of real matrix domains

Maps preserving order exactly between open connected sets of n by n symmetric matrices are classified for n not 1

abstract click to expand

Let $n$ be a positive integer, $n \not=1$, and $S_n$ the set of all $n \times n$ real symmetric matrices. A nonempty subset $\U \subset S_n$ is called a matrix domain if it is open and connected and a map $\phi : \U \to S_n$ is said to be an order emebedding if for every pair $X,Y \in \U$ we have $X \le Y \iff \phi (X) \le \phi(Y)$. We describe the general form of such maps.

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math.RA 2026-06-19

n x n matrix factors as companion pair product iff rank > n-2

by Flavien Mabilat

Product of two matrices similar to companion matrices over sufficiently large fields

The condition holds over any field with at least 2n elements and uses only basic linear algebra.

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In this note, we prove that a square matrix of size $n$ over a field containing at least $2n$ elements can be expressed as the product of two matrices similar to companion matrices, that is to say matrices with the same minimal and characteristic polynomial, if and only if the rank of $A$ is greater than $n-2$, using only elementary facts. We will also give some partial results valid over smaller fields.

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math.RA 2026-06-19

Criteria for NL equaling its pi-sep product given via coefficients

by V. V. Bavula

Explicit descriptions of the subfields (NL)^(pi) and (NL)^(pi)(NL)^(sep) of NL and new explicit criteria for NL = (NL)^(pi)(NL)^(sep)

Descriptions of (NL)^pi and (NL)^pi(NL)^sep use coefficients of f and invariants m_f, m_{f,N} for extensions in characteristic p.

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Let $L=K(\theta)\simeq K[x]/f(x)$ be a simple field extension in prime characteristic $p>0$, $L^{sep}$ and $L^{pi}$ be the maximal separable and purely inseparable subfields of $L$, respectively. Let $N/K$ be a purely inseparable field extension. For the field extensions $L/K$ and $NL/N$, the aim of the paper is to give explicit descriptions of the following subfields and their degrees in terms of the coefficients of the polynomial $f$ and two numerical field invariants $m_f$ and $m_{f,N}$: $L^{pi}$, $L^{pi}L^{sep}$, $(NL)^{pi}$ and $(NL)^{pi}(NL)^{sep}$. From these results, we derive new explicit criteria for $L=L^{pi}L^{sep}$ and $NL=(NL)^{pi}(NL)^{sep}$.

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math.RA 2026-06-19

Nijenhuis Lie 2-algebras equivalent to 2-term Nijenhuis L∞-algebras

by Apurba Das

Nijenhuis Lie 2-algebras

Semidirect products carry the structure and the two representation categories coincide.

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In this paper, we first introduce Nijenhuis Lie 2-algebras as the categorification of Nijenhuis Lie algebras. We prove that the category of Nijenhuis Lie 2-algebras is equivalent to the category of 2-term Nijenhuis $L_\infty$-algebras. Next, given a Nijenhuis Lie algebra, we introduce the notion of a 2-representation and show that the corresponding semidirect product inherits a Nijenhuis Lie 2-algebra structure. On the other hand, we consider a $2$-term representation up to homotopy of a Nijenhuis Lie algebra and obtain a $2$-term Nijenhuis $L_\infty$-algebra as the semidirect product. Finally, we show that the category of $2$-representations and the category of $2$-term representations up to homotopy of a Nijenhuis Lie algebra are equivalent.

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math.LO 2026-06-18

Sheffer function graphs compose all relations on finite domains

by Sergiy Koshkin

Functional completeness and primitive positive decomposition of relations on finite domains

Effective decomposition turns higher-arity relations into binary ones by viewing them as multivalued function graphs.

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We give a new and elementary construction of primitive positive decomposition of higher arity relations into binary relations on finite domains. Such decompositions come up in applications to constraint satisfaction problems, clone theory and relational databases. The construction exploits functional completeness of 2-input functions in many-valued logic by interpreting relations as graphs of partially defined multivalued 'functions'. The 'functions' are then composed from ordinary functions in the usual sense. The construction is computationally effective and relies on well-developed methods of functional decomposition, but reduces relations only to ternary relations. An additional construction then decomposes ternary into binary relations, also effectively, by converting certain disjunctions into existential quantifications. The result gives a uniform proof of Peirce's reduction thesis on finite domains, and shows that the graph of any Sheffer function composes all relations there.

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math.RA 2026-06-18

Restricted Rota-Baxter Lie algebras yield restricted post-Lie algebras

by Yunnan Li, Ke Ou

On restricted Rota-Baxter Lie algebras of arbitrary weight

Arbitrary-weight definition with graph characterization produces splitting and replication in prime characteristic.

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Recently, Ehret and Gilliers introduced the notion of a (trivially) restricted post-Lie algebra, recovering the concepts of a restricted Lie algebra and a restricted pre-Lie algebra. In this paper, we specifically introduce restricted Rota-Baxter Lie algebras of arbitrary weight with an intrinsic graph subalgebra characterization. We show that, via the splitting property, they give rise to restricted post-Lie algebras, and furthermore possess a novel replication property. We then present two natural constructions of such restricted Rota-Baxter structures in prime characteristic: one arising from Rota-Baxter associative algebras of arbitrary weight, and the other from Rota-Baxter Lie algebras of weight $1$. The Rota-Baxter $p$-envelopes of a Rota-Baxter Lie algebra are also examined.

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math.CO 2026-06-18

Cyclic group Schur rings with almost commutative Terwilliger algebras

by Nicholas L. Bastian, Stephen P. Humphries

Schur rings over cyclic groups having Almost Commutative Terwilliger algebras

The classification covers orbit rings and wedge products to give the complete list for any cyclic group.

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Terwilliger algebras are subalgebras of a matrix algebra constructed from an association scheme. Rie Tanaka defined what it means for a Terwilliger algebra to be almost commutative and gave five equivalent conditions in the case where the association scheme is commutative. A sixth condition for a Terwilliger algebra coming from a commutative Schur ring to be almost commutative has since been discovered. In this paper we first provide a classification of orbit Schur rings that produce an almost commutative Terwilliger algebra for a finite cyclic group. In particular, we show that the subgroup of automorphisms used to form the orbit Schur ring is either trivial, or the whole automorphism group when the cyclic group has prime power order. If the cyclic group has order $2^n$, these are the only options. If the cyclic group has order $p^n$, for an odd prime $p$, then the automorphism subgroup of order $p^{n-1}$ also works. If the group has a non-prime power order then the only orbit Schur ring that produces an almost commutative Terwilliger algebra comes from the trivial subgroup of the automorphism group. We then give a condition for when a wedge product of Schur rings produces an almost commutative Terwilliger algebra. This allows us to determine exactly when a Schur ring over a cyclic group produces an almost commutative Terwilliger algebra.

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math.GR 2026-06-18

Engel words cover all non-scalar lifts in SL2 over local rings

by Ayon Roy, Anupam Singh

Surjectivity of Engel Words on SL₂(mathcal{O}) and PSL₂(mathcal{O}₂)

For residue fields larger than a bound depending on m, every such lift is hit by the m-fold commutator; full surjectivity also holds on PSL2

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The study of word maps and Waring-like problems has been widely pursued for finite simple groups, algebraic groups, and Lie groups. In this article, we study Engel word maps $e_{m}(x, y) = \left[\cdots\left[[x, y], y \right], \cdots, y \right]$ on certain linear groups over local rings, namely, $\mathrm{SL}_2(\mathcal R)$ and $\mathrm{PSL}_2(\mathcal R)$. We consider the commutative ring $\mathcal {R} $ to be either a complete, local principal ideal ring $\mathcal O$, or a local principal ideal ring of finite length $\mathcal O_\ell$. Suppose the characteristic of the residue field $k\cong \mathbb F_q$ is $\neq 2$. Under some mild conditions on $q$, we show that there exists a constant $q_0(m)$, such that for all $q \geq q_0(m)$, all lifts in $\mathrm{SL}_2(\mathcal{O})$ of non-scalar elements of $\mathrm{SL}_2(k)$, are in the image of the $m$-th Engel word over $\mathrm{SL}_2(\mathcal{O})$. We further show that all Engel word maps are surjective on $\mathrm{PSL}_2(\mathcal{O}_2)$ where $\mathcal{O}_2$ is a local principal ideal ring of length two. This work generalizes similar results about the Engel word map over fields.

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math.RT 2026-06-17

Radon transform eigenvalues described explicitly for GL_2(F_q) and GL_3(F_q)

by Ivan Motorin, Kai Yamashita

Radon transform for GL_n(mathbb{F}_q)

Integration against unipotent radicals yields a computable spectrum in the lowest ranks of the general linear group over finite fields.

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In this paper, we study the Radon transform associated with the unipotent radical subgroups of $GL_n(\mathbb{F}_q)$. We analyze properties of the Radon transform with a specific emphasis on its eigenvalues. We provide a description of its eigenvalues for the cases of $GL_2(\mathbb{F}_q)$ and $GL_3(\mathbb{F}_q)$.

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math.RT 2026-06-17

Green's formula extends to iterated restriction and induction

by Chao Shen, Jie Xiao

The iterated geometric Green's formula

Categorical isomorphisms now cover compositions of n-1 restrictions with induction and their duals.

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Fang, Lan, and Xiao established the geometric Green's formula as a categorical isomorphism for arbitrary semisimple complexes. In this short note, we generalize their work to multi-step compositions. Specifically, we establish the iterated geometric Green's formulas for the composition of an $(n-1)$-fold restriction and an induction, as well as its dual.

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math.RA 2026-06-16

Non-isomorphic Lie algebras share the same enveloping algebra

by Xabier García-Martínez

Non-isomorphic restricted Lie algebras with isomorphic restricted enveloping algebras

Explicit pairs of p-nilpotent restricted Lie algebras over any field of positive characteristic have isomorphic restricted enveloping algebr

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Let $p$ be a prime. For every field $F$ of characteristic $p$ we exhibit pairs of non-isomorphic finite-dimensional $p$-nilpotent restricted Lie algebras $L$ and $H$ over $F$ whose restricted enveloping algebras $u(L)$ and $u(H)$ are isomorphic as $F$-algebras. Such pairs exist in every dimension at least $p+5$, with $\dim L'=p$ and $\dim H'=p+1$. Thus, the restricted isomorphism problem has a negative answer over every field of positive characteristic, even for $p$-nilpotent algebras over perfect fields.

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math.RA 2026-06-16

Commutative identity component makes graded polynomial image a subspace

by Adison Timótio Silva, Felipe Yukihide Yasumura

A note on the image of graded multilinear polynomials on upper triangular matrices

On upper triangular matrices the set of values is always linear once the identity part of the group grading commutes.

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We investigate the image of polynomials multilinear in graded variables evaluated on the algebra of upper triangular matrices endowed with a group grading. We show that, in general, such an image need not be a vector subspace. However, under the additional assumption that the identity component of the grading is commutative, we prove that the image is always a vector subspace. We further investigate the image of polynomials evaluated on inverse and direct limits of algebras. As a consequence, we prove that the image of a polynomial evaluated on a direct limit of upper triangular matrix algebras whose identity component is commutative is always a vector subspace.

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math.RA 2026-06-16

Finite graded algebras over finite fields have finite identity bases

by Yuri Bahturin, Daniela Martinez Correa +1 more

Finite basis property for finite graded algebras

The finite basis property holds for any finite-dimensional associative algebra graded by a finite group over a finite field.

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Let $G$ be a finite group and let $\F$ be a finite field. We prove that any finite-dimensional $G$-graded associative algebra $A$ over $\F$ has a finite basis for its $G$-graded polynomial identities.

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math.RT 2026-06-12

Poset isomorphisms link silting subcategories to s-torsion pairs

by Liangwei Huang, Haicheng Zhang

Silting subcategories and (co)torsion pairs associated to extended hearts

The bijections also equate hereditary cotorsion pairs and extend to tau-tilting pairs and silting complexes over dg algebras.

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We establish the poset isomorphisms between $(d+1)$-term silting subcategories, functorially finite $s$-torsion pairs in the $d$-extended heart, and hereditary complete cotorsion pairs in a suitable subcategory. As an application, we also give dg algebra versions of these bijections, which establish the poset isomorphisms between $\tau$-tilting pairs, $(d+1)$-term silting complexes, and functorially finite $s$-torsion pairs.

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cs.IT 2026-06-12

New product generalizes duals for LCD and self-dual codes

by Avanish Kumar Chaturvedi, Satyadeep Pandey

W-δ-μ dual codes and LCD codes

The W-δ-μ inner product on finite-field vectors yields explicit dual calculations and existence conditions for standard code classes.

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We introduce a new product on the ambient space $F_q^n$ as a generalization of Euclidean, Hermitian and $\delta$ products. We give some general properties of the dual codes, relation with Euclidean duals, definition and characterization of self orthogonal, self dual, dual containing and LCD codes along with certain existence conditions. Also, we calculate the dual codes of some classes of codes like repetition, binary and $\lambda$-constacyclic codes with respect to this product. Further, we extend and analyse this notion of the product for codes over semisimple rings.

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math.RA 2026-06-12

Semiring polynomials split into linears over finite pair extensions

by Louis Halle Rowen

Roots of polynomials over semirings and hyperfields

A fundamental theorem of algebra holds for pairs with surpassing relation when roots are sufficient.

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We continue our investigation of roots of polynomials over semirings and hyperfields, employing a property on semiring and hyperfield ``pairs'' with a surpassing relation $\preceq,$ which we call $\preceq$-reversibility. There are two kinds of roots generalizing the classical algebraic theory, ``null roots,'' and $\preceq$-roots. The theory works best when all null roots are also $\preceq$-roots. Ensuing results include the fundamental theorem of algebra for pairs, that tangible polynomials with enough roots ``$\preceq$-split,'' at times uniquely, into linear factors over a suitable finite extension of pairs. We also see that polynomials that agree on ``almost'' all null roots are ``almost'' equal. Finally, we obtain roots of integral polynomials over extension pairs, providing a construction of integrally closed pairs over hyperfields and over zero sum free semirings.

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math.RA 2026-06-12

Theorems link clean-like properties to endomorphisms and matrices

by Adel N. Abyzov, Andrey R. Chekhlov +2 more

Rings with Clean-Like Properties: Endomorphism, Matrix and Structural Theorems

Conditions given for weakly strongly k-nil-clean rings, quasi nil-clean matrices over finite fields, and weakly clean endomorphisms of abeli

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We investigate three clean-like properties for arbitrary rings, for endomorphism rings of abelian groups and for matrix rings over finite fields. Specifically, we study and establish when a ring is weakly strongly $k$-nil-clean for some fixed natural number $k\geq 2$, when the matrix ring is either quasi $2$-nil-clean or quasi $3$-nil-clean, and when the endomorphism ring is weakly clean. Our theorems improve substantially on some results due to Goldsmith-V\'amos in Rend. Sem. Mat. Univ. Padova (2007), Breaz {\it et al}. in Linear Algebra \& Appl. (2013), Ko\c{s}an-Zhou in Front. Math. China (2016), Su {\it et al}. in J. Algebra \& Appl. (2027), and some other existing results in this current topic.

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