The isomorphism problem means to decide if two given finite-dimensional simple algebras over the same centre are isomorphic and, if so, to construct an isomorphism between them. A solution to this problem has applications in computational aspects of representation theory, algebraic geometry and Brauer group theory.
The paper presents an algorithm for cyclic algebras that reduces the isomorphism problem to field theory and thus provides a solution if certain field theoretic problems including norm equations can be solved (this is satisfied over number fields). As an application, we can compute all automorphisms of any given cyclic algebra over a number field. A detailed example is provided which serves to construct an explicit example of a noncrossed product division algebra.
Let $p$ be a prime number, $G$ be a finite $p$-group and $K$ be a field of characteristic $p$. The Modular Isomorphism Problem (MIP) asks whether the group algebra $KG$ determines the group $G$. Dealing with MIP, we investigated a question whether the nilpotency class of a finite $p$-group is determined by its modular group algebra over the field of $p$ elements. We give a positive answer to this question provided one of the following conditions holds: (i) $\exp G=p$; (ii) $\cl(G)=2$; (iii) $G'$ is cyclic; (iv) $G$ is a group of maximal class and contains an abelian subgroup of index $p$.
In this paper we study the isomorphism problem for the universal enveloping algebras of nilpotent Lie algebras. We prove that if the characteristic of the underlying field is not~2 or~3, then the isomorphism type of a nilpotent Lie algebra of dimension at most~6 is determined by the isomorphism type of its universal enveloping algebra. Examples show that the restriction on the characteristic is necessary.
The aim of this note is to show that the generalized supertrace, constructed in another paper of the author, inducing an isomorphism between the Hochschild homology of a superalgebra and that of the superalgebra of square supermatrices of a given type over $A$, induces, also, an isomorphism between the cyclic homologies of the two superalgebras.
We describe the graded isomorphisms of rings of endomorphisms of graded flags over graded division algebras. As a consequence describe the isomorphism classes of upper block triangular matrix algebras (over an algebraically closed field of characteristic zero) graded by a finite abelian group.
This paper is a contribution to the isomorphism problem for universal enveloping algebras of finite-dimensional Lie algebras. We focus on solvable Lie algebras of small dimensions over fields of arbitrary characteristic. We prove, over an arbitrary field, that the isomorphism type of a metabelian Lie algebra whose derived subalgebra has codimension one is determined by its universal enveloping algebra. As an application of the results in this paper, we solve the isomorphism problem for solvable Lie algebras of dimension four over fields of characteristic zero and also point out the problems th…
We classify up to isomorphism the quantum generalized Weyl algebras and determine their automorphism groups in all cases in a uniform way, including those where the parameter q is a root of unity, thereby completing the results obtained by [Bavula, V. V.; Jordan, D. A. Isomorphism problems and groups of automorphisms for generalized Weyl algebras. Trans. Amer. Math. Soc. 353 (2001), no. 2, 769-794] and [Richard, L.; Solotar, A. Isomorphisms between quantum generalized Weyl algebras. J. Algebra Appl. 5 (2006), no. 3, 271-285]
For any positive integer $n$ we describe the Leavitt path algebra of the Cayley graph $C_n$ corresponding to the cyclic group $\Z/n\Z$. Using a Kirchberg-Phillips-type realization result, we show that there are exactly four isomorphism classes of such Leavitt path algebras, arising as the algebras corresponding to the graphs $C_i$ ($3\leq i \leq 6$).
Identities and isomorphisms of finite graded matrix algebras
Universidade Federal de Campina Grande (UFCG)（坎皮纳格兰德联邦大学）；Universidade de São Paulo（圣保罗大学）；Universidade Federal de São Carlos (UFSCar)（圣卡洛斯联邦大学）；Universidade Estadual de Campinas (UNICAMP)（坎皮纳斯州立大学）
We present a new algorithm to decide isomorphism between finite graded algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it runs in time polynomial in the order of the input algebras. We introduce heuristics that often dramatically improve the performance of the algorithm and report on an implementation in Magma.
In this paper, we give algorithms for determining the existence of isomorphism between two finite-dimensional Lie algebras and compute such an isomorphism in the affirrmative case. We also provide algorithms for determining algebraic relations of parameters in order to decide whether two parameterized Lie algebras are isomorphic. All of the considered Lie algebras are considered over a field $\F$, where $\F=\C$ or $\F=\R$. Several illustrative examples are given to show the applicability and the effectiveness of the proposed algorithms.
We introduce an algorithm to decide isomorphism between tensors. The algorithm uses the Lie algebra of derivations of a tensor to compress the space in which the search takes place to a so-called densor space. To make the method practicable we give a polynomial-time algorithm to solve a generalization of module isomorphism for a common class of Lie modules. As a consequence, we show that isomorphism testing is in polynomial time for tensors whose derivation algebras are classical Lie algebras and whose densor spaces are 1-dimensional. The method has been implemented in the Magma computer algeb…
Let $X$ be a partially ordered set, $R$ a commutative $2$-torsionfree unital ring and $FI(X,R)$ the finitary incidence algebra of $X$ over $R$. In this note we prove that each $R$-linear Jordan isomorphism of $FI(X,R)$ onto an $R$-algebra $A$ is the near-sum of a homomorphism and an anti-homomorphism.
We prove that if two path algebras with homogeneous relations are isomorphic as algebras, then they are isomorphic as graded path algebras. This extends a result by Bell and Zhang in the connected case.
We introduce certain functors from the category of commutative rings (and related categories) to that of $\mathbb{Z}$-algebras (not necessarily associative or commutative). One of the motivating examples is the Leavitt path algebra functor $R\mapsto L_R(E)$ for a given graph $E$. Our goal is to find "descending" isomorphism results of the type: if $\mathfrak{F},\mathcal{G}$ are algebra functors and $K\subset K'$ a field extension, under what conditions an isomorphism $\mathfrak{F}(K')\cong \mathcal{G}(K')$ of $K'$-algebras implies the existence of an isomorphism $\mathfrak{F}(K)\cong\mathcal{G…
The paper is an implementation in low dimensional cases of the classification method presented before by Rakhimov and Bekbaev. We give a complete classification of a subclass of complex filiform Leibniz algebras obtained from the naturally graded non-Lie filiform Leibniz algebras. A hypothetic formula for the adapted number of isomorphism classes is given.
Let $P$ and $Q$ be finite posets and $R$ a commutative unital ring. In the case where $R$ is indecomposable, we prove that the $R$-linear isomorphisms between partial flag incidence algebras $I^3(P,R)$ and $I^3(Q,R)$ are exactly those induced by poset isomorphisms between $P$ and $Q$. We also show that the $R$-linear derivations of $I^3(P,R)$ are trivial.
We solve the isomorphism problem for non noetherian down-up algebras $A(α,0,γ)$ by lifting isomorphisms between some of their non commutative quotients. The quotients we consider are either quantum polynomial algebras in two variables for $γ= 0$ or quantum versions of the Weyl algebra $A_1$ for non zero $γ$. In particular we obtain that no other down-up algebra is isomorphic to the monomial algebra $A(0,0,0)$. We prove in the second part of the article that this is the only monomial algebra within the family of down-up algebras. Our method uses homological invariants that determine the shape o…
We discuss the Kummer subspaces of tensor products of cyclic algebras, focusing mainly on the case of cyclic algebras of degree 3. We present a family of maximal spaces in the general case, classify all the monomial spaces in the case of tensor products of cyclic algebras of degree 3 using graph theory, and provide an upper bound for the dimension in the generic tensor product of cyclic algebras of degree 3.
The mod-p cohomology ring of a non-trivial finite p-group is an infinite dimensional, finitely presented graded unital algebra over the field with p elements, with generators in positive degrees. We describe an effective algorithm to test if two such algebras are graded isomorphic. As application, we determine all graded isomorphisms between the mod-p cohomology rings of all p-groups of order at most 100.
Leibniz algebras are a non-anticommutative version of Lie algebras. They play an important role in different areas of mathematics and physics and have attracted much attention over the last thirty years. In this paper we investigate whether conditions such as being a Lie algebra, cyclic, simple, semisimple, solvable, supersolvable or nilpotent in such an algebra are preserved by lattice isomorphisms.
The ideals of the Lie algebras of unitriangular polynomial derivations are classified. An isomorphism criterion is given for the Lie factor algebras of the Lie algebras of unitriangular polynomial derivations.
For any countable graph $E$, we investigate the relationship between the Leavitt path algebra $L_{\C}(E)$ and the graph C*-algebra $C^*(E)$. For graphs $E$ and $F$, we examine ring homomorphisms, ring *-homomorphisms, algebra homomorphisms, and algebra *-homomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$. We prove that in certain situations isomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$ yield *-isomorphisms between the corresponding C*-algebras $C^*(E)$ and $C^*(F)$. Conversely, we show that *-isomorphisms between $C^*(E)$ and $C^*(F)$ produce isomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$ …
In this paper we classify the isomorphism classes of four dimensional nilpotent associative algebras over a field F, studying regular subgroups of the affine group AGL_4(F). In particular we provide explicit representatives for such classes when F is a finite field, the real field R or an algebraically closed field.
In this note we solve the isomorphism problem for the multiparameter quantized Weyl algebras, in the case when none of the deformation parameters q_i is a root of unity, over an arbitrary field.
The paper concerns the classification problem of a subclass of complex filiform Leibniz algebras in dimensions 7 and 8. This subclass arises from naturally graded filiform Lie algebras. We give a complete list of isomorphism classes of algebras including Lie case. In parametric families cases, the corresponding orbit functions (invariants) are given. In discrete orbits case, we show representatives of the orbits.
The graph groupoids of directed graphs are topologically isomorphic if and only if there is a diagonal-preserving ring *-isomorphism between the Leavitt path algebras.
Let $KG$ denote the group algebra of the group $G$ over the field $K$ and let $U(KG)$ denote its group of units. Here without the use of a computer we give presentations for the unit groups of all group algebras $KG$, where the size of $KG$ is less than 1024. As a consequence we find the minimum counterexample to the Isomorphism Problem for group algebras.
Work of Jean Renault shows that, for topologically principal étale groupoids, a diagonal-preserving isomorphism of reduced $C^*$-algebras yields an isomorphism of groupoids. Several authors have proved analogues of this result for ample groupoid algebras over integral domains under suitable hypotheses. In this paper, we extend the known results by allowing more general coefficient rings and by weakening the hypotheses on the groupoids. Our approach has the additional feature that we only need to impose conditions on one of the two groupoids. Applications are given to Leavitt path algebras.
Let $K$ be any field, let $L_n$ denote the Leavitt algebra of type $(1,n-1)$ having coefficients in $K$, and let ${\rm M}_d(L_n)$ denote the ring of $d \times d$ matrices over $L_n$. In our main result, we show that ${\rm M}_d(L_n) \cong L_n$ if and only if $d$ and $n-1$ are coprime. We use this isomorphism to answer a question posed in \cite{PS} regarding isomorphisms between various C*-algebras. Furthermore, our result demonstrates that data about the $K_0$ structure is sufficient to distinguish up to isomorphism the algebras in an important class of purely infinite simple $K$-algebras.
Hazrat gave a K-theoretic invariant for Leavitt path algebras as graded algebras. Hazrat conjectured that this invariant classifies Leavitt path algebras up to graded isomorphism, and proved the conjecture in some cases. In this paper, we prove that a weak version of the conjecture holds for all finite essential graphs.
We study Steinberg algebras constructed from ample Hausdorff groupoids over commutative integral domains with identity. We reconstruct (graded) groupoids from (graded) Steinberg algebras and use this to characterise when there is a diagonal-preserving (graded) isomorphism between two (graded) Steinberg algebras. We apply this characterisation to groupoids of directed graphs in order to study diagonal-preserving (graded) isomorphisms of Leavitt path algebras and graph $C^*$-algebras.
There is no systematic general procedure by which isomorphism classes of Hopf algebras that are extensions of $\k F$ by ${\k}^G$ can be found. We develop the general procedure for classification of isomorphism classes of Hopf algebras which are extensions of the group algebra $\k C_p$ by ${\k}^G$ where $C_p$ is a cyclic group of prime order $p$ and ${\k}^G$ is the Hopf algebra dual of $\k G$, $G$ a finite abelian $p$-group and $\k$ is an algebraically closed field of characteristic $0$. We apply the method to calculate the number of isoclasses of commutative extensions and certain extensions o…
In this paper, we study the gravity algebra structure on the negative cyclic homology or the cyclic cohomology of several classes of algebras. These algebras include: Calabi-Yau algebras, symmetric Frobenius algebras, unimodular Poisson algebras, and unimodular Frobenius Poisson algebras. The relationships among these gravity algebras are also discussed under some additional conditions.
The paper aims to investigate the classification problem of low dimensional complex none Lie filiform Leibniz algebras. There are two sources to get classification of filiform Leibniz algebras. The first of them is the naturally graded none Lie filiform Leibniz algebras and the another one is the naturally graded filiform Lie algebras. Here we do consider Leibniz algebras appearing from the naturally graded none Lie filiform Leibniz algebras. This class can be splited into two subclasses. However, isomorphisms within each class there were not investigated. Before U.D.Bekbaev and I.S.Rakhimov s…
Let $L$ be a Lie algebra with its enveloping algebra $U(L)$ over a field. In this paper we survey results concerning the isomorphism problem for enveloping algebras: given another Lie algebra $H$ for which $U(L)$ and $U(H)$ are isomorphic as associative algebras, can we deduce that $L$ and $H$ are isomorphic Lie algebras? Over a field of positive characteristic we consider a similar problem for restricted Lie algebras, that is, given restricted Lie algebras $L$ and $H$ for which their restricted enveloping algebras are isomorphic as algebras, can we deduce that $L$ and $H$ are isomorphic?
In this paper, we investigate the existence of an elementary abelian closure in characteristic not $2$ for biquadratic extensions. We discover that it exists for any non-cyclic extension. We make use of it to obtain a classification for this class of extensions up to isomorphism via descent. This permits us to describe the geometry of this moduli space in group theoretic terms. We also provide two families of polynomials with two parameters that can describe any quartic extensions.
Let $L$ be a separable quadratic extension of either $\mathbb{Q}$ or $\mathbb{F}_q(t)$. We propose efficient algorithms for finding isomorphisms between quaternion algebras over $L$. Our techniques are based on computing maximal one-sided ideals of the corestriction of a central simple $L$-algebra.
Let ${\rm Rep}\,{\cal O}_n$ denote the category of all nondegenerate $^*$ representations of the Cuntz algebra ${\cal O}_n$. For any $2\leq n,m<\infty$, we construct an isomorphism functor $F_{n,m}$ from ${\rm Rep}\,{\cal O}_m$ to ${\rm Rep}\,{\cal O}_n$.
In this paper, we give an important isomorphism between contacyclic codes and cyclic codes over finite principal ideal rings. Necessary and sufficient conditions for the existence of non-trivial cyclic self-dual codes over finite principal ideal rings are given.
We reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for a class of finite dimensional $2$-step nilpotent Lie algebras over a field and for a class of finite $p$-groups. We show that the isomorphism problem for graphs is harder than the two latter isomorphism problems in the sense of Borel reducibility. A computable analogue of Borel reducibility was introduced by S. Coskey, J.D. Hamkins, and R. Miller. A relation of the isomorphism problem for undirected graphs to the well-known problem of classifying pairs of matrices over a field (up to similarit…
Let V_* be the normalized unitary subgroup of the modular group algebra FG of a finite p-group G over a finite field F with the classical involution *. We investigate the isomorphism problem for the group V_*, that asks when the group V_* is determined by its group algebra FG. We confirm it for classes of finite abelian p-groups, 2-groups of maximal class and non-abelian 2-groups of order at most 16.
