Graded identities of matrix algebras and the universal graded algebra

Aljadeff, Eli; Haile, Darrell; Natapov, Michael

doi:10.1090/s0002-9947-10-04811-7

Cited by 31 publications

(59 citation statements)

References 16 publications

(29 reference statements)

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“…However, can this phenomenon happen for two twisted group algebras of the same group? It turns out that, provided the group G is abelian, if C σ G and C ρ G are isomorphic and simple, then they are graded equivalent [4,Theorem 18], [16,Proposition 2.4 (2)] (see the description of finite abelian groups of central type e.g. in [11,Theorem 2.15]).…”

Section: Weak Equivalences Of Graded-simple Algebrasmentioning

confidence: 99%

On weak equivalences of gradings

Gordienko

Schnabel

2018

Journal of Algebra

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When one studies the structure (e.g. graded ideals, graded subspaces, radicals,. . . ) or graded polynomial identities of graded algebras, the grading group itself does not play an important role, but can be replaced by any other group that realizes the same grading. Here we come to the notion of weak equivalence of gradings: two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component. The following question arises naturally: when a group grading on a finite dimensional algebra is weakly equivalent to a grading by a finite group? It turns out that this question can be reformulated purely group theoretically in terms of the universal group of the grading. Namely, a grading is weakly equivalent to a grading by a finite group if and only if the universal group of the grading is residually finite with respect to a special subset of the grading group. The same is true for all the coarsenings of the grading if and only if the universal group of the grading is hereditarily residually finite with respect to the same subset. We show that if n 349, then on the full matrix algebra M n (F ) there exists an elementary group grading that is not weakly equivalent to any grading by a finite (semi)group, and if n 3, then any elementary grading on M n (F ) is weakly equivalent to an elementary grading by a finite group.be two gradings where S and T are (semi)groups and A and B are algebras.The most restrictive case is when we require that both grading (semi)groups coincide: Definition 1.1 (e.g. [11, Definition 1.15]). The gradings (1.1) are isomorphic if S = T and there exists an isomorphism ϕ : A → B of algebras such that ϕ(A (s) ) = B (s) for all s ∈ S. In this case we say that A and B are graded isomorphic. In some cases, such as in [16], less restrictive requirements are more suitable. Definition 1.2 ([16, Definition 2.3]). The gradings (1.1) are equivalent if there exists an isomorphism ϕ : A → B of algebras and an isomorphism ψ : S → T of (semi)groups such that ϕ(A (s) ) = B ψ(s) for all s ∈ S. Remark 1.3. The notion of graded equivalence was considered by Yu. A. Bahturin, S. K. Seghal, and M. V. Zaicev in [6, Remark after Definition 3]. In the paper of V. Mazorchuk and K. Zhao [19] it appears under the name of graded isomorphism. A. Elduque

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Section: Weak Equivalences Of Graded-simple Algebrasmentioning

confidence: 99%

On weak equivalences of gradings

Gordienko

Schnabel

2018

Journal of Algebra

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“…In their paper [3], E. Aljadeff, D. Haile and M. Natapov present a family of groups of central type, which contains all the abelian groups of central type. This family enjoys the property that for any group G in it, Aut(G) acts transitively on the non-degenerate cohomology classes in H 2 (G, C * ) [5,Theorem 18]. One might conjecture that this transitivity property holds for every group of central type.…”

Section: Introductionmentioning

confidence: 99%

Groups of central-type, maximal connected gradings and intrinsic fundamental groups of complex semisimple algebras

Ginosar

Schnabel

2019

Trans. Amer. Math. Soc.

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Abstract. Maximal connected grading classes of a finite-dimensional algebra A are in one-to-one correspondence with Galois covering classes of A which admit no proper Galois covering and therefore are key in computing the intrinsic fundamental group π 1 (A). Our first concern here is the algebras A = Mn(C). Their maximal connected gradings turn out to be in one-to-one correspondence with the Aut(G)-orbits of non-degenerate classes in H 2 (G, C * ), where G runs over all groups of central type whose orders divide n 2 . We show that there exist groups of central type G such that H 2 (G, C * ) admits more than one such orbit of non-degenerate classes. We compute the family Λ of positive integers n such that there is a unique group of central type of order n 2 , namely Cn × Cn. The family Λ is of square-free integers and contains all prime numbers. It is obtained by a full description of all groups of central type whose orders are cube-free. We establish the maximal connected gradings of all finite dimensional semisimple complex algebras using the fact that such gradings are determined by dimensions of complex projective representations of finite groups. In some cases we give a description of the corresponding fundamental groups.

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“…(1) to give an introduction to non-commutative geometry and to the language of Hopf algebras; (2) to build up a theory of non-commutative principal fiber bundles, consider various aspects of these non-commutative objects, highlight the similarities and the differences with their classical counterparts, and illustrate the theory with significant examples. Non-commutative geometry is based on the idea that instead of working with the points of a topological space X (or a C 8 -manifold, or an algebraic variety) we may just as well work with the algebra OpXq of continuous (or C 8 , or regular) functions on X.…”

Section: Introductionmentioning

confidence: 99%

“…Let p : Z pGq Ñ Γ be the homomorphism sending each generator t g to the image of g in Γ. Let Y G be the kernel of p. Then by[2, Prop. 9 and Prop.…”

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confidence: 99%

Principal Fiber Bundles in Non-commutative Geometry

Kassel¹

2017

Quantization, Geometry and Noncommutative Structures in Mathematics and Physics

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Abstract. These are the expanded notes of a course given at the Summer school "Geometric, topological and algebraic methods for quantum field theory" held at Villa de Leyva, Colombia in July 2015. We first give an introduction to noncommutative geometry and to the language of Hopf algebras. We next build up a theory of non-commutative principal fiber bundles and consider various aspects of such objects. Finally, we illustrate the theory using the quantum enveloping algebra U q slp2q and related Hopf algebras.

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Graded identities of matrix algebras and the universal graded algebra

Cited by 31 publications

References 16 publications

On weak equivalences of gradings

On weak equivalences of gradings

Groups of central-type, maximal connected gradings and intrinsic fundamental groups of complex semisimple algebras

Principal Fiber Bundles in Non-commutative Geometry

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