Superinvolutions on upper-triangular matrix algebras

Ioppolo, Antonio; Martino, Fabrizio

doi:10.1016/j.jpaa.2017.08.018

Cited by 14 publications

(10 citation statements)

References 18 publications

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“…A great number of recent researches have been devoted to polynomial identities in algebras with additional structure. To cite just a few, we quote [2,5,6,11,12,[17][18][19][20][21][22]. It turns out that many of these algebras can be viewed and dealt with as -algebras, for an appropriate signature , as we are going to present below.…”

Section: Further Examplesmentioning

confidence: 99%

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Distinguishing simple algebras by means of polynomial identities

Bahturin

Yasumura

2019

São Paulo J. Math. Sci.

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Our main goal is to extend one of classical Razmyslov's Theorem saying that any two simple finite-dimensional-algebras over an algebraically closed field, satisfying the same polynomial identities, are isomorphic. We suggest a method that allows one to reduce problems about identities of algebras with additional structure to the identities of-algebras. For the convenience of the reader, we start with a full detailed proof of Razmyslov's Theorem. Then we describe our method and its consequences for the identities of graded algebras, algebras with involution, and several others. Keywords Graded algebra • Polynomial identity • Universal algebra Mathematics Subject Classification 17A42 • 08B20 • 16R50 1 Introduction: the problem and some cases In this manuscript we consider the following problem. Given two algebras A and B over the same field, suppose that A and B satisfy the same polynomial identities, is it true that A isomorphic to B? Naturally, this question stated in all its generality has counterexamples. Dedicated to Professor Ivan Shestakov on his 70th birthday.

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Section: Further Examplesmentioning

confidence: 99%

“…In the work [22], the authors study graded superinvolutions on arbitrary Z 2 -graded algebras of upper triangular matrices. One of the results of the paper is the computation of all identities with superinvolution for upper triangular matrices of sizes 2 and 3.…”

Section: Theorem 72 Two Finite-dimensional Superalgebras With Involumentioning

confidence: 99%

Distinguishing simple algebras by means of polynomial identities

Bahturin

Yasumura

2019

São Paulo J. Math. Sci.

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“…The superinvolutions on algebras of upper triangular n × n matrices were described in [9] and the corresponding polynomial identities were studied for n = 2, 3. In this paper we determine, up to isomorphism, the superinvolutions on algebras of upper block-triangular matrices.…”

Section: Introductionmentioning

confidence: 99%

“…Now we proceed to construct stable anti-automorphisms on algebras of upper block-triangular matrices that satisfy (11). Let r be a 2m-tuple of non-negative integers that satisfy (9). We consider two types of 2m-tuples.…”

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Superinvolutions on block-triangular matrix algebras

Dias,

Diniz,

Ramos

2020

Preprint

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Superalgebras with Involution or Superinvolution and Almost Polynomial Growth of the Codimensions

Giambruno

Ioppolo

Mattina

2018

Algebr Represent Theor

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Let A be a superalgebra with graded involution or superinvolution * and let c * n (A), n = 1, 2, . . . , be its sequence of * -codimensions. In case A is finite dimensional, in [6,15] it was proved that such a sequence is polynomially bounded if and only if the variety generated by A does not contain the group algebra of Z 2 and a 4-dimensional subalgebra of the 4×4 upper-triangular matrices with suitable graded involutions or superinvolutions.In this paper we study the general case of * -superalgebras satisfying a polynomial identity. As a consequence we classify the varieties of * -superalgebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth, and we give a full classification of their subvarieties which was started in [18].

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Superinvolutions on upper-triangular matrix algebras

Cited by 14 publications

References 18 publications

Distinguishing simple algebras by means of polynomial identities

Distinguishing simple algebras by means of polynomial identities

Superinvolutions on block-triangular matrix algebras

Superalgebras with Involution or Superinvolution and Almost Polynomial Growth of the Codimensions

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