Polynomial identities with involution, superinvolutions and the Grassmann envelope

Aljadeff, Eli; Giambruno, Antonio; Karasik, Yakov

doi:10.1090/proc/13546

Cited by 34 publications

(35 citation statements)

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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%

“…In the work [2], the authors obtain an analogue of Kemer's theory in the case of associative algebras with involution. In this case, if A satisfies a * -identity, then A has the same * -identities as the Grassmann envelope G(B), where B is a finite-dimensional superalgebra with superinvolution.…”

Section: Theorem 72 Two Finite-dimensional Superalgebras With Involumentioning

confidence: 99%

“…Let H = (H , (1) , (2) ) be a triple where H is a unital associative algebra and (1) , (2) are two linear maps, called coproducts (1) , (2) : H → H ⊗ H . Using Sweedler's notation, we can write (i) (h) = h (i) (1) ⊗h (i) (2) , meaning that (i) (h) are arbitrary tensors of degree 2.…”

Section: Algebras With Generalized Actionmentioning

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%

Section: Theorem 72 Two Finite-dimensional Superalgebras With Involumentioning

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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“…Algebras with generalized action. Let H = (H, ∆ (1) , ∆ (2) ) be a triple where H is a unital associative algebra and ∆ (1) , ∆ (2) are two linear maps, called coproducts ∆ (1) , ∆ (2) : H → H ⊗ H. Using Sweedler's notation, we can write ∆ (i) (h) = h (i) (1) ⊗ h (i) (2) , meaning that ∆ (i) (h) are arbitrary tensors of degree 2. In distinction with Hopf algebras, we impose no restrictions on the coproducts.…”

Section: 5mentioning

confidence: 99%

“…An algebra A is called an H-algebra if A is a left H-module via (h, a) → h * a, for any h ∈ H and a ∈ A and for any a, b ∈ A, one has h * (ab) = (h (1) (1) * a)(h (1) (2) * b) + (h (2) (1) * b)(h (2) (2) * a) Such algebras, with a minor modification, have appeared in [6,17]. In the natural way one can define the notions of the homomorphisms of H-algebras, simple Halgebras and so on.…”

Section: 5mentioning

confidence: 99%