Basis of the Identities of the Matrix Algebra of Order Two over a Field of Characteristic p≠2

Koshlukov, Plamen

doi:10.1006/jabr.2000.8738

Cited by 49 publications

(28 citation statements)

References 21 publications

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“…Further on, T (M 2 (K)) was described in [15], see also [16], and in a series of papers by Drensky, see, for example, [8,9]. When char K = p > 2, the same was done in [4,11]. If we add the description of T (E ⊗ E) given in [14], when char K = 0 we shall get the complete list of the nontrivial T -prime algebras whose bases of identities are known.…”

Section: The Algebra a Is T -Prime (Or Verbally Prime) If T (A) Is T mentioning

confidence: 89%

Tensor product theorems in positive characteristic

Azevêdo¹,

Fidelis²,

Koshlukov³

2004

Journal of Algebra

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In this paper we study tensor products of T -prime T -ideals over infinite fields. The behaviour of these tensor products over a field of characteristic 0 was described by Kemer. First we show, using methods due to Regev, that such a description holds if one restricts oneself to multilinear polynomials only. Second, applying graded polynomial identities, we prove that the tensor product theorem fails for the T -ideals of the algebras M 1,1 (E) and E ⊗ E where E is the infinite-dimensional Grassmann algebra; M 1,1 (E) consists of the 2 × 2 matrices over E having even (i.e., central) elements of E, and the other diagonal consisting of odd (anticommuting) elements of E. Note that these proofs do not depend on the structure theory of T -ideals but are "elementary" ones. All this comes to show once more that the structure theory of T -ideals is essentially about the multilinear polynomial identities.

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Section: The Algebra a Is T -Prime (Or Verbally Prime) If T (A) Is T mentioning

confidence: 89%

Tensor product theorems in positive characteristic

Azevêdo¹,

Fidelis²,

Koshlukov³

2004

Journal of Algebra

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“…Отметим, что конечность базиса тождеств алгебры M 2 над бесконечным полем ха-рактеристики p > 2 была доказана Кошлуковым (см. [2]). Следовательно, для мно-гообразий сложности 2 над полем характеристики p > 2 проблема 1 имеет положи-тельное решение.…”

Section: л м самойловunclassified

О Нильиндексе Радикала Относительно Свободной Ассоциативной Алгебры

Самойлов¹

2007

Mat. Zametki

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“…Recall that the polynomial identities of the matrix algebras M n (F ) are known only for n ≤ 2, see [30,6,22]. Their graded counterparts have been extensively studied and described under any reasonable grading, see for example [34,3].…”

Section: Introductionmentioning

confidence: 99%

Graded algebras with polynomial growth of their codimensions

Koshlukov

Mattina

2015

Journal of Algebra

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Let A be an associative algebras over a field of characteristic zero. We prove that the codimensions of A are polynomially bounded if and only if any finite dimensional algebra B with Id(A) = Id(B) has an explicit decomposition into suitable subalgebras; we also give a decomposition of the n-th cocharacter of A into suitable Sn-characters. We give similar characterizations of finite dimensional algebras with invo-lution whose *-codimension sequence is polynomially bounded. In this case we exploit the representation theory of the hyperoctahedral group. §1. Introduction Let F be a field of characteristic zero and F X = F x 1 , x 2 ,. .. the free algebra of countable rank over F. If A is a PI-algebra over F , that is, an algebra satisfying a polynomial identity, we let Id(A) be the T-ideal of F X of identities of A. It is well known that Id(A) is completely determined by the multilinear polynomials it contains; if V n = Span{x σ(1) · · · x σ(n) | σ ∈ S n } is the space of multilinear poly-nomials in x 1 ,. .. , x n , then the sequence c n (A) = dim F Vn Vn∩Id(A) , n = 1, 2,. .. , is called the sequence of codimensions of A and it is an important numerical invariant of Id(A). It was proved by Regev in [R] that for any PI-algebra A, c n (A) is exponentially bounded, i.e., there exist constants a, α > 0 such that c n (A) ≤ aα n for all n. In this paper we study algebras A whose codimension sequence is polynomially bounded i.e., such that for all n, c n (A) ≤ an t for some constants a, t. Kemer in [K1] gave a characterization of such T-ideals in the language of the representation theory of S n. It also follows from [K2] that if c n (A) is polynomially bounded, then Id(A) = Id(B) for a suitable finite dimensional algebra B. For any finite dimensional algebra A over an algebraically closed field we shall prove that A has polynomial growth of the codimensions if and only if A = A 0 ⊕A 1 ⊕ · · · ⊕ A m where A 0 , A 1 ,. .. , A m are F-algebras such that 1) for i = 1,. .. , m, A i = B i + J i , where B i ∼ = F and J i is a nilpotent ideal of A i , 2) A 0 , J 1 ,. .. , J m are nilpotent right ideals of A and 3) A i A k = 0 for all i, k ∈ {1,. .. , m}, i = k and B i A 0 = 0.

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Basis of the Identities of the Matrix Algebra of Order Two over a Field of Characteristic p≠2

Cited by 49 publications

References 21 publications

Tensor product theorems in positive characteristic

Tensor product theorems in positive characteristic

О Нильиндексе Радикала Относительно Свободной Ассоциативной Алгебры

Graded algebras with polynomial growth of their codimensions

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