Central polynomials in the matrix algebra of order two

Colombo, Jones; Koshlukov, Plamen

doi:10.1016/j.laa.2003.07.011

Cited by 25 publications

(36 citation statements)

References 13 publications

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“…Notice that if m is a MU n -graded monomial identity of R, then m follows from (7). The main steps of the proof of Lemmas 6.1 and 6.3 hold also for this grading and we obtain the following result.…”

Section: Lemma 63 If R Does Not Satisfy a Monomial Identity Then T Gmentioning

confidence: 58%

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Graded polynomial identities and central polynomials of matrices over an infinite integral domain

Fonseca

2014

Rend. Circ. Mat. Palermo

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Let K be an infinite integral domain and M n (K ) be the algebra of all n ×n matrices over K . This paper aims for the following goals:• Find a basis for the graded identities for elementary grading in M n (K ) when the neutral component and diagonal coincide; • Describe the Z p -graded central polynomials of M p (K ) when p is a prime number;• Describe the Z-graded central polynomials of M n (K ).

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Section: Lemma 63 If R Does Not Satisfy a Monomial Identity Then T Gmentioning

confidence: 58%

“…When K is a field of characteristic zero, a set of generators may be found for the central polynomials of M 2 (K ) [19]. Colombo and Koshlukov [7] described the central polynomials of M 2 (K ), when K is an infinite field of characteristic p > 2.…”

mentioning

confidence: 99%

Graded polynomial identities and central polynomials of matrices over an infinite integral domain

Fonseca

2014

Rend. Circ. Mat. Palermo

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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: 96%

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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“…When char K = 0, a generating set was produced in [16], and a detailed description of the structure of the central polynomials for M 2 (K ) was given in [9]. When char K = p > 2 a generating set was given in [5].…”

Section: Introductionmentioning

confidence: 99%

Central polynomials for Z2-graded algebras and for algebras with involution

Brandão

Koshlukov

2007

Journal of Pure and Applied Algebra

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We describe the Z 2 -graded central polynomials for the matrix algebra of order two, M 2 (K ), and for the algebras M 1,1 (E) and E ⊗ E over an infinite field K , char K = 2. Here E is the infinite-dimensional Grassmann algebra, and M 1,1 (E) stands for the algebra of the 2 × 2 matrices whose entries on the diagonal belong to E 0 , the centre of E, and the off-diagonal entries lie in E 1 , the anticommutative part of E. It turns out that in characteristic 0 the graded central polynomials for M 1,1 (E) and E ⊗ E are the same (it is well known that these two algebras satisfy the same polynomial identities when char K = 0). On the contrary, this is not the case in characteristic p > 2. We describe systems of generators for the Z 2 -graded central polynomials for all these algebras.Finally we give a generating set of the central polynomials with involution for M 2 (K ). We consider the transpose and the symplectic involutions.MSC: 16R10; 16R20; 16R40; 15A75 IntroductionAn important task in the study of algebras with polynomial identities is the description of the central polynomials in a given algebra. The existence of central polynomials for the matrix algebras M n (K ) over a field K was proved by means of direct construction by Formanek [8] and by Razmyslov [19]. Note that Razmyslov also found central polynomials for other important classes of algebras. These are the following. Let E be the infinite dimensional Grassmann algebra of a vector space V with a basis e 1 , e 2 , . . .; then E has a basis consisting of 1 and the products e i 1 e i 2 . . . e i k , i 1 < i 2 < · · · < i k , k ≥ 1. The multiplication in E is induced by e i e j = −e j e i and e 2 i = 0. The algebra E has a natural Z 2 -grading defined as follows: E = E 0 ⊕ E 1 where E i is the span of all basic elements such that k ≡ i (mod 2), i = 0, 1. Now let M n (E) be the n × n matrix algebra over E. Set M a,b the subalgebra of M a+b (E) that consistis of the block matrices having two blocks on the main diagonal, of sizes a × a and b × b whose elements * Corresponding author.

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Central polynomials in the matrix algebra of order two

Cited by 25 publications

References 13 publications

Graded polynomial identities and central polynomials of matrices over an infinite integral domain

Graded polynomial identities and central polynomials of matrices over an infinite integral domain

Tensor product theorems in positive characteristic

Central polynomials for Z2-graded algebras and for algebras with involution

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