On the identities of the Grassmann algebras in characteristicp&gt;0

Giambruno, Antonio; Koshlukov, Plamen

doi:10.1007/bf02809905

Cited by 49 publications

(33 citation statements)

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“…(a) If n is odd, then the set (4) {f r,σ | σ ∈ A n and 0 ≤ r ≤ n − 4} ∪ {g σ | σ ∈ A n } spans all central A-polynomials of degree n for G. Futhermore…”

Section: Theorem 2 ([5]mentioning

confidence: 99%

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The central polynomials for the Grassmann algebra

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“…(a) If n is odd, then the set (4) {f r,σ | σ ∈ A n and 0 ≤ r ≤ n − 4} ∪ {g σ | σ ∈ A n } spans all central A-polynomials of degree n for G. Futhermore…”

Section: Theorem 2 ([5]mentioning

confidence: 99%

“…The polynomial identities for G were described in [9] by Krakowski and Regev when char(F )=0, and by various authors in the general case (see [4] and [10]). The central polynomials for the Grassmann algebra were described independently by several authors, see for example [1], [2] and [6].…”

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The central polynomials for the Grassmann algebra

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“…The identities of E were described for any field K in [1,17]. (One can find a brief account of the results of the last two papers in [6].) The identities satisfied by the algebra UT n (K) of the upper triangular matrices over K are well known as well (see [5,Chapter 5] for further information).…”

Section: Introductionmentioning

confidence: 99%

Central polynomials in the matrix algebra of order two

Colombo

Koshlukov

2004

Linear Algebra and its Applications

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We exhibit minimal bases of the polynomial identities for the matrix algebra M 2 (K) of order two over an infinite field K of characteristic p / = 2. We show that when p = 3 the T -ideal of this algebra is generated by three independent identities, and when p > 3 one needs only two identities: the standard identity of degree four and the Hall identity. Note that the same holds when the base field is of characteristic 0. Furthermore, using the exact form of the basis of the identities for M 2 (K) we give finite minimal set of generators of the T -space of the central polynomials for the algebra M 2 (K). The set of generators depends on the characteristic of the field as well.

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“…The identities of the Grassmann algebra E are known in detail, see [14] for the case of characteristic 0, and the references of [10] for the remaining cases. The identities of M 2 (K ) were described in [18] when char K = 0, and in [12] when K is infinite and char K = p > 2.…”

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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On the identities of the Grassmann algebras in characteristicp>0

Cited by 49 publications

References 11 publications

The central polynomials for the Grassmann algebra

The central polynomials for the Grassmann algebra

Central polynomials in the matrix algebra of order two

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

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