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On the graded identities of the Grassmann algebra
No full textWe survey the results concerning the graded identities of the infinite dimensional Grassmann algebra
On some recent results about the graded Gelfand-Kirillov dimension of graded PI-algebras
Full text link2010 Mathematics Subject Classification: 16R10, 16W55, 15A75.We survey some recent results on graded Gelfand-Kirillov dimension of PI-algebras over a field F of characteristic 0. In particular, we focus on verbally prime algebras with the grading inherited by that of Vasilovsky and upper triangular matrices, i.e., UTn(F), UTn(E) and UTa,b(E), where E is the infinite dimensional Grassmann algebra
The image of Lie polynomials on real Lie algebras of dimension up to 3
No full textLet be the free Lie algebra over of rank generated by , and let f\inF_n' be a multilinear Lie polynomial contained in the commutator ideal
of . In this paper, we determine the image for Lie algebras of dimension , and of the Lie algebra of dimension 4 stated in a paper of Baker dating back to 1901. In all the cases studied, the L'vov-Kaplansky Conjecture has a positive answer
The quotient algebra of an H-module Lie algebra
No full textIn this paper we define the H-module Lie algebra of quotients for an H-semiprime Lie algebra L, where H is a cocommutative Hopf algebra and we compute the maximal H-module Lie algebra of quotients of L, say QmH(L). As an application, we compute the Gelfand-Kirillov dimension (GK) of QmH(L) showing it equals the GK dimension of L
Graded Identities of Several Tensor Products of the Grassmann Algebra
No full textLet F be an infinite field of characteristic different from two and E be the unitary Grassmann algebra of an infinite dimensional F-vector space L. Denote by Egr an arbitrary Z_2-grading on E such that the subspace L is homogeneous. We consider Egr x E^n as a (Z_2xZ_2^n)-graded algebra, where the grading on E is supposed to be the canonical one, and we find its graded ideal of identities
A negative answer to a Bahturin-Regev Conjecture about regular algebras in positive characteristic
No full textLet A = A1 ⊕···⊕Ar be a decomposition of the algebra A as a direct sum of vector subspaces. If for every choice of the indices 1 ≤ ij ≤ r there exist aij ∈ Aij such that the product ai1 ···ain = 0, and for every 1 ≤ i,j ≤ r there is a constant β(i,j)= 0 with aiaj = β(i, j)ajai for ai ∈ Ai, aj ∈ Aj, the above decomposition is regular. Bahturin and Regev raised the following conjecture: suppose the regular decomposition comes from a group grading on A, and form the r × r matrix whose (i,j)th entry equals β(i,j). Then this matrix is invertible if and only if the decomposition is minimal (that is one cannot get a regular decomposition of A by coarsening the decomposition). Aljadeff and David proved that the conjecture is true in the case the base field is of characteristic 0. We prove that the conjecture does not hold for algebras over fields of positive characteristic, by constructing algebras with minimal regular decompositions such that the associated matrix is singular
Gradings and graded identities of null-filiform Leibniz algebras
No full textWe classify gradings on null-filiform Leibniz algebras up to equivalence over arbitrary fields. Furthermore, we provide a basis for the graded identities and determine a basis of the relatively free algebra. As a consequence, we establish that the ideal of all graded identities of null-filiform Leibniz algebras satisfy the Specht property. Finally, we extend these results to infinite-dimensional analogs of null-filiform Leibniz algebras
Y-proper graded cocharacters of the algebra UT(F) of m × m upper triangular matrices over F
No full textLet F be a field of characteristic 0 and let be the algebra of upper triangular matrices of order m with entries from F. In this paper we give a description of the Y-proper graded cocharacters of equipped with any elementary G-grading, where G is a finite group
Images of graded polynomials on matrix algebras
Full text linkThe aim of this paper is to start the study of images of graded polynomials on full matrix algebras. We work with the matrix algebra over a field K endowed with its canonical -grading (Vasilovsky's grading). We explicitly determine the possibilities for the linear span of the image of a multilinear graded polynomial over the field of rational numbers and state an analogue of the L'vov-Kaplansky conjecture about images of multilinear graded polynomials on matrices, where n is a prime number. We confirm such conjecture for polynomials of degree 2 over when K is a quadratically closed field of characteristic zero or greater than n and for polynomials of arbitrary degree over matrices of order 2. We also determine all the possible images of semi-homogeneous graded polynomials evaluated on
Cocharacters of UT_n(E)
No full textLet F be a field of characteristic 0 and let E be the infinite-dimensional Grassmann algebra over F. In the first part of this paper we give an algorithm calculating the generating function of the cocharacter sequence of the n x n upper triangular matrix algebra UTn(E) with entries in E, lying in a strip of a fixed size. In the second part we compute the double Hilbert series H(E; T_k, Y_l) of E, then we define the (k, l)-multiplicity series of any PI-algebra. As an application, we derive from H(E; T_k, Y_l) an easy algorithm determining the (k, l)-multiplicity series of UT_n(E)
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