32 research outputs found

    A model for the relatively free graded algebra of block triangular matrices with entries from a graded algebra

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    Let G be a group and A be a G-graded algebra satisfying a polynomial identity. We buid up a model for the relative free G-graded algebra and we obtain, as an application, the "factoring" property for the T_G-ideals of block triangular matrices with entries from the finite dimensional Grassmann algebra E for some particular Z_2-grading

    Images of multilinear polynomials on n×nn\times n upper triangular matrices over infinite fields

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    In this paper we prove that the image of multilinear polynomials evaluated on the algebra UTn(K)UT_n(K) of n×nn\times n upper triangular matrices over an infinite field KK equals JrJ^r, a power of its Jacobson ideal J=J(UTn(K))J=J(UT_n(K)). In particular, this shows that the analogue of the Lvov-Kaplansky conjecture for UTn(K)UT_n(K) is true, solving a conjecture of Fagundes and de Mello. To prove that fact, we introduce the notion of commutator-degree of a polynomial and characterize the multilinear polynomials of commutator-degree rr in terms of its coefficients. It turns out that the image of a multilinear polynomial ff on UTn(K)UT_n(K) is JrJ^r if and only if ff has commutator degree rr.Comment: To appear in Israel Journal of Mathematic

    A new approach to the Lvov-Kaplansky conjecture through gradings

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    In this paper we consider images of (ordinary) noncommutative polynomials on matrix algebras endowed with a graded structure. We give necessary and sufficient conditions to verify that some multilinear polynomial is a central polynomial, or a trace zero polynomial, and we use this approach to present an equivalent statement to the Lvov-Kaplansky conjecture.Comment: 10 pages; comments are welcom

    On star-homogeneous-graded polynomial identities of upper triangular matrices over an arbitrary field

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    We study the graded polynomial identities with a homogeneous involution on the algebra of upper triangular matrices endowed with a fine group grading. We compute their polynomial identities and a basis of the relatively free algebra, considering an arbitrary base field. We obtain the asymptotic behaviour of the codimension sequence when the characteristic of the base field is zero. As a consequence, we compute the exponent and the second exponent of the same algebra endowed with any group grading and any homogeneous involution

    Graded identities of block-triangular matrices

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    Let FF be an infinite field and UT(d1,…,dn)UT(d_1,\dots, d_n) be the algebra of upper block-triangular matrices over FF. In this paper we describe a basis for the GG-graded polynomial identities of UT(d1,…,dn)UT(d_1,\dots, d_n), with an elementary grading induced by an nn-tuple of elements of a group GG such that the neutral component corresponds to the diagonal of UT(d1,…,dn)UT(d_1,\dots,d_n). In particular, we prove that the monomial identities of such algebra follow from the ones of degree up to 2n−12n-1. Our results generalize for infinite fields of arbitrary characteristic, previous results in the literature which were obtained for fields of characteristic zero and for particular GG-gradings. In the characteristic zero case we also generalize results for the algebra UT(d1,…,dn)⊗CUT(d_1,\dots, d_n)\otimes C with a tensor product grading, where CC is a color commutative algebra generating the variety of all color commutative algebras.Comment: 24 pages and 39 references. We have added section 5 in the text about tensor products by color commutative superalgebra

    Polynomial identities and images of polynomials on null-filiform Leibniz algebras

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    In this paper we study identities and images of polynomials on null-filiform Leibniz algebras. If LnL_n is an nn-dimensional null-filiform Leibniz algebra, we exhibit a finite minimal basis for \mbox{Id}(L_n), the polynomial identities of LnL_n, and we explicitly compute the images of multihomogeneous polynomials on LnL_n. We present necessary and sufficient conditions for the image of a multihomogeneous polynomial ff to be a subspace of LnL_n. For the particular case of multilinear polynomials, we prove that the image is always a vector space, showing that the analogue of the L'vov-Kaplansky conjecture holds for LnL_n. We also prove similar results for an analog of null-filiform Leibniz algebras in the infinite-dimensional case.Comment: 13 pages; comments are welcom

    Graded monomial identities and almost non-degenerate gradings on matrices

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    Let FF be a field of characteristic zero, GG be a group and RR be the algebra Mn(F)M_n(F) with a GG-grading. Bahturin and Drensky proved that if RR is an elementary and the neutral component is commutative then the graded identities of RR follow from three basic types of identities and monomial identities of length ≥2\geq 2 bounded by a function f(n)f(n) of nn. In this paper we prove the best upper bound is f(n)=nf(n)=n, more generally we prove that all the graded monomial identities of an elementary GG-grading on Mn(F)M_n(F) follow from those of degree at most nn. We also study gradings which satisfy no monomial identities but the trivial ones, which we call almost non-degenerate gradings. The description of non-degenerate elementary gradings on matrix algebras is reduced to the description of non-degenerate elementary gradings on matrix algebras that have commutative neutral component. We provide necessary conditions so that the grading on RR is almost non-degenerate and we apply the results on monomial identities to describe all almost non-degenerate Z\mathbb{Z}-gradings on Mn(F)M_n(F) for n≤5n\leq 5.Comment: 21 pages. Corrections and improvements of some result
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