32 research outputs found
A model for the relatively free graded algebra of block triangular matrices with entries from a graded algebra
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 upper triangular matrices over infinite fields
In this paper we prove that the image of multilinear polynomials evaluated on
the algebra of upper triangular matrices over an infinite
field equals , a power of its Jacobson ideal . In
particular, this shows that the analogue of the Lvov-Kaplansky conjecture for
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 in terms of
its coefficients. It turns out that the image of a multilinear polynomial
on is if and only if has commutator degree .Comment: To appear in Israel Journal of Mathematic
A new approach to the Lvov-Kaplansky conjecture through gradings
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
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
Let be an infinite field and be the algebra of upper
block-triangular matrices over . In this paper we describe a basis for the
-graded polynomial identities of , with an elementary
grading induced by an -tuple of elements of a group such that the
neutral component corresponds to the diagonal of . In
particular, we prove that the monomial identities of such algebra follow from
the ones of degree up to . 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 -gradings. In
the characteristic zero case we also generalize results for the algebra
with a tensor product grading, where 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
In this paper we study identities and images of polynomials on null-filiform
Leibniz algebras. If is an -dimensional null-filiform Leibniz algebra,
we exhibit a finite minimal basis for \mbox{Id}(L_n), the polynomial
identities of , and we explicitly compute the images of multihomogeneous
polynomials on . We present necessary and sufficient conditions for the
image of a multihomogeneous polynomial to be a subspace of . 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 . 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
Let be a field of characteristic zero, be a group and be the
algebra with a -grading. Bahturin and Drensky proved that if is
an elementary and the neutral component is commutative then the graded
identities of follow from three basic types of identities and monomial
identities of length bounded by a function of . In this
paper we prove the best upper bound is , more generally we prove that
all the graded monomial identities of an elementary -grading on
follow from those of degree at most . 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 is almost non-degenerate and we
apply the results on monomial identities to describe all almost non-degenerate
-gradings on for .Comment: 21 pages. Corrections and improvements of some result