Search results for "Graded ring"
showing 10 items of 16 documents
Graded polynomial identities and codimensions: Computing the exponential growth
2010
Abstract Let G be a finite abelian group and A a G-graded algebra over a field of characteristic zero. This paper is devoted to a quantitative study of the graded polynomial identities satisfied by A. We study the asymptotic behavior of c n G ( A ) , n = 1 , 2 , … , the sequence of graded codimensions of A and we prove that if A satisfies an ordinary polynomial identity, lim n → ∞ c n G ( A ) n exists and is an integer. We give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple G × Z 2 -graded algebra related to A.
Graded algebras with polynomial growth of their codimensions
2015
Abstract Let A be an algebra over a field of characteristic 0 and assume A is graded by a finite group G . We study combinatorial and asymptotic properties of the G -graded polynomial identities of A provided A is of polynomial growth of the sequence of its graded codimensions. Roughly speaking this means that the ideal of graded identities is “very large”. We relate the polynomial growth of the codimensions to the module structure of the multilinear elements in the relatively free G -graded algebra in the variety generated by A . We describe the irreducible modules that can appear in the decomposition, we show that their multiplicities are eventually constant depending on the shape obtaine…
Identities of PI-Algebras Graded by a Finite Abelian Group
2011
We consider associative PI-algebras over an algebraically closed field of zero characteristic graded by a finite abelian group G. It is proved that in this case the ideal of graded identities of a G-graded finitely generated PI-algebra coincides with the ideal of graded identities of some finite dimensional G-graded algebra. This implies that the ideal of G-graded identities of any (not necessary finitely generated) G-graded PI-algebra coincides with the ideal of G-graded identities of the Grassmann envelope of a finite dimensional (G × ℤ2)-graded algebra, and is finitely generated as GT-ideal. Similar results take place for ideals of identities with automorphisms.
Multialternating graded polynomials and growth of polynomial identities
2012
Let G be a finite group and A a finite dimensional G-graded algebra over a field of characteristic zero. When A is simple as a G-graded algebra, by mean of Regev central polynomials we construct multialternating graded polynomials of arbitrarily large degree non vanishing on A. As a consequence we compute the exponential rate of growth of the sequence of graded codimensions of an arbitrary G-graded algebra satisfying an ordinary polynomial identity. In particular we show it is an integer. The result was proviously known in case G is abelian.
Integral curves of derivations
1988
We integrate, by a constructive method, derivations of even degree on the sections of an exterior bundle by families of Z 2-graded algebra automorphisms, dependent on a real parameter, and which satisfy a flow condition. We also study the case of local endomorphisms when their components of degree zero and derivations and with no component of negative degree, but then we have integral families of R-linear automorphisms. This integration method can be applied to the Frolicher—Nijenhuis derivations on the Cartan algebra of differential forms, and to the integration of superfields on graded manifolds.
The graded identities of upper triangular matrices of size two
2002
AbstractLet UT2 be the algebra of 2×2 upper triangular matrices over a field F. We first classify all possible gradings on UT2 by a group G. It turns out that, up to isomorphism, there is only one non-trivial grading and we study all the graded polynomial identities for such algebra. In case F is of characteristic zero we give a complete description of the space of multilinear graded identities in the language of Young diagrams through the representation theory of the hyperoctahedral group. We finally establish a result concerning the rate of growth of the identities for such algebra by proving that its sequence of graded codimensions has almost polynomial growth.
Graded polynomial identities and Specht property of the Lie algebrasl2
2013
Abstract Let G be a group. The Lie algebra sl 2 of 2 × 2 traceless matrices over a field K can be endowed up to isomorphism, with three distinct non-trivial G-gradings induced by the groups Z 2 , Z 2 × Z 2 and Z . It has been recently shown (Koshlukov, 2008 [8] ) that for each grading the ideal of G-graded identities has a finite basis. In this paper we prove that when char ( K ) = 0 , the algebra sl 2 endowed with each of the above three gradings has an ideal of graded identities Id G ( sl 2 ) satisfying the Specht property, i.e., every ideal of graded identities containing Id G ( sl 2 ) is finitely based.
Gradings on the algebra of upper triangular matrices and their graded identities
2004
Abstract Let K be an infinite field and let UT n ( K ) denote the algebra of n × n upper triangular matrices over K . We describe all elementary gradings on this algebra. Further we describe the generators of the ideals of graded polynomial identities of UT n ( K ) and we produce linear bases of the corresponding relatively free graded algebras. We prove that one can distinguish the elementary gradings by their graded identities. We describe bases of the graded polynomial identities in several “typical” cases. Although in these cases we consider elementary gradings by cyclic groups, the same methods serve for elementary gradings by any finite group.
On the graded identities and cocharacters of the algebra of 3×3 matrices
2004
Abstract Let M2,1(F) be the algebra of 3×3 matrices over an algebraically closed field F of characteristic zero with non-trivial Z 2 -grading. We study the graded identities of this algebra through the representation theory of the hyperoctahedral group Z 2 ∼S n . After splitting the space of multilinear polynomial identities into the sum of irreducibles under the Z 2 ∼S n -action, we determine all the irreducible Z 2 ∼S n -characters appearing in this decomposition with non-zero multiplicity. We then apply this result in order to study the graded cocharacter of the Grassmann envelope of M2,1(F). Finally, using the representation theory of the general linear group, we determine all the grade…
The nearly Gorenstein property for numerical duplications and semitrivial extensions
2022
In this thesis we present and study the ideal duplication, a new construction within the class of the relative ideals of a numerical semigroup $S$, that, under specific assumptions, produces a relative ideal of the numerical duplication $SJoin^b E$, for some ideal $E$ of $S$. We prove that every relative ideal of the numerical duplication can be uniquely written as the ideal duplication of two relative ideals of $S$; this allows us to better understand how the basic operations of the class of the relative ideals of $SJoin^b E$ work. In particular, we characterize the ideals $E$ such that $SJoin^b E$ is nearly Gorenstein. With the aim to generalize this construction to commutative rings with…