Let K be an infinite field of characteristic different from two and let U1 be the Lie algebra of the derivations of the algebra of Laurent polynomials Kt,t−1. The algebra U1 admits a natural ...Z-grading. We provide a basis for the graded identities of U1 and prove that they do not admit any finite basis. Moreover, we provide a basis for the identities of certain graded Lie algebras with a grading such that every homogeneous component has dimension ≤1, if a basis of the multilinear graded identities is known. As a consequence of this latter result we are able to provide a basis of the graded identities of the Lie algebra W1 of the derivations of the polynomial ring Kt. The Z-graded identities for W1, in characteristic 0, were described in 8. As a consequence of our results, we give an alternative proof of the main result, Theorem 1, in 8, and generalize it to positive characteristic. We also describe a basis of the graded identities for the special linear Lie algebra slq(K) with the Pauli gradings where q is a prime number.
Let K be a field and let Jn,k be the Jordan algebra of a degenerate symmetric bilinear form b of rank n−k over K. Then one can consider the decomposition Jn,k=Bn−k⊕Dk, where Bn−k represents the ...corresponding Jordan algebra, denoted as Bn−k=K⊕V. In this algebra, the restriction of b on the (n−k)-dimensional subspace V is non-degenerate, while Dk accounts for the degenerate part of Jn,k. This paper aims to provide necessary and sufficient conditions to check if a given multilinear polynomial is an identity for Jn,k. As a consequence of this result and under certain hypothesis on the base field, we exhibit a finite basis for the T-ideal of polynomial identities of Jn,k. Over a field of characteristic zero, we also prove that the ideal of identities of Jn,k satisfies the Specht property. Moreover, similar results are obtained for weak identities, trace identities and graded identities with a suitable Z2-grading as well. In all of these cases, we employ methods and results from Invariant Theory. Finally, as a consequence from the trace case, we provide a counterexample to the embedding problem given in 8 in case of infinite dimensional Jordan algebras with trace.
Let F be a field of characteristic 0 and let UTm(F) 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 UTm(F) equipped with any elementary G-grading, where G is a finite group.
Let K be a field of characteristic 0 and let W1 be the Lie algebra of the derivations of the polynomial ring Kt. The algebra W1 admits a natural Z-grading. We describe the graded identities of W1 for ...this grading. It turns out that all these Z-graded identities are consequences of a collection of polynomials of degree 1, 2 and 3 and that they do not admit a finite basis. Recall that the “ordinary” (non-graded) identities of W1 coincide with the identities of the Lie algebra of the vector fields on the line and it is a long-standing open problem to find a basis for these identities. We hope that our paper might be a step to solving this problem.
Kac–Moody algebras, g(A), are Lie algebras defined by generators and relations given by generalized Cartan matrices A. In this paper, we study the graded identities for Kac–Moody algebras when the ...matrix A is diagonal. More precisely, we provide a basis for the graded identities of g(A) equipped with its natural grading, the grading of Cartan type. These results are obtained over an arbitrary infinite field. We also compute the graded codimensions for these algebras and provide a basis for the vector space of the multihomogeneous polynomials of any given multidegree in the relatively free algebra. As the base field is infinite we have a vector space basis of the relatively free algebra. As a consequence of our results, we give an alternative proof of Theorem 17 in 15, and generalize it to characteristic two. Finally, we also describe a basis of the graded identities for the Heisenberg algebra with its natural grading, over any field.
Let
P
a locally finite partially ordered set,
F
a field,
G
a group, and
I
(
P
,
F
)
the incidence algebra of
P
over
F
. We describe all the inequivalent elementary
G
-gradings on this algebra. If
P
...is bounded,
F
is an infinite field of characteristic zero, and
A
,
B
are both elementary
G
-graded incidence algebras of
P
satisfying the same
G
-graded polynomial identities, and the automorphism group of
P
acts transitively on the maximal chains of
P
, we show that
A
and
B
are graded isomorphic.
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 obtained by the corresponding multipartition after removing its first row. We relate, moreover, the polynomial growth to the colengths. Finally we describe in detail the algebras whose graded codimensions are of linear growth.
Let F be a field of characteristic 0. We consider the algebra UTm(F) of upper triangular matrices of order m endowed with an elementary Zm-grading induced by the m-tuple ϕ=(0,0,1,…,m−2), then we ...compute its Y-proper graded cocharacter sequence and we give the explicit formulas for the multiplicities in the case m=2,3,4,5.
Let F be an infinite field of characteristic different from two and E be the infinite dimensional Grassmann algebra over F. We consider the upper triangular matrix algebra UT2(E) with entries in E ...endowed with the Z2-grading inherited by the natural Z2-grading of E and we study its ideal of Z2-graded polynomial identities (TZ2-ideal) and its relatively free algebra. In particular we show that the set of Z2-graded polynomial identities of UT2(E) does not depend on the characteristic of the field. Moreover we compute the Z2-graded Hilbert series of UT2(E) and its Z2-graded Gelfand–Kirillov dimension.
PDF
