Let E be the Grassmann algebra of an infinite-dimensional vector space L over a field of characteristic zero. In this paper, we study the
-gradings on E having the form
, in which each element of a ...basis of L has
-degree
, or
. We provide a criterion for the support of this structure to coincide with a subgroup of the group
, and we describe the graded identities for the corresponding gradings. We strongly use Elementary Number Theory as a tool, providing an interesting connection between this classical part of Mathematics, and PI Theory. Our results are generalizations of the approach presented in Brandão A, Fidelis C, Guimarães A.
-gradings of full support on the Grassmann algebra. J Algebra. 2022;601:332-353. DOI:10.1016/j.jalgebra.2022.03.014. See also in arXiv preprint, arXiv:2009.01870v1, 2020.
Let G be a group and F an infinite field. Assume that A is a finite dimensional F-algebra with an elementary G-grading. In this paper, we study the graded identities satisfied by the tensor product ...grading on the F-algebra A⊗C, where C is an H-graded colour β-commutative algebra. More precisely, under a technical condition, we provide a basis for the TG-ideal of graded polynomial identities of A⊗C, up to graded monomial identities. Furthermore, the F-algebra of upper block-triangular matrices UT(d1,…,dn), as well as the matrix algebra Mn(F), with an elementary grading such that the neutral component corresponds to its diagonal, are studied. As a consequence of our results, a basis for the graded identities, up to graded monomial identities of degrees ≤2d−1, for Md(E) and Mq(F)⊗UT(d1,…,dn), with a tensor product grading, is exhibited. In this latter case, d=d1+…+dn. Here E denotes the infinite dimensional Grassmann algebra with its natural Z2-grading, and the grading on Mq(F) is Pauli grading. The results presented in this paper generalize results from 14 and from other papers which were obtained for fields of characteristic zero.
Let E be the infinite dimensional Grassmann algebra over an infinite field of characteristic different from 2. Let
be a n-induced
-grading on E and let
be its support. In this paper, we study the ...properties of the set
as a subset of
Assuming that this set is bounded, we describe the graded identities of the respective graded algebra. We also show that
is a subgroup of
if and only if it is cyclic, and we describe the graded identities in this case.
Denote by Mat
k,l
(
F
) the algebra
M
n
(
F
) of matrices of order
n
=
k
+
l
with the grading (Mat
k,l
0
(
F
),Mat
k,l
1
(
F
)), where Mat
k,l
0
(
F
) admits the basis
and Mat
k,l
1
(
F
) admits the ...basis
. Denote by
M
k,l
(
F
) the Grassmann envelope of the superalgebra Mat
k,l
(
F
).
In the paper, bases of the graded identities of the superalgebras Mat
1,2
(
F
) and
M
1,2
(
F
) are described.
