Let
F
be a field of characteristic zero and let
R
be an algebra that admits a regular grading by an abelian group
H
. Moreover, we consider
G
a group and let
A
be an algebra with a grading by the ...group
G
×
H
, we define the
R
-hull of
A
as the
G
×
H
-graded algebra given by
R
(
A
)
=
⊕
(
g
,
h
)
∈
G
×
H
A
(
g
,
h
)
⊗
R
h
. In this paper we provide a basis for the graded identities (resp. central polynomials) of the
R
-hull of
A
, assuming that a (suitable) basis for the graded identities (resp. central polynomials) of the
G
×
H
-graded algebra A is known. In particular, for any
a
,
b
∈
ℕ
, we find a basis for the graded identities and the graded central polynomials for the algebra
M
a
,
b
(
E
), graded by the group
G
×
ℤ
2
. Here
E
is the Grassmann algebra of an infinite dimensional
F
-vector space, equipped with its natural
ℤ
2
-grading and the matrix algebra
M
a
+
b
(
F
) is equipped with an elementary grading by the group
G
×
ℤ
2
, so that its neutral component coincides with the subspace of the diagonal matrices. We describe the isomorphism classes of gradings on
M
a
,
b
(
E
) that arise in this way and count the isomorphism classes of such gradings. Moreover, we give an alternative proof of the fact that the tensor product
M
a
,
b
(
E
) ⊗
M
r
,
s
(
E
) is PI-equivalent to
M
a
r
+
b
s
,
a
s
+
b
r
(
E
). Finally, when the grading group is
ℤ
3
×
ℤ
2
(resp.
ℤ
×
ℤ
2
), we present a complete description of a basis for the graded central polynomials for the algebra
M
2,1
(
E
) (resp.
M
a
,
b
(
E
) in the case
b
= 1).
Let
K
be a field of characteristic zero. We study the graded identities of the special linear Lie algebra with the Pauli and Cartan gradings. Given a prime number
p
we provide a finite basis for the ...graded identities of
s
l
p
(
K
)
with the Pauli grading by the group ℤ
p
× ℤ
p
and compute its graded codimensions. We also prove that
var
ℤ
p
×
ℤ
p
(
s
l
p
(
K
)
)
is a minimal variety and satisfies the Specht property. As a by-product we determine a basis for the identities of certain graded Lie algebras with a grading in which every homogeneous subspace has dimension ≤ 1. For
s
l
m
(
K
)
with the Cartan grading a finite basis for the graded identities is determined, moreover a basis for the subspace of the multilinear polynomials in the relatively free algebra
L
〈
X
G
〉
/
T
G
(
s
l
m
(
K
)
)
, as a vector space, is exhibited. As a consequence we compute the graded codimensions for
m
= 2 and provide bases for the graded identities and for the subspace of the multilinear polynomials in the relatively free algebra of certain Lie subalgebras of
M
m
(
K
)
(
−
)
with the Cartan grading.
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 F be an infinite field. The primeness property for central polynomials of Mn(F) was established by A. Regev, i.e., if the product of two polynomials in distinct variables is central then each ...factor is also central. In this paper we consider the analogous property for Mn(F) and determine, within the elementary gradings with commutative neutral component, the ones that satisfy this property, namely the crossed product gradings. Next we consider Mn(R), where R admits a regular grading, with a grading such that Mn(F) is a homogeneous subalgebra and provide sufficient conditions – satisfied by Mn(E) with the trivial grading – to prove that Mn(R) has the primeness property if Mn(F) does. We also prove that the algebras Ma,b(E) satisfy this property for ordinary central polynomials. Hence we conclude that, over a field of characteristic zero, every verbally prime algebra has the primeness property.
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.
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