In the present paper it is proved that a variety of associative PI-superalgebras with graded involution of finite basic rank over a field of characteristic zero is minimal of fixed *-graded exponent ...if, and only if, it is generated by a subalgebra of an upper block triangular matrix algebra equipped with a suitable elementary ℤ
2
-grading and graded involution.
Let F be an algebraically closed field of characteristic zero and G a finite cyclic group. Let
be a variety of associative G-graded PI-algebras over F of finite basic rank. In this paper, we prove ...that if
is minimal with respect to a given G-exponent, then there exist finite-dimensional G-simple F-algebras
such that
is generated by a suitable G-graded upper block triangular matrix algebra
endowed with an elementary grading and where the diagonal blocks are given by the
's. Moreover, for a fixed m-tuple
of finite-dimensional G-simple F-algebras, we prove the converse of the above result for some important classes of G-graded algebras
endowed with elementary gradings. In particular, we conclude that the variety generated by A is minimal when A has one or two G-simple blocks as well whenever all (except for at most one) the G-simple components of A are G-regular.
Let
F
be an infinite field of characteristic different from two and
E
be the unitary Grassmann algebra of an infinite dimensional
F
-vector space
L
. Denote by
E
g
r
an arbitrary
ℤ
2
-grading on
E
...such that the subspace
L
is homogeneous. We consider
E
g
r
⊗
E
⊗
n
as a
(
ℤ
2
×
ℤ
2
n
)
-graded algebra, where the grading on
E
is supposed to be the canonical one, and we find its graded ideal of identities.
Let F be an algebraically closed field of characteristic zero and let
be an n-tuple of simple superalgebras such that
and there exists
such that
, with
, whereas
for all remaining indices. In this ...paper, we classify the minimal F-superalgebras such that its graded semisimple part is given by
. Moreover, we obtain necessary and sufficient conditions to the supervariety generated by such minimal superalgebra be or not minimal of superexponent
It has been shown that in characteristic zero the generators of the minimal supervarieties of finite basic rank belong to the class of
minimal superalgebras
introduced by Giambruno and Zaicev (Trans ...Am Math Soc 355:5091–5117,
2003
). In the present paper the complete list of minimal supervarieties generated by minimal superalgebras whose maximal semisimple homogeneous subalgebra is the sum of three graded simple algebras is provided. As a consequence, we negatively answer the question of whether any minimal superalgebra generates a minimal supervariety.
It has been recently proved that a variety of associative PI-superalgebras with graded involution of finite basic rank over a field of characteristic zero is minimal of fixed ⁎-graded exponent if, ...and only if, it is generated by a subalgebra of an upper block triangular matrix algebra, A:=UTZ2⁎(A1,…,Am), equipped with a suitable elementary Z2-grading and graded involution. Here we give necessary and sufficient conditions so that IdZ2⁎(A) factorizes in the product of the ideals of ⁎-graded polynomial identities of its ⁎-graded simple components Ai.
Let F be an algebraically closed field of characteristic zero and G be an arbitrary finite cyclic group. In this paper, given an m-tuple (A1,…,Am) of finite dimensional G-simple algebras, we focus on ...the study of the factorability of the TG-ideals IdG((UT(A1,…,Am),α˜)) of the G-graded upper block triangular matrix algebras UT(A1,…,Am) endowed with elementary G-gradings induced by some maps α˜.
When G is a cyclic p-group we prove that the factorability of the ideal IdG((UT(A1,…,Am),α˜) is equivalent to the G-regularity of all (except for at most one) the G-simple components A1,…,Am, as well to the existence of a unique isomorphism class of α˜-admissible elementary G-gradings for UT(A1,…,Am). Moreover, we present some necessary and sufficient conditions to the factorability of IdG((UT(A1,A2),α˜)), even in case G is not a p-group, with some stronger assumptions on the gradings of the algebras A1 and A2.
Given a minimal superalgebra A=Ass⊕J(A), any subsequence of the graded simple summands of Ass determines a homogeneous subalgebra of A which is still a minimal superalgebra. In the present paper we ...provide a sufficient condition so that the TZ2-ideal of graded polynomial identities satisfied by A factorizes as the product of the TZ2-ideals associated to its suitable homogeneous subalgebras of such a type. We use this fact to show that in this event A generates a minimal supervariety of fixed superexponent.
Let p be an odd prime number, F a field of characteristic zero, and let Ebe the unitary Grassmann algebra generated by the infinite-dimensionalF-vector space L. We determine the bases of the ℤ
p
...-graded identities.Moreover we compute the ℤ
p
-graded codimension and cocharacter sequences for the algebra E endowed with any ℤ
p
-grading such that L is a homogeneous subspace.
