The trace codimensions give a quantitative description of the identities satisfied by an algebra with trace. Here we study the asymptotic behaviour of the sequence of trace codimensions
c
n
tr
(
A
) ...and of pure trace codimensions
c
n
ptr
(
A
) of a finite-dimensional algebra
A
over a field of characteristic zero. We find an upper and lower bound of both codimensions and as a consequence we get that the limits
lim
n
→
∞
c
n
t
r
(
A
)
n
and
lim
n
→
∞
c
n
p
t
r
(
A
)
n
always exist and are integers. This result gives a positive answer to a conjecture of Amitsur in this setting. Finally we characterize the varieties of algebras whose exponential growth is bounded by 2.
In this paper we prove that if
A
is any algebra with involution * satisfying a non-trivial polynomial identity, then its sequence of *-codimensions is eventually non-decreasing. Furthermore, by ...making use of the *-exponent we reconstruct the only two *-algebras, up to
T*
-equivalence, generating varieties of almost polynomial growth. As a third result we characterize the varieties of algebras with involution whose exponential growth is bounded by 2.
Let
A
be a superalgebra with graded involution or superinvolution ∗ and let
c
n
∗
(
A
)
,
n
= 1,2,…, be its sequence of ∗-codimensions. In case
A
is finite dimensional, in Giambruno et al. (Algebr. ...Represent. Theory
19
(3), 599–611
2016
, Linear Multilinear Algebra
64
(3), 484–501
2016
) it was proved that such a sequence is polynomially bounded if and only if the variety generated by
A
does not contain the group algebra of
ℤ
2
and a 4-dimensional subalgebra of the 4 × 4 upper-triangular matrices with suitable graded involutions or superinvolutions. In this paper we study the general case of ∗-superalgebras satisfying a polynomial identity. As a consequence we classify the varieties of ∗-superalgebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth, and we give a full classification of their subvarieties which was started in Ioppolo and La Mattina (J. Algebra
472
, 519–545
2017
).
Let F be an algebraically closed field of characteristic zero. We give a complete classification of finite dimensional simple superalgebras over F endowed with a graded automorphism or a ...pseudoautomorphism.
To any superalgebra A is attached a numerical sequence cnsup(A), n≥1, called the sequence of supercodimensions of A. In characteristic zero its asymptotics are an invariant of the superidentities ...satisfied by A. It is well-known that for a PI-superalgebra such sequence is exponentially bounded and expsup(A)=limn→∞cnsup(A)n is an integer that can be explicitly computed.
Here we introduce a notion of fundamental superalgebra over a field of characteristic zero. We prove that if A is such an algebra, then(1)C1ntexpsup(A)n≤cnsup(A)≤C2ntexpsup(A)n, where C1>0,C2,t are constants and t is a half integer that can be explicitly written as a linear function of the dimension of the even part of A and the nilpotency index of the Jacobson radical. We also give a characterization of fundamental superalgebras through the representation theory of the symmetric group.
As a consequence we prove that if A is a finitely generated PI-superalgebra, then the inequalities in (1) still hold and t is a half integer. It follows that limn→∞logncnsup(A)expsup(A)n exists and is a half integer.
Let
A
be an associative algebra over a field
F
of characteristic zero and let
c
n
(
A
)
,
n
=
1
,
2
,
…
, be the sequence of codimensions of
A
. It is well-known that
c
n
(
A
)
,
n
=
1
,
2
,
…
, ...cannot have intermediate growth, i.e., either is polynomially bounded or grows exponentially. Here we present some results on algebras whose sequence of codimensions is polynomially bounded.
Let A be an associative algebra over a field F of characteristic zero, and let χ
n
(A), n = 1,2,..., be the sequence of cocharacters of A. For every n ≥ 1, let l
n
(A) denote the nth colength of A, ...counting the number of S
n
-irreducibles appearing in χ
n
(A). In this article, we classify the algebras A such that the sequence of colengths l
n
(A), n = 1,2,..., is bounded by four. Moreover we construct a finite number of algebras A
1
,..., A
d
, such that l
n
(A) ≤ 4 if and only if A
1
,..., A
d
∉ var(A).
Let G be the infinite dimensional Grassmann algebra over a field F of characteristic zero and UT2 the algebra of 2 × 2 upper triangular matrices over F. The relevance of these algebras in PI-theory ...relies on the fact that they generate the only two varieties of almost polynomial growth, i.e., they grow exponentially but any proper subvariety grows polynomially. In this paper we completely classify, up to PI-equivalence, the associative algebras A such that A ∈ Var(G) or A ∈ Var(UT2).
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