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Finite basis problem for identities with involution
We consider associative algebras with involution over a field of
characteristic zero. We proved that any algebra with involution satisfies the
same identities with involution as the Grassmann envelope of some finite
dimensional -graded algebra with graded involution. As a consequence we
obtain the positive solution of the Specht problem for identities with
involution: any associative algebra with involution over a field of
characteristic zero has a finite basis of identities with involution. These
results are analogs of theorems of A.R.Kemer for ordinary identities
The centre of generic algebras of small PI algebras
Verbally prime algebras are important in PI theory. They are well known over
a field of characteristic zero: 0 and (the trivial ones), ,
, . Here is the free associative algebra with free
generators , is the infinite dimensional Grassmann algebra over ,
and are the matrices over and over ,
respectively. Moreover are certain subalgebras of ,
defined below. The generic algebras of these algebras have been studied
extensively. Procesi gave a very tight description of the generic algebra of
. The situation is rather unclear for the remaining nontrivial verbally
prime algebras.
In this paper we study the centre of the generic algebra of in
two generators. We prove that this centre is a direct sum of the field and a
nilpotent ideal (of the generic algebra). We describe the centre of this
algebra. As a corollary we obtain that this centre contains nonscalar elements
thus we answer a question posed by Berele.Comment: 15 pages. Misprints corrected. Provisionally accepted to publication
in Journal of Algebr
On the polynomial identities of the algebra
Verbally prime algebras are important in PI theory. They were described by
Kemer over a field of characteristic zero: 0 and (the trivial ones),
, , . Here is the free associative algebra of
infinite rank, with free generators , denotes the infinite dimensional
Grassmann algebra over , and are the matrices
over and over , respectively. The algebras are subalgebras
of , see their definition below. The generic (also called
relatively free) algebras of these algebras have been studied extensively.
Procesi described the generic algebra of and lots of its properties.
Models for the generic algebras of and are also known but
their structure remains quite unclear.
In this paper we study the generic algebra of in two generators,
over a field of characteristic 0. In an earlier paper we proved that its centre
is a direct sum of the field and a nilpotent ideal (of the generic algebra),
and we gave a detailed description of this centre. Those results were obtained
assuming the base field infinite and of characteristic different from 2. In
this paper we study the polynomial identities satisfied by this generic
algebra. We exhibit a basis of its polynomial identities. It turns out that
this algebra is PI equivalent to a 5-dimensional algebra of certain upper
triangular matrices. The identities of the latter algebra have been studied;
these were described by Gordienko. As an application of our results we describe
the subvarieties of the variety of unitary algebras generated by the generic
algebra in two generators of . Also we describe the polynomial
identities in two variables of the algebra .Comment: 21 page
Semigroup graded algebras and codimension growth of graded polynomial identities
We show that if is any of four semigroups of two elements that are not
groups, there exists a finite dimensional associative -graded algebra over a
field of characteristic such that the codimensions of its graded polynomial
identities have a non-integer exponent of growth. In particular, we provide an
example of a finite dimensional graded-simple semigroup graded algebra over an
algebraically closed field of characteristic with a non-integer graded
PI-exponent, which is strictly less than the dimension of the algebra. However,
if is a left or right zero band and the -graded algebra is unital, or
is a cancellative semigroup, then the -graded algebra satisfies the
graded analog of Amitsur's conjecture, i.e. there exists an integer graded
PI-exponent. Moreover, in the first case it turns out that the ordinary and the
graded PI-exponents coincide. In addition, we consider related problems on the
structure of semigroup graded algebras.Comment: 21 pages; Minor misprints are correcte
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