101 research outputs found
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
PolynomiaI Identities in T-prime AIgebras
We survey results concerning the polynornial identities satisfied by "important" algebras. We discuss classical and new facts about the polynomial identities satisfied by the matrix algebra of order two, by the Grassmann (or exterior) algebra, and by its tensor square
A Basis for the Graded Identities of the Pair (M2(K), gl2(K))
2010 Mathematics Subject Classification: 16R10, 17B01.Let M2(K) be the algebra of 2×2 matrices over an infinite integral domain K. In this note we describe a basis for the Z2-graded identities of the pair (M2(K),gl2(K)).∗ Partially supported by CNPq (Grant 304003/2011-5) and FAPESP (Grant 2010/50347-9).
∗∗ Partially supported by CNPq, DPP/UnB and by CNPq-FAPDF PRONEX grant 2009/00091-0 (193.000.580/2009)
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
Gradings and Graded Identities for the Matrix Algebra of Order Two in Characteristic 2
2010 Mathematics Subject Classification: 16R10, 16R99, 16W50.Let K be an infinite field and let M2(K) be the matrix algebra of
order two over K. The polynomial identities of M2(K) are known whenever
the characteristic of K is different from 2. The algebra M2(K) admits a
natural grading by the cyclic group of order 2; the graded identities for
this grading are known as well. But M2(K) admits other gradings that
depend on the field and on its characteristic. Here we describe the graded
identities for all nontrivial gradings by the cyclic group of order 2 when the
characteristic of K equals 2. It turns out that there is only one grading to
consider. This grading is not elementary. On the other hand the graded
identities are the same as for the elementary grading.∗ Partially supported by grants from CNPq (No. 304003/2011-5), and from FAPESP (No. 2010/50347-9)
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