46 research outputs found
Noncommutative Grobner basis, Hilbert series, Anick's resolution and BERGMAN under MS-DOS
The definition and main results connected with Grцbner basis, Hilbert series and Anick's resolution are formulated. The method of the infinity behavior prediction of Grцbner basis in noncommutative case is presented. The extensions of BERGMAN package for IBM PC compatible computers are described
Defining Relations of Noncommutative Trace Algebra of Two Matrices
The noncommutative (or mixed) trace algebra is generated by
generic matrices and by the algebra generated by all
traces of products of generic matrices, . It is known that over a
field of characteristic 0 this algebra is a finitely generated free module over
a polynomial subalgebra of the center . For and we have
found explicitly such a subalgebra and a set of free generators of the
-module . We give also a set of defining relations of as an
algebra and a Groebner basis of the corresponding ideal. The proofs are based
on easy computer calculations with standard functions of Maple, the explicit
presentation of in terms of generators and relations, and methods of
representation theory of the general linear group.Comment: 19 page
Gr\"obner-Shirshov bases for Lie algebras over a commutative algebra
In this paper we establish a Gr\"{o}bner-Shirshov bases theory for Lie
algebras over commutative rings. As applications we give some new examples of
special Lie algebras (those embeddable in associative algebras over the same
ring) and non-special Lie algebras (following a suggestion of P.M. Cohn (1963)
\cite{Conh}). In particular, Cohn's Lie algebras over the characteristic
are non-special when . We present an algorithm that one can check
for any , whether Cohn's Lie algebras is non-special. Also we prove that any
finitely or countably generated Lie algebra is embeddable in a two-generated
Lie algebra
Groebner bases of ideals invariant under endomorphisms
We introduce the notion of Groebner S-basis of an ideal of the free
associative algebra K over a field K invariant under the action of a
semigroup S of endomorphisms of the algebra. We calculate the Groebner S-bases
of the ideal corresponding to the universal enveloping algebra of the free
nilpotent of class 2 Lie algebra and of the T-ideal generated by the polynomial
identity [x,y,z]=0, with respect to suitable semigroups S. In the latter case,
if |X|>2, the ordinary Groebner basis is infinite and our Groebner S-basis is
finite. We obtain also explicit minimal Groebner bases of these ideals.Comment: 15 page
Finite Gr\"obner--Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids
This paper shows that every Plactic algebra of finite rank admits a finite
Gr\"obner--Shirshov basis. The result is proved by using the combinatorial
properties of Young tableaux to construct a finite complete rewriting system
for the corresponding Plactic monoid, which also yields the corollaries that
Plactic monoids of finite rank have finite derivation type and satisfy the
homological finiteness properties left and right . Also, answering a
question of Zelmanov, we apply this rewriting system and other techniques to
show that Plactic monoids of finite rank are biautomatic.Comment: 16 pages; 3 figures. Minor revision: typos fixed; figures redrawn;
references update
On the Cancellation Rule in the Homogenization
We consider the possible ways of the homogenization of non-graded non-commutative algebra and show that it should be combined with the cancellation rule to get the mathematically adequate correspondence between graded and non-graded algebras
Gröbner Basis Approach to Some Combinatorial Problems
We consider several simple combinatorial problems and discuss different ways to express them using polynomial equations and try to describe the Gröbner basis of the corresponding ideals. The main instruments are complete symmetric polynomials that help to express different conditions in rather compact way