15 research outputs found
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
Gröbner–Shirshov basis of the Adyan extension of the Novikov group
AbstractThe goal of this paper is to give a comparatively short and simple analysis of the Adyan origional group constraction (S.I. Adyan, Unsolvability of some algorithmic problems in the theory of groups, Trudy MMO 6 (1957) 231–298)
Composition-Diamond Lemma for Tensor Product of Free Algebras
In this paper, we establish Composition-Diamond lemma for tensor product of two free algebras over a field. As an application, we
construct a Groebner-Shirshov basis in by lifting a
Groebner-Shirshov basis in , where is a commutative
algebra.Comment: 19 page
Constructions of free commutative integro-differential algebras
In this survey, we outline two recent constructions of free commutative
integro-differential algebras. They are based on the construction of free
commutative Rota-Baxter algebras by mixable shuffles. The first is by
evaluations. The second is by the method of Gr\"obner-Shirshov bases.Comment: arXiv admin note: substantial text overlap with arXiv:1302.004
Lyndon-Shirshov basis and anti-commutative algebras
Chen, Fox, Lyndon 1958 \cite{CFL58} and Shirshov 1958 \cite{Sh58} introduced
non-associative Lyndon-Shirshov words and proved that they form a linear basis
of a free Lie algebra, independently. In this paper we give another approach to
definition of Lyndon-Shirshov basis, i.e., we find an anti-commutative
Gr\"{o}bner-Shirshov basis of a free Lie algebra such that is the
set of all non-associative Lyndon-Shirshov words, where is the set of
all monomials of , a basis of the free anti-commutative algebra on ,
not containing maximal monomials of polynomials from . Following from
Shirshov's anti-commutative Gr\"{o}bner-Shirshov bases theory \cite{S62a2}, the
set is a linear basis of a free Lie algebra
Geodesic rewriting systems and pregroups
In this paper we study rewriting systems for groups and monoids, focusing on
situations where finite convergent systems may be difficult to find or do not
exist. We consider systems which have no length increasing rules and are
confluent and then systems in which the length reducing rules lead to
geodesics. Combining these properties we arrive at our main object of study
which we call geodesically perfect rewriting systems. We show that these are
well-behaved and convenient to use, and give several examples of classes of
groups for which they can be constructed from natural presentations. We
describe a Knuth-Bendix completion process to construct such systems, show how
they may be found with the help of Stallings' pregroups and conversely may be
used to construct such pregroups.Comment: 44 pages, to appear in "Combinatorial and Geometric Group Theory,
Dortmund and Carleton Conferences". Series: Trends in Mathematics.
Bogopolski, O.; Bumagin, I.; Kharlampovich, O.; Ventura, E. (Eds.) 2009,
Approx. 350 p., Hardcover. ISBN: 978-3-7643-9910-8 Birkhause