Gröbner-Shirshov bases for Lie algebras over a commutative algebra

Details Gröbner-Shirshov bases for Lie algebras over a commutative algebra Gröbner-Shirshov bases for Lie algebras over a commutative algebra

2010-06-16

arxiv.org/abs/1006.3217v3

Journal of Algebra 337 (2011) 82--102

Mathematics

Rings and Algebras

Scientific paper

10.1016/j.jalgebra.2011.04.020

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 $p$ are non-special when $p=2,\ 3,\ 5$. We present an algorithm that one can check for any $p$, 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.

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