Speciality problem for Malcev algebras and Poisson Malcev algebras. (English) Zbl 0971.17019

Costa, Roberto (ed.) et al., Nonassociative algebra and its applications. Proceedings of the fourth international conference, São Paulo, Brazil. New York, NY: Marcel Dekker (ISBN 978-0-8247-0406-3/pbk; 978-1-138-40176-1/hbk; 978-0-429-18767-4/ebook). Lect. Notes Pure Appl. Math. 211, 365-371 (2000).
What are nowadays called Maltsev algebras were introduced by Maltsev in 1955. They generalize Lie algebras and have been studied extensively in recent years. Given any alternative algebra \(A\), the new algebra defined on \(A\) by the commutator \([x,y]=xy-yx\) is a Maltsev algebra, denoted by \(A^-\). But while any Lie algebra over a field can be embedded, by the well-known Poincaré-Birkhoff-Witt theorem, as a subalgebra of \(A^-\) for a suitable associative algebra \(A\), it is not known whether any Maltsev algebra over a field of characteristic \(\neq 2,3\) can be embedded as a subalgebra of \(A^-\) for an alternative algebra \(A\). If this is the case, the Maltsev algebra is said to be special. This is the main open problem for Maltsev algebras and the paper under review presents a very interesting approach to it.
Given a Lie algebra \(L\), its symmetric algebra \(S(L)\) is a Poisson-Lie algebra. The author defines more general Poisson algebras and constructs a “Poisson-Maltsev algebra” \(\widetilde S(M)\) attached to any Maltsev algebra \(M\). By standard arguments one can also construct a universal alternative enveloping algebra \(U(M)\), which is filtered, and its associated graded algebra is \(\text{gr} U(M)\).
It turns out that whether or not \(M\) is special is closely related to the possibility of \(\widetilde S(M)\) and \(\text{gr} U(M)\) admitting a so-called “algebraic quantization deformation”. This allows the author to give some necessary conditions for a Maltsev algebra to be special and, although the known examples of Maltsev algebras satisfy these conditions (which are very difficult to check), there are candidates to be non-special.
For the entire collection see [Zbl 0940.00027].


17D10 Mal’tsev rings and algebras
17B63 Poisson algebras