On Lie \(k\)-algebras. (English) Zbl 0844.17001

Hanlon, Phil; Wachs, Michelle

Adv. Math. 113, No. 2, 206-236 (1995).

A Lie \(k\)-algebra is a bigraded vector space with a \((k + 1)\)-ary skew-symmetric operation which preserves the grading, which is alternating and which satisfies an identity extending the Jacobi identity of Lie superalgebras (the case \(k = 1)\). The symmetric group \(S_n\) acts on the multilinear part of degree \(n\) of the free Lie \(k\)-algebra. On the other hand, \(S_n\) acts on the top homology of the poset of partitions of \(\{1, \dots, n\}\) whose blocks have cardinality \(\equiv 1 \bmod k\). The main result is that when \(n \equiv 1 \bmod k\), these two characters coincide. A basis of the multilinear part is also constructed, extending the basis of left to right bracketings in the free Lie algebra. Furthermore, the authors extend the Koszul complex of Lie algebras, and compute the homology of the evenly generated free Lie \(k\)-algebra.

Reviewer: C.Reutenauer (Montreal)

MSC:

17B01

Identities, free Lie (super)algebras

Keywords:

free Lie \(k\)-algebra; symmetric group; Koszul complex; Lie algebras; homology