On Lie k-Algebras

Abstract

We define the notion of a "Lie k-algebra" to be a ( k + 1)-ary skew-symmetric operation on a bigraded vector space which satisfies a certain relation of degree 2 k + 1. The notion of Lie 1-algebra coincides with the notion of Lie superalgebra. An ordinary Lie algebra is precisely a Lie 1-algebra with odd elements. We show first that the boundary map in the Koszul complex (constructed as the Koszul complex for ordinary Lie algebras) squares to zero. We then show that the 1 nk +1 homogeneous part of the free Lie k-algebra with ( nk + 1) even generators is isomorphic, as an S nk+1 -module, to the cohomology of Π (1) nk +1 , the poset of all partitions of nk + 1 in which every block size is congruent to 1 mod k. This result is analogous to a classical result relating the free Lie algebra with n generators to the cohomology of the partition lattice. We also construct an explicit basis for the 1 nk +1 homogeneous part of the free Lie k-algebra with nk + 1 even generators and for the cohomology of Π (1) nk +1 . Lastly, we compute the Lie k-algebra homology of the free Lie k-algebra.