ON LIE K-ALGEBRAS

We define the notion of a ''Lie k-algebra'' to be a (k + 1)-ary skew-s ymmetric operation on a bigraded vector space which satisfies a certai n relation of degree 2k + 1. The notion of Lie 1-algebra coincides wit h the notion of Lie superalgebra. An ordinary Lie algebra is precisely a Lie 1-algebra with odd elements. We show first that the boundary ma p in the Koszul complex (constructed as the Koszul complex for ordinar y Lie algebras) squares to zero. We then show that the 1(nk+1) homogen eous part of the free Lie k-algebra with (nk + 1) even generators is i somorphic, as an S-nk+1-module, to the cohomology of Pi(nk+1)((1)), th e poset of all partitions of nk + 1 in which every block size is congr uent to 1 mod k. This result is analogous to a classical result relati ng the free Lie algebra with n generators to the cohomology of the par tition lattice. We also construct an explicit basis for the 1(nk+1) ho mogeneous part of the free Lie k-algebra with nk + 1 even generators a nd for the cohomology of Pi(nk+1)((1)). Lastly, we compute the Lie k-a lgebra homology of the free Lie k-algebra. (C) 1995 Academic Press, In c.