On the adjoint homology of 2-step nilpotent Lie algebras.

CAGLIERO, LEANDRO; TIRAO, PAULO

BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY

AUSTRALIAN MATHEMATICS PUBL ASSOC INC

Año: 2005 vol. 71 p. 177 - 182

0004-9727

Let ${germ n}$ be a 2-step nilpotent Lie algebra and $H_{*}({germ n},{germ n})$ the homology spaces of the adjoint representation of ${germ n}$. The main results of the paper under review give a lower bound and an upper bound for $dim H_{*}({germ n},{germ n})$. Thus, Theorem 1.1 in the paper says that $dim H_{*}({germ n},{germ n})ge2^{z+[v/2]}·c$, where $c=dim{germ n}$ if $v$ is even and $c=dim{germ n}+ert v-zert$ if $v$ is odd. Here $z$ is the dimension of the center of ${germ n}$ and $v=dim{germ n}-z$. On the other hand, $dim H_{*}({germ n},{germ n})ledim H_{*}({germ n})·dim{germ n}$, which actually holds for all finite-dimensional nilpotent Lie algebras. The examples of Heisenberg Lie algebras and free 2-step nilpotent Lie algebras are then analyzed, and these examples support the conjecture that $dim H_{*}({germ n},{germ n})$ is asymptotically equivalent to $dim H_{*}({germ n})·dim{germ n}$ when we let $dim{germ n} oinfty$.