The adjoint homology of the free 2-step nilpotent Lie algebra

L. CAGLIERO; P. TIRAO

QUARTERLY JOURNAL OF MATHEMATICS

Oxford University Press,

Año: 2002 vol. 53 p. 125 - 145

0033-5606

Let $V$ be a vector space over $Bbb C$ of dimension $r$. The free 2-step complex Lie algebra of rank $r$ is $Cal L(r)=V oplus wedge ^2V$, where $wedge ^2V$ is the centre of $Cal L(r)$ and for $u,~vin V ~ [u,v]=uwedge v$. The authors investigate the $GL(V)$-structure of $H_*(Cal L(r),Cal L(r))$. They prove (Th. 4.4) that $$H_p(Cal L(r),Cal L(r))equiv rac{H_p(Cal L(r))otimes wedge ^2V} {F_p}oplus rac{H_p(Cal L(r))otimes V}{E_p}$$ as $GL(V)$-modules. Here $E_p$ and $F_{p-1}$ are maximal isomorphic submodules of $H_p(Cal L(r))otimes V$ and $H_{p-1}(Cal L(r)) otimes wedge ^2V$, respectively. The explicit formula for $ dim H_*(Cal L(r),Cal L(r))$ is given.