Faithful representations of minimal dimension of current Heisenberg Lie algebras

LEANDRO CAGLIERO AND NADINA ROJAS

INTERNATIONAL JOURNAL OF MATHEMATICS

WORLD SCIENTIFIC PUBL CO PTE LTD

Lugar: -; Año: 2009 vol. 20 p. 1347 - 1362

0129-167X

egin{abstract}Given a Lie algebra $g$ over a field of characteristic zero $k$,let $mu(g)=min{dim pi: pi ext{ is a faithful representation of }g}$.Let $h_{m}$ bethe Heisenberg Lie algebra of dimension $2m+1$ over $k$ and let$k[t]$ be the polynomial algebra in one variable. Given$minmathbb{N}$ and $pink[t]$, let $h_{m,p}=h_motimesk[t]/(p)$ be the current Lie algebra associated to $h_m$ and$k[t]/(p)$, where $(p)$ is the principal ideal in $k[t]$ generated by $p$.In this paper we prove that $ mu(h_{m,p}) = m deg p + leftlceil 2sqrt{deg p} ight ceil. $end{abstract}