IJPAM: Volume 53, No. 4 (2009)

UNIVERSAL CENTRAL EXTENSIONS OF COMPLEX SIMPLE
LIE ALGEBRAS EXTENDED OVER ${\mathbb{C}}[SL_{2}]$

Luca Guerrini
Department of Mathematics for Economic and Social Sciences
University of Bologna
5, Viale Filopanti, Bologna, 40126, ITALY
e-mail: guerrini@rimini.unibo.it

Abstract.In this paper, a vector bundle $SL_{2}(\mathbb{C}) {\times }_{B_{2}} \mathbb{C}[B_{2}]$ over $X=SL_{2}/B_{2}$ with fiber $\mathbb{C}[B_{2}]=\mathbb{C}[t_{1},t_{2}^{\pm 1}]$ is constructed, where $\mathbb{C}[SL_{2}]$ is the algebra of regular functions on the linear algebraic group $SL_{2}(\mathbb{C})$ , and $B_{2}$ denotes a Borel subgroup of $SL_{2}(\mathbb{C})$ . As well, an imbedding of $\mathbb{C}[SL_{2}]$ into ${\mathbb{H}^{0}}={H}^{0}\left(X,SL_{2}(\mathbb{C})\right.$ $\left.{\times }_{B_{2}} \mathbb{C}[B_{2}]\right)$ , the algebra of holomorphic sections of the previous bundle, is explicitly given. Moreover, an inclusion of the universal central extension $\mathbf{\widehat{\mathcal{G}}}_{\mathbb{C}[SL_{2}]}$ of the Lie algebra ${\mathbb{C}}[SL_{2}]{\times }_{{\mathbb{C}}\,} {\mathcal{G}}$ , with $\mathcal{G}$ a simple complex finite dimensional Lie algebra, into the universal central extension of the Lie algebra ${\mathbb{H}^{0}}$ is proved.