A Characterization of the Block Lie Algebra and Itsq-Forms in Characteristic 0

Abstract

The Block algebraLreferred to here is the Lie algebra over a fieldFof characteristic 0 with basis {er,s∣(r,s)∈Z×Z\\{(0,0)}} and subject to the commutation relations [eh,k,er,s]=(rk−sh)eh+r,k+s. Let 0,1≠q∈F. Theq-formL(q) ofLis the Lie algebra with basis {er,s(q)∣(r,s)∈Z×Z\\{(0,0)}} and subject to the commutation relations [eh,k(q),er,s(q)]=(qrk−qsh)eh+r,k+s(q). We study here aZ×Z-graded Lie algebra[formula] withA0,0=0 and dimAi,j≤1 for all (i,j)∈Z×Z\\{(0,0)}. In this paper, we show thatAis necessarily isomorphic toLorL′(q)=[L(q),L(q)] for someq∈FifAsatisfies a certain nondegeneracy condition.