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

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 dim Ai, 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.