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

✍ Scribed by J.Marshall Osborn; Kaiming Zhao

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 394 KB
Volume: 207
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1998.7494

No coin nor oath required. For personal study only.

✦ Synopsis

The Block algebra L referred to here is the Lie algebra over a field F of Ä Ž . ÄŽ .44 characteristic 0 with basis e N r, s g Z = Z _ 0, 0 and subject to the comr, s w

x Ž . mutation relations e , e s rk y sh e . Let 0, 1 / q g F. The q-form h, k r, s h qr, kqs Ž . Ä Ž . Ž . ÄŽ .44 L q of L is the Lie algebra with basis e q N r, s g Z = Z _ 0, 0 and r, s w Ž . Ž .x Ž r k sh . Ž . subject to the commutation relations e q , e q s q y q e q .

h, k r , s h qr, kqs

We study here a Z = Z-graded Lie algebra A s A with A s 0 and [i, j g Z i, j 0, 0 Ž . ÄŽ .4 dim A F 1 for all i, j g Z = Z _ 0, 0 . In this paper, we show that A is i, j Ž . w Ž . Ž .x necessarily isomorphic to L or LЈ q s L q , L q for some q g F if A satisfies a certain nondegeneracy condition.

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