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The Lie algebra preserving a degenerate bilinear form

Lookup NU author(s): Dr James Waldron


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Let k be an arbitrary field and d a positive integer. For each degenerate symmetric or antisymmetric bilinear form M on k^d we determine the structure of the Lie algebra of matrices that preserve M, and of the Lie algebra of matrices that preserve the subspace spanned by M. We show that these Lie algebras are semidirect products of classical Lie algebras and certain representations, and determine their radicals, derived series and semisimple quotients. Our main motivation and application is to determine the structure of the graded Lie algebra of derivations of each commutative or graded commutative algebra with Hilbert polynomial 1 + dt + t^2. Some of our results apply to more general bilinear forms and graded algebras.

Publication metadata

Author(s): Waldron J

Publication type: Article

Publication status: In Press

Journal: Journal of Lie Theory

Year: 2022

Acceptance date: 08/05/2022

ISSN (electronic): 0949-5932

Publisher: Heldermann Verlag