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Journal of Lie Theory 33 (2023), No. 2, 477--495
Copyright Heldermann Verlag 2023

The Lie Algebra Preserving A Degenerate Bilinear Form

James Waldron
School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, England

Let k be an arbitrary field and d a positive integer. For each degenerate symmetric or antisymmetric bilinear form M on kd 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+t2. Some of our results apply to more general bilinear forms and graded algebras.

Keywords: Graded algebras, graded Lie algebras, biliner forms, derivations.

MSC: 17B05,13N15,17B70.

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