The Jacobian Module of a Lie Algebra

Abstract

ABSTRACT. There is a natural way to associate to the commuting variety C(A)

Orthogonal Bundles and Skew-Hamiltonian Matrices

“…The reasoning is somewhat similar to what is done in the unstructured case treated in [Hul80]. There one reduces to prove the irreducibility of pairs of n × n matrices (A, B) whose commutator [A, B] has constant rank r. This result also has a symmetric analogue proved in [Bas00,BPV90], which is used in [Ott07] to show irreducibility in the symplectic case.…”

A note on the variety of pairs of matrices whose product is symmetric

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We give different short proofs for a result proved by C. Mueller in [9]: Over an algebraically closed field pairs of n × n matrices whose product is symmetric form an irreducible, reduced, and complete intersection variety of dimension (3n 2 + n)/2. Our work is connected to the work of Brennan, Pinto, and Vasconcelos in [2].

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“…Working from the fact that V n is irreducible, we will use a technique introduced in [2] to show that the variety of pairs of matrices whose product is symmetric is reduced and a complete intersection. This technique relies on the notion of Jacobian module and some other related results.…”

“…In char k = 2 we can give another proof for the fact that the variety of pairs of n × n matrices whose product is symmetric is a complete intersection. This proof is based on results from [2] which we summarize in the following theorem:…”

“…In [2] a special commuting variety is studied, the variety of commuting pairs of symmetric matrices. Partially inspired by this work, we consider the ideal J generated by entries of XY −Y t X t in the polynomial ring R = k[x 11 , .…”