Omni-Lie Algebras
نویسنده
چکیده
Without the factor of 1 2 , this would be the semidirect product Lie algebra for the usual action of gl(n,R) on R. With the factor of 1 2 , the bracket does not satisfy the Jacobi identity. Nevertheless, it does satisfy the Jacobi identity on many subspaces which are closed under the bracket. In fact, we will see that any Lie algebra structure on R is realized on such a subspace. If B is any bilinear operation on R, we define the adjoint operator adB : R n → gl(n,R) by adB(v)(w) = B(v, w), and we denote by FB ⊂ En the graph of this operator. A simple calculation shows:
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