Lie Superalgebras Based on gln Associated to the Adjoint Representation, and Invariant Geometric Structures Defined on Them

Finite-dimensional real and complex Lie superalgebras whose underlying Lie algebra is gln and whose odd module is gln itself under the adjoint representation are classified up to isomorphism. It is shown that for n ≥ 3 there are one-parameter families of nonisomorphic such Lie superalgebras, plus another set of finitely many different isomorphism classes. For n = 2 there are 10 different isomorphism classes over the real field, and 8 different over the complex numbers. For n = 1 there are 2 different isomorphism classes over either ground field. Representatives on each isomorphism class are given, and their automorphism groups are determined. The question as to which representatives admit ℤ2-graded, ad-invariant geometric structures (of orthogonal or symplectic type) is also addressed, and a precise list of which of such geometric structures can be defined on each isomorphism class is given. In particular, it is shown that Z2-homogeneous, orthogonal, ad-invariant geometric structures must be odd. The case of gl 2 over the real field is further analyzed in order to determine for which of the equivalence classes that admit such a structure, that structure can be induced by an underlying Minkowski metric on the 4-dimensional (nongraded) gl2.

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