Geometric differences between the use of Lie algebras and Lie superalgebras (Q423988)

In this paper, the authors identify exactly to what extent the geometry defined on a given vector space formed as the direct sum of two subspaces \(U\) and \(W\) decides what is the best-suited algebraic structure to use in \(\text{End}(U \bigoplus W)\): either its Lie algebra structure or its Lie superalgebra structure. To do this they consider the linear maps that preserve \(B\), the geometry defined on \(U \bigoplus W\) via \(B = B_U \bigoplus B_W\), where \(B_U\) and \(B_W\) are geometries on \(U\) and \(W\), respectively. It turns out that the Lie algebra structure on \(\text{End}(U \bigoplus W)\) closes with nontrivial maps \(U \mapsto W\) and \(W \mapsto U\) that are consistent with the given geometries on them, exactly when these geometries are of the same type, that is, both orthogonal, or both symplectic or both unitary or both anti-unitary, respectively. On the other hand, the Lie superalgebra requires to combine the geometry-types of \(U\) and \(W\) in such a way that one is orthogonal and the other symplectic, or one is unitary and the other anti-unitary. The authors also address the question of defining a geometry \(B\) on \(U \bigoplus W\) of odd degree and they also determine the Lie algebra and superalgebras structures of the subspace of \(\text{End}(U \bigoplus W)\) that preserve such a \(B\).