Exponentials of Bivectors and their Symplectic Counterparts

Abstract

Let V be a vector space of finite dimension over the field of real or complex numbers, q a nondegenerate quadratic form on V, φ the symmetric bilinear form such that q(x) = φ(x, x) for all x ∈ V, and Cl(V, q) the Clifford algebra provided with a Clifford multiplication and an exterior multiplication such that xy = x ∧ y + φ(x, y) for all x, y ∈V. First I shall explain that the theorem of Lipschitz about exterior exponentials of bivectors leads to an easy calculation of the Clifford exponential of a bivector u by means of the exterior exponential of another bivector v.