Clifford and Heisenberg algebras

Abstract

This chapter recalls various results on Clifford algebras and Heisenberg algebras. It first introduces the Clifford algebra of a vector space V equipped with a symmetric bilinear form B and then specializes the construction of the Clifford algebra to the case of V ⊕ V*. Next, the chapter argues that, if (V,ω‎) is a symplectic vector space, then the associated Heisenberg algebra is constructed and then specialized to the case of V ⊕ V*. Hereafter, the chapter considers the combination of the Clifford and Heisenberg algebras for V ⊕ V*, and constructs the complex Λ‎· (V*) ⊗ S· (V*), ̄ƌ) which is the subcomplex of polynomial forms in the de Rham complex. Finally, when V is equipped with a scalar product, this complex is related to a Witten complex over V.