On a construction of the Cliffordean algebra (Q1923222)

Let \(V\) be a vector space over a field \(K\) with characteristic zero, and \(\langle , \rangle\) be a symmetric bilinear form. The Cliffordean algebra \(C(V)\) is a factor algebra \(T(V)/J\), where \(T(V)\) is the tensor algebra of \(V\) and \(J\) is the ideal generated by tensors of the type \({\mathfrak x} \otimes {\mathfrak x} - \langle {\mathfrak x}, {\mathfrak x} \rangle\). Denote by \(\widetilde {\mathfrak p}\) the canonical homomorphism \(T(V) \to C(V)\). The purpose of this paper is to construct a new kind of linear map (or Cliffordean alternation) \(A:T(V) \to T(V)\) and an Grassmann algebra \((\text{Im} (A),.)\) in such a way that the diagram \[ \begin{matrix} && T(V) \\ & \overset \widetilde {\mathfrak p} { \swarrow} && \overset {\widetilde A} {\searrow } \\ C(V) && @<<\widetilde {\mathfrak p} \mid \text{Im} (A)< && \text{Im} (A) \end{matrix} \] will be commutative and \(\widetilde {\mathfrak p} \mid \text{Im}(A)\) will be an isomorphism of algebras.