Exterior Algebra - The Alternating Tensor Algebra

The Alternating Tensor Algebra

If K is a field of characteristic 0, then the exterior algebra of a vector space V can be canonically identified with the vector subspace of T(V) consisting of antisymmetric tensors. Recall that the exterior algebra is the quotient of T(V) by the ideal I generated by x ⊗ x.

Let Tr(V) be the space of homogeneous tensors of degree r. This is spanned by decomposable tensors

The antisymmetrization (or sometimes the skew-symmetrization) of a decomposable tensor is defined by

where the sum is taken over the symmetric group of permutations on the symbols {1,...,r}. This extends by linearity and homogeneity to an operation, also denoted by Alt, on the full tensor algebra T(V). The image Alt(T(V)) is the alternating tensor algebra, denoted A(V). This is a vector subspace of T(V), and it inherits the structure of a graded vector space from that on T(V). It carries an associative graded product defined by

Although this product differs from the tensor product, the kernel of Alt is precisely the ideal I (again, assuming that K has characteristic 0), and there is a canonical isomorphism