Associative Algebra - Representations

Representations

A representation of a unital algebra A is a unital algebra homomorphism ρ: A → End(V) from A to the endomorphism algebra of some vector space (or module) V. The property of ρ being a unital algebra homomorphism means that ρ preserves the multiplicative operation (that is, ρ(xy)=ρ(x)ρ(y) for all x and y in A), and that ρ sends the unity of A to the unity of End(V) (that is, to the identity endomorphism of V).

If A and B are two algebras, and ρ: A → End(V) and τ: B → End(W) are two representations, then it is easy to define a (canonical) representation A ⊗ B → End(V ⊗ W) of the tensor product algebra A ⊗ B on the vector space V ⊗ W. Note, however, that there is no natural way of defining a tensor product of two representations of a single associative algebra in such a way that the result is still a representation of that same algebra (not of its tensor product with itself), without somehow imposing additional conditions. Here, by tensor product of representations, the usual meaning is intended: the result should be a linear representation of the same algebra on the product vector space. Imposing such additional structure typically leads to the idea of a Hopf algebra or a Lie algebra, as demonstrated below.