Invariant Subspace - Left Ideals

Left Ideals

If A is an algebra, one can define a left regular representation Φ on A: Φ(a)b = ab is a homomorphism from A to L(A), the algebra of linear transformations on A

The invariant subspaces of Φ are precisely the left ideals of A. A left ideal M of A gives a subrepresentation of A on M.

If M is a left ideal of A. Consider the quotient vector space A/M. The left regular representation Φ on M now descends to a representation Φ' on A/M. If denotes an equivalence class in A/M, Φ'(a) = . The kernel of the representation Φ' is the set {a ∈ A| ab ∈ M for all b}.

The representation Φ' is irreducible if and only if M is a maximal left ideal, since a subspace V ⊂ A/M is an invariant under {Φ'(a)| a ∈ A} if and only if its preimage under the quotient map, V + M, is a left ideal in A.