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.
Read more about this topic: Invariant Subspace
Famous quotes containing the words left and/or ideals:
“Ah! How neatly tied, in these people, is the umbilical cord of morality! Since they left their mothers they have never sinned, have they? They are apostles, they are the descendants of priests; one can only wonder from what source they draw their indignation, and above all how much they have pocketed to do this, and in any case what it has done for them.”
—Antonin Artaud (1896–1948)
“The old ideals are dead as nails—nothing there. It seems to me there remains only this perfect union with a woman—sort of ultimate marriage—and there isn’t anything else.”
—D.H. (David Herbert)