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:
“The names of those who in their lives fought for life,
Who wore at their hearts the fire’s centre.
Born of the sun they traveled a short while towards the sun,
And left the vivid air signed with their honour.”
—Stephen Spender (1909–1995)
“We want our children to become warm, decent human beings who reach out generously to those in need. We hope they find values and ideals to give their lives purpose so they contribute to the world and make it a better place because they have lived in it. Intelligence, success, and high achievement are worthy goals, but they mean nothing if our children are not basically kind and loving people.”
—Neil Kurshan (20th century)