Editing Invariant subspace (section)

== Left ideals ==

If ''A'' is an [[algebra over a field|algebra]], one can define a [[regular representation|''left regular representation'']] Φ on ''A'': Φ(''a'')''b'' = ''ab'' is a [[algebra homomorphism|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 [[Algebra_over_a_field#Subalgebras_and_ideals|ideal]] of ''A'' then the left regular representation Φ on ''M'' now descends to a representation Φ' on the [[quotient vector space]] ''A''/''M''. If [''b''] denotes an [[equivalence class]] in ''A''/''M'', Φ'(''a'')[''b''] = [''ab'']. The kernel of the representation Φ' is the set {''a'' ∈ ''A'' | ''ab'' ∈ ''M'' for all ''b''}.

The representation Φ' is [[irreducible representation|irreducible]] if and only if ''M'' is a [[maximal ideal|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''.