The notion is closely related to that of a representation of a Lie group.
on tangent spaces at the identities is a Lie algebra homomorphism.
[1] In quantum theory, one considers "observables" that are self-adjoint operators on a Hilbert space.
The angular momentum operators, for example, satisfy the commutation relations Thus, the span of these three operators forms a Lie algebra, which is isomorphic to the Lie algebra so(3) of the rotation group SO(3).
is any subspace of the quantum Hilbert space that is invariant under the angular momentum operators,
An understanding of the representation theory of so(3) is of great help in, for example, analyzing Hamiltonians with rotational symmetry, such as the hydrogen atom.
Many other interesting Lie algebras (and their representations) arise in other parts of quantum physics.
Indeed, the history of representation theory is characterized by rich interactions between mathematics and physics.
Similarly, many other constructions from module theory in abstract algebra carry over to this setting: submodule, quotient, subquotient, direct sum, Jordan-Hölder series, etc.
A simple but useful tool in studying irreducible representations is Schur's lemma.
Then V is said to be completely reducible (or semisimple) if it is isomorphic to a direct sum of irreducible representations (cf.
A Lie algebra is said to be reductive if the adjoint representation is semisimple.
In the other direction, the definition of a reductive Lie algebra means that it decomposes as a direct sum of ideals (i.e., invariant subspaces for the adjoint representation) that have no nontrivial sub-ideals.
Some of these ideals will be one-dimensional and the rest are simple Lie algebras.
over a field k, one can associate a certain ring called the universal enveloping algebra of
Specifically, the finite-dimensional irreducible representations are constructed as quotients of Verma modules, and Verma modules are constructed as quotients of the universal enveloping algebra.
be the quotient ring of T by the ideal generated by elements of the form There is a natural linear map from
The PBW theorem implies that the canonical map is actually injective.
But one can also use the left and right regular representation to make the enveloping algebra a
-module induced by W. It satisfies (and is in fact characterized by) the universal property: for any
be a finite-dimensional semisimple Lie algebra over a field of characteristic zero.
(in the solvable or nilpotent case, one studies primitive ideals of the enveloping algebra; cf.
turns out to be too large especially for homological algebra methods to be useful: it was realized that a smaller subcategory category O is a better place for the representation theory in the semisimple case in zero characteristic.
For instance, the category O turned out to be of a right size to formulate the celebrated BGG reciprocity.
[citation needed] One of the most important applications of Lie algebra representations is to the representation theory of real reductive Lie groups.
is a Hilbert-space representation of, say, a connected real semisimple linear Lie group G, then it has two natural actions: the complexification
-module structure allows harmonic analysis to be carried out in a way similar to that on connected compact semisimple Lie groups.
If we have a Lie superalgebra L, then a representation of L on an algebra is a (not necessarily associative) Z2 graded algebra A which is a representation of L as a Z2 graded vector space and in addition, the elements of L acts as derivations/antiderivations on A.
More specifically, if H is a pure element of L and x and y are pure elements of A, Also, if A is unital, then Now, for the case of a representation of a Lie algebra, we simply drop all the gradings and the (−1) to the some power factors.
This is a representation on an algebra: the (anti)derivation property is the superJacobi identity.