Lie algebra representation

In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space formula_1 together with a collection of operators on formula_1 satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.
The notion is closely related to that of a representation of a Lie group. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra.
In the study of representations of a Lie algebra, a particular ring, called the universal enveloping algebra, associated with the Lie algebra plays an important role. The universality of this ring says that the category of representations of a Lie algebra is the same as the category of modules over its enveloping algebra.
Formal definition.
Let formula_3 be a Lie algebra and let formula_1 be a vector space. We let formula_5 denote the space of endomorphisms of formula_1, that is, the space of all linear maps of formula_1 to itself. We make formula_5 into a Lie algebra with bracket given by the commutator: formula_9 for all "ρ,σ" in formula_5. Then a representation of formula_3 on formula_1 is a Lie algebra homomorphism
Explicitly, this means that formula_14 should be a linear map and it should satisfy
for all "X, Y" in formula_3. The vector space "V", together with the representation "ρ", is called a formula_3-module. (Many authors abuse terminology and refer to "V" itself as the representation).
The representation formula_14 is said to be faithful if it is injective.
One can equivalently define a formula_3-module as a vector space "V" together with a bilinear map formula_20 such that
for all "X,Y" in formula_3 and "v" in "V". This is related to the previous definition by setting "X" ⋅ "v" = "ρ"("X")("v").
Examples.
Adjoint representations.
The most basic example of a Lie algebra representation is the adjoint representation of a Lie algebra formula_23 on itself:
Indeed, by virtue of the Jacobi identity, formula_25 is a Lie algebra homomorphism.
Infinitesimal Lie group representations.
A Lie algebra representation also arises in nature. If formula_26: "G" → "H" is a homomorphism of (real or complex) Lie groups, and formula_3 and formula_28 are the Lie algebras of "G" and "H" respectively, then the differential formula_29 on tangent spaces at the identities is a Lie algebra homomorphism. In particular, for a finite-dimensional vector space "V", a representation of Lie groups
determines a Lie algebra homomorphism
from formula_3 to the Lie algebra of the general linear group GL("V"), i.e. the endomorphism algebra of "V".
For example, let formula_33. Then the differential of formula_34 at the identity is an element of formula_35. Denoting it by formula_36 one obtains a representation formula_37 of "G" on the vector space formula_23. This is the adjoint representation of "G". Applying the preceding, one gets the Lie algebra representation formula_39. It can be shown that formula_40, the adjoint representation of formula_3.
A partial converse to this statement says that every representation of a finite-dimensional (real or complex) Lie algebra lifts to a unique representation of the associated simply connected Lie group, so that representations of simply-connected Lie groups are in one-to-one correspondence with representations of their Lie algebras.
In quantum physics.
In quantum theory, one considers "observables" that are self-adjoint operators on a Hilbert space. The commutation relations among these operators are then an important tool. 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). Then if formula_1 is any subspace of the quantum Hilbert space that is invariant under the angular momentum operators, formula_1 will constitute a representation of the Lie algebra so(3). 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.
Basic concepts.
Invariant subspaces and irreducibility.
Given a representation formula_45 of a Lie algebra formula_23, we say that a subspace formula_47 of formula_1 is invariant if formula_49 for all formula_50 and formula_51. A nonzero representation is said to be irreducible if the only invariant subspaces are formula_1 itself and the zero space formula_53. The term "simple module" is also used for an irreducible representation.
Homomorphisms.
Let formula_23 be a Lie algebra. Let "V", "W" be formula_23-modules. Then a linear map formula_56 is a homomorphism of formula_23-modules if it is formula_23-equivariant; i.e., formula_59 for any formula_60. If "f" is bijective, formula_61 are said to be equivalent. Such maps are also referred to as intertwining maps or morphisms.
Similarly, many other constructions from module theory in abstract algebra carry over to this setting: submodule, quotient, subquotient, direct sum, Jordan-Hölder series, etc.
Schur's lemma.
A simple but useful tool in studying irreducible representations is Schur's lemma. It has two parts:
Complete reducibility.
Let "V" be a representation of a Lie algebra formula_23. Then "V" is said to be completely reducible (or semisimple) if it is isomorphic to a direct sum of irreducible representations (cf. semisimple module). If "V" is finite-dimensional, then "V" is completely reducible if and only if every invariant subspace of "V" has an invariant complement. (That is, if "W" is an invariant subspace, then there is another invariant subspace "P" such that "V" is the direct sum of "W" and "P".)
If formula_23 is a finite-dimensional semisimple Lie algebra over a field of characteristic zero and "V" is finite-dimensional, then "V" is semisimple; this is Weyl's complete reducibility theorem. Thus, for semisimple Lie algebras, a classification of irreducible (i.e. simple) representations leads immediately to classification of all representations. For other Lie algebra, which do not have this special property, classifying the irreducible representations may not help much in classifying general representations.
A Lie algebra is said to be reductive if the adjoint representation is semisimple. Certainly, every (finite-dimensional) semisimple Lie algebra formula_3 is reductive, since "every" representation of formula_3 is completely reducible, as we have just noted. 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. Thus, a reductive Lie algebra is a direct sum of a commutative algebra and a semisimple algebra.
Invariants.
An element "v" of "V" is said to be formula_23-invariant if formula_73 for all formula_74. The set of all invariant elements is denoted by formula_75.
Basic constructions.
Tensor products of representations.
If we have two representations of a Lie algebra formula_23, with "V"1 and "V"2 as their underlying vector spaces, then the tensor product of the representations would have "V"1 ⊗ "V"2 as the underlying vector space, with the action of formula_23 uniquely determined by the assumption that
for all formula_79 and formula_80.
In the language of homomorphisms, this means that we define formula_81 by the formula 
In the physics literature, the tensor product with the identity operator is often suppressed in the notation, with the formula written as
where it is understood that formula_84 acts on the first factor in the tensor product and formula_85 acts on the second factor in the tensor product. In the context of representations of the Lie algebra su(2), the tensor product of representations goes under the name "addition of angular momentum." In this context, formula_86 might, for example, be the orbital angular momentum while formula_87 is the spin angular momentum.
Dual representations.
Let formula_23 be a Lie algebra and formula_89 be a representation of formula_23. Let formula_91 be the dual space, that is, the space of linear functionals on formula_1. Then we can define a representation formula_93 by the formula
where for any operator formula_95, the transpose operator formula_96 is defined as the "composition with formula_97" operator:
The minus sign in the definition of formula_99 is needed to ensure that formula_99 is actually a representation of formula_23, in light of the identity formula_102
If we work in a basis, then the transpose in the above definition can be interpreted as the ordinary matrix transpose.
Representation on linear maps.
Let formula_61 be formula_23-modules, formula_23 a Lie algebra. Then formula_106 becomes a formula_23-module by setting formula_108. In particular, formula_109; that is to say, the formula_23-module homomorphisms from formula_1 to formula_47 are simply the elements of formula_106 that are invariant under the just-defined action of formula_23 on formula_106. If we take formula_47 to be the base field, we recover the action of formula_23 on formula_91 given in the previous subsection.
Representation theory of semisimple Lie algebras.
See Representation theory of semisimple Lie algebras.
Enveloping algebras.
To each Lie algebra formula_23 over a field "k", one can associate a certain ring called the universal enveloping algebra of formula_23 and denoted formula_121. The universal property of the universal enveloping algebra guarantees that every representation of formula_23 gives rise to a representation of formula_121. Conversely, the PBW theorem tells us that formula_23 sits inside formula_121, so that every representation of formula_121 can be restricted to formula_23. Thus, there is a one-to-one correspondence between representations of formula_23 and those of formula_121.
The universal enveloping algebra plays an important role in the representation theory of semisimple Lie algebras, described above. 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.
The construction of formula_121 is as follows. Let "T" be the tensor algebra of the vector space formula_23. Thus, by definition, formula_132 and the multiplication on it is given by formula_133. Let formula_121 be the quotient ring of "T" by the ideal generated by elements of the form 
There is a natural linear map from formula_23 into formula_121 obtained by restricting the quotient map of formula_138 to degree one piece. The PBW theorem implies that the canonical map is actually injective. Thus, every Lie algebra formula_23 can be embedded into an associative algebra formula_140in such a way that the bracket on formula_23 is given by formula_142 in formula_97.
If formula_23 is abelian, then formula_121 is the symmetric algebra of the vector space formula_23.
Since formula_23 is a module over itself via adjoint representation, the enveloping algebra formula_121 becomes a formula_23-module by extending the adjoint representation. But one can also use the left and right regular representation to make the enveloping algebra a formula_23-module; namely, with the notation formula_151, the mapping formula_152 defines a representation of formula_23 on formula_121. The right regular representation is defined similarly.
Induced representation.
Let formula_23 be a finite-dimensional Lie algebra over a field of characteristic zero and formula_156 a subalgebra. formula_157 acts on formula_121 from the right and thus, for any formula_159-module "W", one can form the left formula_121-module formula_161. It is a formula_23-module denoted by formula_163 and called the formula_23-module induced by "W". It satisfies (and is in fact characterized by) the universal property: for any formula_23-module "E"
Furthermore, formula_167 is an exact functor from the category of formula_159-modules to the category of formula_23-modules. These uses the fact that formula_121 is a free right module over formula_157. In particular, if formula_163 is simple (resp. absolutely simple), then "W" is simple (resp. absolutely simple). Here, a formula_23-module "V" is absolutely simple if formula_174 is simple for any field extension formula_175.
The induction is transitive: formula_176
for any Lie subalgebra formula_177 and any Lie subalgebra formula_178. The induction commutes with restriction: let formula_156 be subalgebra and formula_180 an ideal of formula_23 that is contained in formula_159. Set formula_183 and formula_184. Then formula_185.
Infinite-dimensional representations and "category O".
Let formula_23 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. Dixmier for the definitive account.)
The category of (possibly infinite-dimensional) modules over formula_23 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.
(g,K)-module.
One of the most important applications of Lie algebra representations is to the representation theory of real reductive Lie group. The application is based on the idea that if formula_188 is a Hilbert-space representation of, say, a connected real semisimple linear Lie group "G", then it has two natural actions: the complexification formula_23 and the connected maximal compact subgroup "K". The formula_23-module structure of formula_188 allows algebraic especially homological methods to be applied and formula_192-module structure allows harmonic analysis to be carried out in a way similar to that on connected compact semisimple Lie groups.
Representation on an algebra.
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.
A Lie (super)algebra is an algebra and it has an adjoint representation of itself. This is a representation on an algebra: the (anti)derivation property is the superJacobi identity.
If a vector space is both an associative algebra and a Lie algebra and the adjoint representation of the Lie algebra on itself is a representation on an algebra (i.e., acts by derivations on the associative algebra structure), then it is a Poisson algebra. The analogous observation for Lie superalgebras gives the notion of a Poisson superalgebra.