Lie algebra representation

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 together with a collection of operators on 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 be a Lie algebra and let be a vector space. We let denote the space of endomorphisms of , that is, the space of all linear maps of to itself. We make into a Lie algebra with bracket given by the commutator: for all ρ,σ in . Then a representation of on is a Lie algebra homomorphism
.
Explicitly, this means that should be a linear map and it should satisfy
for all X, Y in . The vector space V, together with the representation ρ, is called a -module. (Many authors abuse terminology and refer to V itself as the representation).
The representation is said to be faithful if it is injective.
One can equivalently define a -module as a vector space V together with a bilinear map such that
for all X,Y in 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 on itself:
Indeed, by virtue of the Jacobi identity, is a Lie algebra homomorphism.
Infinitesimal Lie group representations
A Lie algebra representation also arises in nature. If : G → H is a homomorphism of (real or complex) Lie groups, and and are the Lie algebras of G and H respectively, then the differential 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 to the Lie algebra of the general linear group GL(V), i.e. the endomorphism algebra of V.
For example, let . Then the differential of at the identity is an element of . Denoting it by one obtains a representation of G on the vector space . This is the adjoint representation of G. Applying the preceding, one gets the Lie algebra representation . It can be shown that , the adjoint representation of .
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 is any subspace of the quantum Hilbert space that is invariant under the angular momentum operators, 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 of a Lie algebra , we say that a subspace of is invariant if for all and . A nonzero representation is said to be irreducible if the only invariant subspaces are itself and the zero space . The term simple module is also used for an irreducible representation.
Homomorphisms
Let be a Lie algebra. Let V, W be -modules. Then a linear map is a homomorphism of -modules if it is -equivariant; i.e., for any . If f is bijective, 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:
If V, W are irreducible -modules and is a homomorphism, then is either zero or an isomorphism.
If V is an irreducible -module over an algebraically closed field and is a homomorphism, then is a scalar multiple of the identity.
Complete reducibility
Let V be a representation of a Lie algebra . 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 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 is reductive, since every representation of 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 -invariant if for all . The set of all invariant elements is denoted by .
Basic constructions
Tensor products of representations
If we have two representations of a Lie algebra , with V1 and V2 as their underlying vector spaces, then the tensor product of the representations would have V1 ⊗ V2 as the underlying vector space, with the action of uniquely determined by the assumption that
for all and .
In the language of homomorphisms, this means that we define by the formula
. This is called the Kronecker sum of and , defined in Matrix addition#Kronecker_sum and Kronecker product#Properties, and more specifically in Tensor product of representations.
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 acts on the first factor in the tensor product and 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, might, for example, be the orbital angular momentum while is the spin angular momentum.
Dual representations
Let be a Lie algebra and be a representation of . Let be the dual space, that is, the space of linear functionals on . Then we can define a representation by the formula
where for any operator , the transpose operator is defined as the "composition with " operator:
The minus sign in the definition of is needed to ensure that is actually a representation of , in light of the identity
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 be -modules, a Lie algebra. Then becomes a -module by setting . In particular, ; that is to say, the -module homomorphisms from to are simply the elements of that are invariant under the just-defined action of on . If we take to be the base field, we recover the action of on given in the previous subsection.
Representation theory of semisimple Lie algebras
See Representation theory of semisimple Lie algebras.
Enveloping algebras
To each Lie algebra over a field k, one can associate a certain ring called the universal enveloping algebra of and denoted . The universal property of the universal enveloping algebra guarantees that every representation of gives rise to a representation of . Conversely, the PBW theorem tells us that sits inside , so that every representation of can be restricted to . Thus, there is a one-to-one correspondence between representations of and those of .
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 is as follows. Let T be the tensor algebra of the vector space . Thus, by definition, and the multiplication on it is given by . Let be the quotient ring of T by the ideal generated by elements of the form
.
There is a natural linear map from into obtained by restricting the quotient map of to degree one piece. The PBW theorem implies that the canonical map is actually injective. Thus, every Lie algebra can be embedded into an associative algebra in such a way that the bracket on is given by in .
If is abelian, then is the symmetric algebra of the vector space .
Since is a module over itself via adjoint representation, the enveloping algebra becomes a -module by extending the adjoint representation. But one can also use the left and right regular representation to make the enveloping algebra a -module; namely, with the notation , the mapping defines a representation of on . The right regular representation is defined similarly.
Induced representation
Let be a finite-dimensional Lie algebra over a field of characteristic zero and a subalgebra. acts on from the right and thus, for any -module W, one can form the left -module . It is a -module denoted by and called the -module induced by W. It satisfies (and is in fact characterized by) the universal property: for any -module E
.
Furthermore, is an exact functor from the category of -modules to the category of -modules. These uses the fact that is a free right module over . In particular, if is simple (resp. absolutely simple), then W is simple (resp. absolutely simple). Here, a -module V is absolutely simple if is simple for any field extension .
The induction is transitive:
for any Lie subalgebra and any Lie subalgebra . The induction commutes with restriction: let be subalgebra and an ideal of that is contained in . Set and . Then .
Infinite-dimensional representations and "category O"
Let 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 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 groups. The application is based on the idea that if is a Hilbert-space representation of, say, a connected real semisimple linear Lie group G, then it has two natural actions: the complexification and the connected maximal compact subgroup K. The -module structure of allows algebraic especially homological methods to be applied and -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,
H[xy] = (H[x])y + (−1)xHx(H[y])
Also, if A is unital, then
H[1] = 0
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.
See also
Representation of a Lie group
Weight (representation theory)
Weyl's theorem on complete reducibility
Root system
Weyl character formula
Representation theory of a connected compact Lie group
Whitehead's lemma (Lie algebras)
Kazhdan–Lusztig conjectures
Quillen's lemma - analog of Schur's lemma
Notes
References
Bernstein I.N., Gelfand I.M., Gelfand S.I., "Structure of Representations that are generated by vectors of highest weight," Functional. Anal. Appl. 5 (1971)
.
A. Beilinson and J. Bernstein, "Localisation de g-modules," Comptes Rendus de l'Académie des Sciences, Série I, vol. 292, iss. 1, pp. 15–18, 1981.
D. Gaitsgory, Geometric Representation theory, Math 267y, Fall 2005
Ryoshi Hotta, Kiyoshi Takeuchi, Toshiyuki Tanisaki, D-modules, perverse sheaves, and representation theory; translated by Kiyoshi Takeuch
N. Jacobson, Lie algebras, Courier Dover Publications, 1979.
(elementary treatment for SL(2,C))
Further reading