Lie algebra representation
|Group theory → Lie groups
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.
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 a decisive 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.
- 1 Formal definition
- 2 Examples
- 3 Basic concepts
- 4 Basic constructions
- 5 Enveloping algebras
- 6 Induced representation
- 7 Representations of a semisimple Lie algebra
- 8 (g,K)-module
- 9 Classifying finite-dimensional representations of Lie algebras
- 10 Representation on an algebra
- 11 See also
- 12 Notes
- 13 References
- 14 Further reading
Explicitly, this means that ρ is a linear map that satisfies
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 = ρx (v).
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 φ: 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.
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. Similarly, many other constructions from module theory in abstract algebra carry over to this setting: submodule, quotient, subquotient, direct sum, Jordan-Hölder series, etc.
Let V be a -module. Then V is said to be semisimple or completely reducible if it satisfies the following equivalent conditions: (cf. semisimple module)
- V is a direct sum of simple modules.
- V is the sum of its simple submodules.
- Every submodule of V is a direct summand: for every submodule W of V, there is a complement P such that V = W ⊕ P.
If is a finite-dimensional semisimple Lie algebra over a field of characteristic zero and V is finite-dimensional, then V is semisimple (Weyl's complete reducibility theorem). A Lie algebra is said to be reductive if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive. An element v of V is said to be -invariant if for all . The set of all invariant elements is denoted by . is a left-exact functor.
If we have two representations, with V1 and V2 as their underlying vector spaces and ·[·]1 and ·[·]2 as the representations, then the product of both representations would have V1 ⊗ V2 as the underlying vector space and
If L is a real Lie algebra and ρ: L × V→ V is a complex representation of it, we can construct another representation of L called its dual representation as follows.
Let V∗ be the dual vector space of V. In other words, V∗ is the set of all linear maps from V to C with addition defined over it in the usual linear way, but scalar multiplication defined over it such that for any z in C, ω in V∗ and X in V. This is usually rewritten as a contraction with a sesquilinear form 〈·,·〉. i.e. 〈ω,X〉 is defined to be ω[X].
We define as follows:
- 〈(A)[ω],X〉 + 〈ω, ρA[X]〉 = 0,
for any A in L, ω in V∗ and X in V. This defines uniquely.
Let be -modules, a Lie algebra. Then becomes a -module by setting . In particular, . Since any field becomes a -module with a trivial action, taking W to be the base field, the dual vector space becomes a -module.
To each Lie algebra over a field k, one can associate a certain ring called the universal enveloping algebra of . The construction is universal and consequently (along with the PBW theorem) representations of corresponds in one-to-one with algebra representations of universal enveloping algebra of . The construction 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 . Since is an associative algebra over the field k, it can be turned into a Lie algebra via the commutator (omitting from the notation). There is a canonical morphism of Lie algebras obtained by restricting to degree one piece. The PBW theorem implies that the canonical map is actually injective. Note 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.
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 .
Representations of a semisimple Lie algebra
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 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.
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 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.
Classifying finite-dimensional representations of Lie algebras
The case of sl(2,C)
The Lie algebra sl(2,C) of the special linear group SL(2,C) is the space of 2x2 trace-zero matrices with complex entries. The following elements form a basis:
These satisfy the commutation relations
Every finite-dimensional representation of sl(2,C) decomposes as a direct sum of irreducible. This claim follows from the general result on complete reducibility of semisimple Lie algebras, or from the fact that sl(2,C) is the complexification of the Lie algebra of the simply connected compact group SU(2). The irreducible representations , in turn, can be classified by the largest eigenvalue of , which must be a non-negative integer m. The irreducible representation with largest eigenvalue m has dimension and is spanned by eigenvectors for with eigenvalues . The operators and move up and down the chain of eigenvectors, respectively.
The case of sl(3,C)
There is a similar theory classifying the irreducible representations of sl(3,C). The Lie algebra sl(3,C) is eight dimensional. We may work with a basis consisting of the following two diagonal elements
together with six other matrices and each of which as a 1 in an off-diagonal entry and zeros elsewhere. (The 's have a 1 above the diagonal and the 's have a 1 below the diagonal.)
The strategy is then to simultaneously diagonalize and in each irreducible representation . Recall that in the sl(2,C) case, the action of and raise and lower the eigenvalues of . Similarly, in the sl(3,C) case, the action of and "raise" and "lower" the eigenvalues of and . The irreducible representations are then classified by the largest eigenvalues and of and , respectively, where and are non-negative integers.
Unlike the representations of sl(2,C), the representation of sl(3,C) cannot be described explicitly in general. Thus, it requires an argument to show that every pair actually arises the highest weight of some irreducible representation. This can be done as follows. First, we construct the "fundamental representations", with highest weights (1,0) and (0,1). These are the three-dimensional standard representation (in which ) and the dual of the standard representation. Then one takes a tensor product of copies of the standard representation and copies of the dual of the standard representation, and extracts an irreducible invariant subspace.
Although the representations cannot be described explicitly, there is a lot of useful information describing their structure. For example, the dimension of the irreducible representation with highest weight is given by
There is also a simple pattern to the multiplicities of the various weight spaces.
The case of a general semisimple Lie algebras
Similarly to how semisimple Lie algebras can be classified, the finite-dimensional representations of semisimple Lie algebras can be classified. This is a beautiful, classical theory, described in several textbooks, including (Fulton & Harris 1992), (Hall 2015), and (Humphreys 1972). Let be a semisimple Lie algebra and let be a Cartan subalgebra of , that is, a maximal commutative subalgebra with the property that adH is diagonalizable for all H in . As an example, we may consider the case where is sl(n,C), the algebra of n by n traceless matrices, and is the subalgebra of traceless diagonal matrices. We then let R denote the associated root system. We then choose a base (or system of positive simple roots) for R.
We choose an inner product on that is invariant under the action of the Weyl group of R, which we use to identify with its dual space. If is a representation of , we define a weight of V to be a an element in with the property that for some nonzero v in V, we have for all H in . We then define one weight to be higher than another weight if is expressible as a linear combination of elements of with non-negative real coefficients. A weight is called a highest weight if is higher than every other weight of . Finally, if is a weight, we say that is dominant if it has non-negative inner product with each element of and we say that is integral if is an integer for each in R.
Finite-dimensional representations of a semisimple Lie algebra are completely reducible, so it suffices to classify irreducible (simple) representations. The irreducible representations, in turn, may be classified by the "theorem of the highest weight" as follows:
- Every irreducible, finite-dimensional representation of has a highest weight, and this highest weight is dominant and integral.
- Two irreducible, finite-dimensional representations with the same highest weight are isomorphic.
- Every dominant integral element arises as the highest weight of some irreducible, finite-dimensional representation of .
This classification generalizes the more elementary representation theory of sl(2;C), described above, where the irreducible representations are classified by the largest eigenvalue of the diagonal element H, which is a non-negative integer. The last point of the theorem is the most difficult one. In the case of the Lie algebra sl(3;C), the construction can be done in an elementary way. In general, the construction of the representations may be given by using Verma modules.
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.
- H[xy] = (H[x])y + (−1)xHx(H[y])
Also, if A is unital, then
- H = 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.
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.
- Quillen's lemma - analog of Schur's lemma
- Verma module
- Kazhdan–Lusztig conjectures
- Whitehead's lemma (Lie algebras)
- Hall 2015 Theorem 5.6
- Dixmier 1977, Theorem 1.6.3
- Jacobson 1962
- Hall 2015 Section 10.3
- Hall 2015 Theorems 4.28 and 5.6
- Hall 2015 Section 4.6
- Hall 2015 Chapter 6
- Hall 2015 Theorem 6.7
- Hall 2015 Proposition 6.17
- Hall 2015 Theorem 6.27
- Hall 2015 Section 7.7.1
- Hall 2015 Theorems 9.4 and 9.5
- Hall 2015 Proposition 6.17
- Hall 2015 Sections 9.5-9.7
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- D. Gaitsgory, Geometric Representation theory, Math 267y, Fall 2005
- Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer
- Ryoshi Hotta, Kiyoshi Takeuchi, Toshiyuki Tanisaki, D-modules, perverse sheaves, and representation theory; translated by Kiyoshi Takeuch
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- Garrett Birkhoff; Philip M. Whitman (1949). "Representation of Jordan and Lie Algebras" (PDF). Trans. Amer. Math. Soc. 65: 116–136.
- Ben-Zvi, David; Nadler, David (2012). "Beilinson-Bernstein localization over the Harish-Chandra center". arXiv: .