Lie algebra representation

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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 $V$ together with a collection of operators on $V$ 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

A representation of a Lie algebra ${\mathfrak {g}}$ is a Lie algebra homomorphism

\rho \colon {\mathfrak {g}}\to {\mathfrak {gl}}(V)

from ${\mathfrak {g}}$ to the Lie algebra of endomorphisms on a vector space V (with the commutator as the Lie bracket), sending an element x of ${\mathfrak {g}}$ to an element ρ_x of ${\mathfrak {gl}}(V)$ .

Explicitly, this means that ρ is a linear map that satisfies

\rho _{[x,y]}=[\rho _{x},\rho _{y}]=\rho _{x}\rho _{y}-\rho _{y}\rho _{x}\,

for all x,y in ${\mathfrak {g}}$ . The vector space V, together with the representation ρ, is called a ${\mathfrak {g}}$ -module. (Many authors abuse terminology and refer to V itself as the representation).

The representation $\rho$ is said to be faithful if it is injective.

One can equivalently define a ${\mathfrak {g}}$ -module as a vector space V together with a bilinear map ${\mathfrak {g}}\times V\to V$ such that

[x,y]\cdot v=x\cdot (y\cdot v)-y\cdot (x\cdot v)

for all x,y in ${\mathfrak {g}}$ 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 ${\mathfrak {g}}$ on itself:

{\textrm {ad}}:{\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}}),\quad x\mapsto \operatorname {ad} _{x},\quad \operatorname {ad} _{x}(y)=[x,y].

Indeed, by virtue of the Jacobi identity, $\operatorname {ad}$ 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 ${\mathfrak {g}}$ and ${\mathfrak {h}}$ are the Lie algebras of G and H respectively, then the differential $d_e \phi: \mathfrak g \to \mathfrak h$ 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

\phi :G\to \mathrm {GL} (V)\,

determines a Lie algebra homomorphism

d\phi :{\mathfrak {g}}\to {\mathfrak {gl}}(V)

from ${\mathfrak {g}}$ to the Lie algebra of the general linear group GL(V), i.e. the endomorphism algebra of V.

For example, let $c_{g}(x)=gxg^{-1}$ . Then the differential of $c_{g}:G\to G$ at the identity is an element of $\mathrm {GL} ({\mathfrak {g}})$ . Denoting it by $\operatorname {Ad} (g)$ one obtains a representation $\operatorname {Ad}$ of G on the vector space ${\mathfrak {g}}$ . This is the adjoint representation of G. Applying the preceding, one gets the Lie algebra representation $d\operatorname {Ad}$ . It can be shown that $d_{e}\operatorname {Ad} =\operatorname {ad}$ , the adjoint representation of ${\mathfrak {g}}$ .

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.^[1]

Basic concepts

Homomorphisms

Let ${\mathfrak {g}}$ be a Lie algebra. Let V, W be ${\mathfrak {g}}$ -modules. Then a linear map $f:V\to W$ is a homomorphism of ${\mathfrak {g}}$ -modules if it is ${\mathfrak {g}}$ -equivariant; i.e., $f(x\cdot v)=x\cdot f(v)$ for any $x\in {\mathfrak {g}},v\in V$ . If f is bijective, $V,W$ 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.

Complete reducibility

Let V be a ${\mathfrak {g}}$ -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 ${\mathfrak {g}}$ 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).^[2]

A Lie algebra is said to be reductive if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive.

Invariants

An element v of V is said to be ${\mathfrak {g}}$ -invariant if $x\cdot v=0$ for all $x\in {\mathfrak {g}}$ . The set of all invariant elements is denoted by $V^{\mathfrak {g}}$ . The map $V\mapsto V^{\mathfrak {g}}$ is a left-exact functor.

Basic constructions

Tensor products of representations

If we have two representations of a Lie algebra ${\mathfrak {g}}$ , 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 ${\mathfrak {g}}$ uniquely determined by the assumption that

x\cdot (v_{1}\otimes v_{2})=(x\cdot v_{1})\otimes v_{2}+v_{1}\otimes (x\cdot v_{2}).

for all $v_{1}\in V_{1}$ and $v_{2}\in V_{2}$ .

In the language of homomorphisms, this means that we define $\rho _{1}\otimes \rho _{2}:{\mathfrak {g}}\rightarrow {\mathfrak {gl}}(V_{1}\otimes V_{2})$ by the formula $(\rho _{1}\otimes \rho _{2})(x)=\rho _{1}(x)\otimes I+I\otimes \rho _{2}(x)$ .

Dual representations

Let ${\mathfrak {g}}$ be a Lie algebra and $\rho :{\mathfrak {g}}\rightarrow {\mathfrak {gl}}(V)$ be a representation of ${\mathfrak {g}}$ . Let $V^{*}$ be the dual space, that is, the space of linear functionals on $V$ . Then we can define a representation $\rho ^{*}:{\mathfrak {g}}\rightarrow {\mathfrak {gl}}(V^{*})$ by the formula

\rho ^{*}(x)=-(\rho (x))^{\mathrm {tr} }

where for any operator $A:V\rightarrow V$ , the transpose operator $A^{\mathrm {tr} }:V^{*}\rightarrow V^{*}$ is defined as the "composition with $A$ " operator:

(A^{\mathrm {tr} }\phi )(v)=\phi (Av)

The minus sign in the definition of $\rho ^{*}$ is needed to ensure that $\rho ^{*}$ is actually a representation of ${\mathfrak {g}}$ , in light of the identity $(AB)^{\mathrm {tr} }=B^{\mathrm {tr} }A^{\mathrm {tr} }$ .

Representation on linear maps

Let $V,W$ be ${\mathfrak {g}}$ -modules, ${\mathfrak {g}}$ a Lie algebra. Then $\operatorname {Hom} (V,W)$ becomes a ${\mathfrak {g}}$ -module by setting $(x\cdot f)(v)=xf(v)-f(xv)$ . In particular, $\operatorname {Hom} _{\mathfrak {g}}(V,W)=\operatorname {Hom} (V,W)^{\mathfrak {g}}$ ; that is to say, the ${\mathfrak {g}}$ -module homomorphisms from $V$ to $W$ are simply the elements of $\operatorname {Hom} (V,W)$ that are invariant under the just-defined action of ${\mathfrak {g}}$ on $\operatorname {Hom} (V,W)$ . If we take $W$ to be the base field, we recover the action of ${\mathfrak {g}}$ on $V^{*}$ given in the previous subsection.

Classifying finite-dimensional representations of Lie algebras

There is a beautiful theory classifying the finite-dimensional representations of a semisimple Lie algebra over $\mathbb {C}$ . First, as already noted, every such representation decomposes as a direct sum of irreducible. The finite-dimensional irreducible representations are then described by a "theorem of the highest weight." The theory is described in various textbooks, including those by Fulton and Harris, Hall, and Humphreys listed in the references section.

We now describe the theory in increasing generality, starting with two simple cases that can be done "by hand" and then proceeding to the general result.

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:

X={\begin{pmatrix}0&1\\0&0\end{pmatrix}}\qquad Y={\begin{pmatrix}0&0\\1&0\end{pmatrix}}\qquad H={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}~,

These satisfy the commutation relations

[H,X]=2X,\quad [H,Y]=-2Y,\quad [X,Y]=H

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^[3], or from the fact that sl(2,C) is the complexification of the Lie algebra of the simply connected compact group SU(2).^[4] The irreducible representations $\pi$ , in turn, can be classified^[5] by the largest eigenvalue of $\pi (H)$ , which must be a non-negative integer m. The irreducible representation with largest eigenvalue m has dimension $m+1$ and is spanned by eigenvectors for $\pi (H)$ with eigenvalues $m,m-2,\ldots ,-m+2,-m$ . The operators $\pi (X)$ and $\pi (Y)$ move up and down the chain of eigenvectors, respectively.

The case of sl(3,C)

Example of the weights of a representation of the Lie algebra sl(3,C), with the highest weight circled

The eight-dimensional adjoint representation of sl(3,C), referred to as the "eightfold way" in particle physics

There is a similar theory^[6] 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

H_{1}={\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}},\quad H_{2}={\begin{pmatrix}0&0&0\\0&1&0\\0&0&-1\end{pmatrix}}

together with six other matrices $X_{1},\,X_{2},\,X_{3}$ and $Y_{1},\,Y_{2},\,Y_{3}$ each of which as a 1 in an off-diagonal entry and zeros elsewhere. (The $X_{i}$ 's have a 1 above the diagonal and the $Y_{i}$ 's have a 1 below the diagonal.)

The strategy is then to simultaneously diagonalize $\pi (H_{1})$ and $\pi (H_{2})$ in each irreducible representation $\pi$ . Recall that in the sl(2,C) case, the action of $\pi (X)$ and $\pi (Y)$ raise and lower the eigenvalues of $\pi (H)$ . Similarly, in the sl(3,C) case, the action of $\pi (X_{i})$ and $\pi (Y_{i})$ "raise" and "lower" the eigenvalues of $\pi (H_{1})$ and $\pi (H_{2})$ . The irreducible representations are then classified^[7] by the largest eigenvalues $m_{1}$ and $m_{2}$ of $\pi (H_{1})$ and $\pi (H_{2})$ , respectively, where $m_{1}$ and $m_{2}$ 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 $(m_{1},m_{2})$ 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 $\pi (X)=X$ ) and the dual of the standard representation. Then one takes a tensor product of $m_{1}$ copies of the standard representation and $m_{2}$ copies of the dual of the standard representation, and extracts an irreducible invariant subspace.^[8]

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 $(m_{1},m_{2})$ is given by^[9]

\mathrm {dim} (m_{1},m_{2})={\frac {1}{2}}(m_{1}+1)(m_{2}+1)(m_{1}+m+2)

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 by means of a "theorem of the highest weight.: This is a beautiful, classical theory, described in several textbooks, including (Fulton & Harris 1992), (Hall 2015), and (Humphreys 1972). Let ${\mathfrak {g}}$ be a semisimple Lie algebra and let ${\mathfrak {h}}$ be a Cartan subalgebra of ${\mathfrak {g}}$ , that is, a maximal commutative subalgebra with the property that ad_H is diagonalizable for all H in ${\mathfrak {h}}$ . As an example, we may consider the case where ${\mathfrak {g}}$ is sl(n,C), the algebra of n by n traceless matrices, and ${\mathfrak {h}}$ is the subalgebra of traceless diagonal matrices.^[10] We then let R denote the associated root system. We then choose a base (or system of positive simple roots) $\Delta$ for R.

We now briefly summarize the structures needed to state the theorem of the highest weight; more details can be found here. We choose an inner product on ${\mathfrak {h}}$ that is invariant under the action of the Weyl group of R, which we use to identify ${\mathfrak {h}}$ with its dual space. If $(\pi ,V)$ is a representation of ${\mathfrak {g}}$ , we define a weight of V to be a an element $\lambda$ in ${\mathfrak {h}}$ with the property that for some nonzero v in V, we have $\pi (H)v=\langle \lambda ,H\rangle v$ for all H in ${\mathfrak {h}}$ . We then define one weight $\lambda$ to be higher than another weight $\mu$ if $\lambda -\mu$ is expressible as a linear combination of elements of $\Delta$ with non-negative real coefficients. A weight $\mu$ is called a highest weight if $\mu$ is higher than every other weight of $\pi$ . Finally, if $\lambda$ is a weight, we say that $\lambda$ is dominant if it has non-negative inner product with each element of $\Delta$ and we say that $\lambda$ is integral if $2\langle \lambda ,\alpha \rangle /\langle \alpha ,\alpha \rangle$ is an integer for each $\alpha$ 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:^[11]

Every irreducible, finite-dimensional representation of ${\mathfrak {g}}$ 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 ${\mathfrak {g}}$ .

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.^[12] In general, the construction of the representations may be given by using Verma modules.^[13]

Enveloping algebras

To each Lie algebra ${\mathfrak {g}}$ over a field k, one can associate a certain ring called the universal enveloping algebra of ${\mathfrak {g}}$ and denoted $U({\mathfrak {g}})$ . The universal property of the universal enveloping algebra guarantees that every representation of ${\mathfrak {g}}$ gives rise to a representation of $U({\mathfrak {g}})$ . Conversely, the PBW theorem tells us that ${\mathfrak {g}}$ sits inside $U({\mathfrak {g}})$ , so that every representation of $U({\mathfrak {g}})$ can be restricted to ${\mathfrak {g}}$ . Thus, there is a one-to-one correspondence between representations of ${\mathfrak {g}}$ and those of $U({\mathfrak {g}})$ .

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.^[14]

The construction of $U({\mathfrak {g}})$ is as follows.^[15] Let T be the tensor algebra of the vector space ${\mathfrak {g}}$ . Thus, by definition, $T=\oplus _{n=0}^{\infty }\otimes _{1}^{n}{\mathfrak {g}}$ and the multiplication on it is given by $\otimes$ . Let $U({\mathfrak {g}})$ be the quotient ring of T by the ideal generated by elements of the form

[x,y]-(x\otimes y-y\otimes x)

There is a natural linear map from ${\mathfrak {g}}$ into $U({\mathfrak {g}})$ obtained by restricting the quotient map of $T\to U({\mathfrak {g}})$ to degree one piece. The PBW theorem implies that the canonical map is actually injective. Thus, every Lie algebra ${\mathfrak {g}}$ can be embedded into an associative algebra $A=U({\mathfrak {g}})$ in such a way that the bracket on ${\mathfrak {g}}$ is given by $[x,y]=xy-yx$ in $A$ .

If ${\mathfrak {g}}$ is abelian, then $U({\mathfrak {g}})$ is the symmetric algebra of the vector space ${\mathfrak {g}}$ .

Since ${\mathfrak {g}}$ is a module over itself via adjoint representation, the enveloping algebra $U({\mathfrak {g}})$ becomes a ${\mathfrak {g}}$ -module by extending the adjoint representation. But one can also use the left and right regular representation to make the enveloping algebra a ${\mathfrak {g}}$ -module; namely, with the notation $l_{x}(y)=xy,x\in {\mathfrak {g}},y\in U({\mathfrak {g}})$ , the mapping $x\mapsto l_{x}$ defines a representation of ${\mathfrak {g}}$ on $U({\mathfrak {g}})$ . The right regular representation is defined similarly.

Induced representation

Let ${\mathfrak {g}}$ be a finite-dimensional Lie algebra over a field of characteristic zero and ${\mathfrak {h}}\subset {\mathfrak {g}}$ a subalgebra. $U({\mathfrak {h}})$ acts on $U({\mathfrak {g}})$ from the right and thus, for any ${\mathfrak {h}}$ -module W, one can form the left $U({\mathfrak {g}})$ -module $U({\mathfrak {g}})\otimes _{U({\mathfrak {h}})}W$ . It is a ${\mathfrak {g}}$ -module denoted by $\operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}W$ and called the ${\mathfrak {g}}$ -module induced by W. It satisfies (and is in fact characterized by) the universal property: for any ${\mathfrak {g}}$ -module E

\operatorname {Hom} _{\mathfrak {g}}(\operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}W,E)\simeq \operatorname {Hom} _{\mathfrak {h}}(W,\operatorname {Res} _{\mathfrak {h}}^{\mathfrak {g}}E)

Furthermore, $\operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}$ is an exact functor from the category of ${\mathfrak {h}}$ -modules to the category of ${\mathfrak {g}}$ -modules. These uses the fact that $U({\mathfrak {g}})$ is a free right module over $U({\mathfrak {h}})$ . In particular, if $\operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}W$ is simple (resp. absolutely simple), then W is simple (resp. absolutely simple). Here, a ${\mathfrak {g}}$ -module V is absolutely simple if $V\otimes _{k}F$ is simple for any field extension $F/k$ .

The induction is transitive: $\operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}\simeq \operatorname {Ind} _{\mathfrak {h'}}^{\mathfrak {g}}\circ \operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {h'}}$ for any Lie subalgebra ${\mathfrak {h'}}\subset {\mathfrak {g}}$ and any Lie subalgebra ${\mathfrak {h}}\subset {\mathfrak {h}}'$ . The induction commutes with restriction: let ${\mathfrak {h}}\subset {\mathfrak {g}}$ be subalgebra and ${\mathfrak {n}}$ an ideal of ${\mathfrak {g}}$ that is contained in ${\mathfrak {h}}$ . Set ${\mathfrak {g}}_{1}={\mathfrak {g}}/{\mathfrak {n}}$ and ${\mathfrak {h}}_{1}={\mathfrak {h}}/{\mathfrak {n}}$ . Then $\operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}\circ \operatorname {Res} _{\mathfrak {h}}\simeq \operatorname {Res} _{\mathfrak {g}}\circ \operatorname {Ind} _{\mathfrak {h_{1}}}^{\mathfrak {g_{1}}}$ .

Representations of a semisimple Lie algebra

Let ${\mathfrak {g}}$ 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 ${\mathfrak {g}}$ 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.^[16]

(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 $\pi$ is a Hilbert-space representation of, say, a connected real semisimple linear Lie group G, then it has two natural actions: the complexification ${\mathfrak {g}}$ and the connected maximal compact subgroup K. The ${\mathfrak {g}}$ -module structure of $\pi$ allows algebraic especially homological methods to be applied and $K$ -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 super Jacobi 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.

Notes

↑ Hall 2015 Theorem 5.6
↑ Dixmier 1977, Theorem 1.6.3
↑ 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
↑ Hall 2015 Section 9.5
↑ Jacobson 1962
↑ http://mathoverflow.net/questions/64931/why-the-bgg-category-o

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)
Dixmier, J. (1977), Enveloping Algebras, Amsterdam, New York, Oxford: North-Holland, ISBN 0-444-11077-1 .
A. Beilinson and J. Bernstein, "Localisation de g-modules," C. R. Acad. Sci. Paris Sér. I Math., vol. 292, iss. 1, pp. 15–18, 1981.
Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. ISBN 978-0-387-97495-8. MR 1153249.
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
Humphreys, James (1972), Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, 9, Springer
N. Jacobson, Lie algebras, Courier Dover Publications, 1979.
Garrett Birkhoff; Philip M. Whitman (1949). "Representation of Jordan and Lie Algebras" (PDF). Trans. Amer. Math. Soc. 65: 116–136.