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 $V$ 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

Let ${\mathfrak {g}}$ be a Lie algebra and let $V$ be a vector space. We let ${\mathfrak {gl}}(V)$ denote the space of endomorphisms of $V$ , that is, the space of all linear maps of $V$ to itself. We make ${\mathfrak {gl}}(V)$ into a Lie algebra with bracket given by the commutator: $[X,Y]=XY-YX$ . Then a representation of ${\mathfrak {g}}$ on $V$ is a Lie algebra homomorphism

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

.

Explicitly, this means that $\rho$ should be a linear map and it should satisfy

\rho ([X,Y])=\rho (X)\rho (Y)-\rho (Y)\rho (X)

for all $X$ and $Y$ 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]

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

[L_{x},L_{y}]=i\hbar L_{z},\;\;[L_{y},L_{z}]=i\hbar L_{x},\;\;[L_{z},L_{x}]=i\hbar L_{y},

.

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).^[2] Then if $V$ is any subspace of the quantum Hilbert space that is invariant under the angular momentum operators, $V$ 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 $\rho :{\mathfrak {g}}\rightarrow \mathrm {End} (V)$ of a Lie algebra ${\mathfrak {g}}$ , we say that a subspace $W$ of $V$ is invariant if $\rho (X)w\in W$ for all $w\in W$ and $X\in {\mathfrak {g}}$ . A nonzero representation is said to be irreducible if the only invariant subspaces are $V$ itself and the zero space $\{0\}$ . The term simple module is also used for an irreducible representation.

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.

Schur's lemma

A simple but useful tool in studying irreducible representations is Schur's lemma. It has two parts:^[3]

If V, W are irreducible ${\mathfrak {g}}$ -modules and $f:V\to W$ is a homomorphism, then $f$ is either zero or an isomorphism.
If V is an irreducible ${\mathfrak {g}}$ -module over an algebraically closed field and $f:V\to V$ is a homomorphism, then $f$ is a scalar multiple of the identity.

Complete reducibility

Let V be a representation of a Lie algebra ${\mathfrak {g}}$ . 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 ${\mathfrak {g}}$ 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.^[4] 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 ${\mathfrak {g}}$ is reductive, since every representation of ${\mathfrak {g}}$ 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 ${\mathfrak {g}}$ -invariant if $x\cdot v=0$ for all $x\in {\mathfrak {g}}$ . The set of all invariant elements is denoted by $V^{\mathfrak {g}}$ .

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)

.^[5]

In the physics literature, the tensor product with the identity operator is often suppressed in the notation, with the formula written as

(\rho _{1}\otimes \rho _{2})(X)=\rho _{1}(X)+\rho _{2}(X)

,

where it is understood that $\rho _{1}(x)$ acts on the first factor in the tensor product and $\rho _{2}(x)$ 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, $\rho _{1}(X)$ might, for example, be the orbital angular momentum while $\rho _{2}(X)$ is the spin angular momentum.

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 $(AB)^{\mathrm {tr} }=B^{\mathrm {tr} }A^{\mathrm {tr} }$ .

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 $V,W$ 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 $\mathbb {C}$ . The finite-dimensional irreducible representations are described by a theorem of the highest weight. The theory is described in various textbooks, including Fulton & Harris (1991), Hall (2015), and Humphreys (1972).

Following an overview, the theory is described in increasing generality, starting with two simple cases that can be done "by hand" and then proceeding to the general result. The emphasis here is on the representation theory; for the geometric structures involving root systems needed to define the term "dominant integral element," follow the above link on weights in representation theory.

Overview

Classification of the finite-dimensional irreducible representations of a semisimple Lie algebra ${\mathfrak {g}}$ over $\mathbb {R}$ or $\mathbb {C}$ generally consists of two steps. The first step amounts to analysis of hypothesized representations resulting in a tentative classification. The second step is actual realization of these representations.

A real Lie algebra is usually complexified enabling analysis in an algebraically closed field. Working over the complex numbers in addition admits nicer bases. The following theorem applies: A real-linear finite-dimensional representations of a real Lie algebra extends to a complex-linear representations of its complexification. The real-linear representation is irreducible if and only if the corresponding complex-linear representation is irreducible.^[6] Moreover, a complex semisimple Lie algebra has the complete reducibility property. This means that every finite-dimensional representation decomposes as a direct sum of irreducible representations.

Conclusion: Classification amounts to studying irreducible complex linear representations of the (complexified) Lie algebra.

Classification: Step One

The first step is to hypothesize the existence of irreducible representations. That is to say, one hypothesizes that one has an irreducible representation $\pi$ of a complex semisimple Lie algebra $\mathfrak g,$ without worrying about how the representation is constructed. The properties of these hypothetical representations are investigated,^[7] and conditions necessary for the existence of an irreducible representation are then established.

The properties involve the weights of the representation. Here is the simplest description.^[8] Let ${\mathfrak {h}}$ be a Cartan subalgebra of ${\mathfrak {g}}$ , that is a maximal commutative subalgebra with the property that $\mathrm {ad} _{H}$ is diagonalizable for each $H\in {\mathfrak {h}}$ ,^[9] and let $H_{1},\ldots ,H_{n}$ be a basis for ${\mathfrak {h}}$ . A weight $\lambda$ for a representation $(\pi ,V)$ of ${\mathfrak {g}}$ is a collection of simultaneous eigenvalues

(\lambda _{1},\ldots ,\lambda _{n})

for the commuting operators $\pi (H_{1}),\ldots ,\pi (H_{n})$ . In basis-independent language, $\lambda$ is a linear functional on ${\mathfrak {h}}$ .

A partial ordering on the set of weights is defined, and the notion of highest weight in terms of this partial ordering is established for any set of weights. Using the structure on the Lie algebra, the notions dominant element and integral element are defined. Every finite-dimensional representation must have a maximal weight $\lambda$ , i.e., one for which no strictly higher weight occurs. If $V$ is irreducible and $v$ is a weight vector with weight $\lambda$ , then the entire space $V$ must be generated by the action of ${\mathfrak {g}}$ on $v$ . Thus, $(\pi ,V)$ is a "highest weight cyclic" representation. One then shows that the weight $\lambda$ is actually the highest weight (not just maximal) and that every highest weight cyclic representation is irreducible. One then shows that two irreducible representations with the same highest weight are isomorphic. Finally, one shows that the highest weight $\lambda$ must be dominant and integral.

Conclusion: Irreducible representations are classified by their highest weights, and the highest weight is always a dominant integral element.

Step One has the side benefit that the structure of the irreducible representations is better understood. Representations decompose as direct sums of weight spaces, with the weight space corresponding to the highest weight one-dimensional. Repeated application of the representatives of certain elements of the Lie algebra called lowering operators yields a set of generators for the representation as a vector space. The application of one such operator on a vector with definite weight results either in zero or a vector with strictly lower weight. Raising operators work similarly, but results in a vector with strictly higher weight or zero. The representatives of the Cartan subalgebra acts diagonally in a basis of weight vectors.

Classification: Step Two

Step Two is concerned with constructing the representations that Step One allows for. That is to say, we now fix a dominant integral element $\lambda$ and try to construct an irreducible representation with highest weight $\lambda$ .

There are several standard ways of constructing irreducible representations:

Construction using Verma modules. This approach is purely Lie algebraic. (Generally applicable to complex semisimple Lie algebras.)^[10]^[11]
The compact group approach using the Peter–Weyl theorem. If, for example, ${\mathfrak {g}}=\mathrm {sl} (n;\mathbb {C} )$ , one would work with the simply connected compact group $\mathrm{SU}(n)$ . (Generally applicable to complex semisimple Lie algebras.)^[12]^[13]
Construction using the Borel–Weil theorem, in which holomorphic representations of the group $G$ corresponding to ${\mathfrak {g}}$ are constructed. (Generally applicable to complex semisimple Lie algebras.)^[13]
Performing standard operations on known representations, in particular applying Clebsch–Gordan decomposition to tensor products of representations. (Not generally applicable.)^{[nb 1]} In the case ${\mathfrak {g}}=\mathrm {sl} (3;\mathbb {C} )$ , this construction is described below.
In the simplest cases, construction from scratch.^[14]

Conclusion: Every dominant integral element of a complex semisimple Lie algebra gives rise to an irreducible, finite-dimensional representation. These are the only irreducible representations.

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,^[15] or from the fact that sl(2,C) is the complexification of the Lie algebra of the simply connected compact group SU(2).^[16] The irreducible representations $\pi$ , in turn, can be classified^[17] by the largest eigenvalue of $\pi (H)$ , which must be a non-negative integer m. That is to say, in this case, a "dominant integral element" is simply a non-negative integer. 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. This analysis is described in detail in the representation theory of SU(2) (from the point of the view of the complexified Lie algebra).

One can give a concrete realization of the representations (Step Two in the overview above) in either of two ways. First, in this simple example, it is not hard to write down an explicit basis for the representation and an explicit formula for how the generators $X,Y,H$ of the Lie algebra act on this basis.^[18] Alternatively, one can realize the representation^[19] with highest weight $m$ by letting $V_{m}$ denote the space of homogeneous polynomials of degree $m$ in two complex variables, and then defining the action of $X$ , $Y$ , and $H$ by

\pi _{m}(X)=-z_{2}{\frac {\partial }{\partial z_{1}}};\quad \pi _{m}(Y)=-z_{1}{\frac {\partial }{\partial z_{2}}};\quad \pi _{m}(H)=-z_{1}{\frac {\partial }{\partial z_{1}}}+z_{2}{\frac {\partial }{\partial z_{2}}}.

Note that the formulas for the action of $X$ , $Y$ , and $H$ do not depend on $m$ ; the subscript in the formulas merely indicates that we are restricting the action of the indicated operators to the space of homogeneous polynomials of degree $m$ in $z_{1}$ and $z_{2}$ .

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^[20] classifying the irreducible representations of sl(3,C), which is the complexified Lie algebra of the group SU(3). 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^[21] 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. That is to say, in this setting, a "dominant integral element" is precisely a pair of 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 (Step Two in the overview above). 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.^[22]

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^[23]

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

There is also a simple pattern to the multiplicities of the various weight spaces. Finally, the irreducible representations with highest weight $(0,m)$ can be realized concretely on the space of homogeneous polynomials of degree $m$ in three complex variables.^[24]

The case of a general semisimple Lie algebras

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.^[25] 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 in the article on weights in representation theory. 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 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:^[26]

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 ${\mathfrak {g}}$ .

The last point of the theorem (Step Two in the overview above) is the most difficult one. In the case of the Lie algebra sl(3;C), the construction can be done in an elementary way, as described above. In general, the construction of the representations may be given by using Verma modules.^[27]

Additional properties of the representations

Much is known about the representations of a complex semisimple Lie algebra ${\mathfrak {g}}$ , besides the classification in terms of highest weights. We mention a few of these briefly. We have already alluded to Weyl's theorem, which states that every finite-dimensional representation of ${\mathfrak {g}}$ decomposes as a direct sum of irreducible representations. There is also the Weyl character formula, which leads to the Weyl dimension formula (a formula for the dimension of the representation in terms of its highest weight), the Kostant multiplicity formula (a formula for the multiplicities of the various weights occurring in a representation). Finally, there is also a formula for the eigenvalue of the Casimir element, which acts as a scalar in each irreducible representation.

Lie group representations and Weyl's unitarian trick

Although it is possible to develop the representation theory of Lie algebras—let's say complex semisimple Lie algebras for definiteness—in a self-contained way, it can be illuminating to bring in a perspective using Lie groups. This approach is particularly helpful in understanding Weyl's theorem on complete reducibility. It is known that every complex semisimple Lie algebra ${\mathfrak {g}}$ has a compact real form ${\mathfrak k}$ .^[28] This means first that ${\mathfrak {g}}$ is the complexification of ${\mathfrak k}$ :

{\mathfrak {g}}={\mathfrak {k}}+i{\mathfrak {k}}

and second that there exists a simply connected compact group $K$ whose Lie algebra is ${\mathfrak k}$ . As an example, we may consider ${\mathfrak {g}}=\mathrm {sl} (n;\mathbb {C} )$ , in which case $K$ may be taken to be the special unitary group SU(n).

Given a finite-dimensional representation $V$ of ${\mathfrak {g}}$ , we can restrict it to ${\mathfrak k}$ . Then since $K$ is simply connected, we can integrate the representation to the group $K$ .^[29] The method of averaging over the group shows that there is an inner product on $V$ that is invariant under the action of $K$ ; that is, the action of $K$ on $V$ is unitary. At this point, we may use unitarity to see that $V$ decomposes as a direct sum of irreducible representations.^[30] This line of reasoning is called the unitarian trick and was Weyl's original argument for what is now called Weyl's theorem. There is also a purely algebraic argument for the complete reducibility of representations of semisimple Lie algebras.

If ${\mathfrak {g}}$ is a complex semisimple Lie algebra, there is a unique complex semisimple Lie group $G$ with Lie algebra ${\mathfrak {g}}$ , in addition to the simply connected compact group $K$ . (If ${\mathfrak {g}}=\mathrm {sl} (n;\mathbb {C} )$ then $G=\mathrm {SL} (n;\mathbb {C} )$ .) Then we have the following result about finite-dimensional representations.^[31]

Statement: The objects in the following list are in one-to-one correspondence:

Smooth representations of $K$
Holomorphic representations of $G$
Real linear representations of $\mathfrak{k}$
Complex linear representations of ${\mathfrak {g}}$

Conclusion: The representation theory of compact Lie groups can shed light on the representation theory of complex semisimple Lie algebras.

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.^[32]

The construction of $U({\mathfrak {g}})$ is as follows.^[33] 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 $\operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}\circ \operatorname {Res} _{\mathfrak {h}}\simeq \operatorname {Res} _{\mathfrak {g}}\circ \operatorname {Ind} _{\mathfrak {h_{1}}}^{\mathfrak {g_{1}}}$ .

Infinite-dimensional representations and "category O"

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 (possibly infinite-dimensional) 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.^[34]

(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.

Remarks

↑ This approach is used heavily for classical Lie algebras in Fulton & Harris (1991).

Notes

↑ Hall 2015 Theorem 5.6
↑ Hall 2013 Section 17.3
↑ Hall 2015 Theorem 4.29
↑ Dixmier 1977, Theorem 1.6.3
↑ Hall 2015 Section 4.3
↑ Hall 2015, Proposition 4.6.
↑ See Section 6.4 of Hall 2015 in the case of sl(3,C)
↑ Hall 2015, Section 6.2. (There specialized to $\mathrm {sl} (3;\mathbb {C}$ )
↑ Hall 2015, Section 7.2.
↑ Bäuerle, de Kerf & ten Kroode 1997, Chapter 20.
↑ Hall 2015, Sections 9.5–9.7
↑ Hall 2015, Chapter 12.
1 2 Rossmann 2002, Chapter 6.
↑ This approach for $\mathrm {sl} (2;\mathbb {C} )$ can be found in Example 4.10. of Hall, 2015 & Section 4.2.
↑ Hall 2015 Section 10.3
↑ Hall 2015 Theorems 4.28 and 5.6
↑ Hall 2015 Section 4.6
↑ Hall 2015 Equation 4.16
↑ Hall 2015 Example 4.10
↑ Hall 2015 Chapter 6
↑ Hall 2015 Theorem 6.7
↑ Hall 2015 Proposition 6.17
↑ Hall 2015 Theorem 6.27
↑ Hall 2015 Exercise 6.8
↑ Hall 2015 Section 7.7.1
↑ Hall 2015 Theorems 9.4 and 9.5
↑ Hall 2015 Sections 9.5-9.7
↑ Knapp 2002 Section VI.1
↑ Hall 2015 Theorem 5.6
↑ Hall 2015 Section 4.4
↑ Knapp 2001, Section 2.3.
↑ Hall 2015 Section 9.5
↑ Jacobson 1962
↑ http://mathoverflow.net/questions/64931/why-the-bgg-category-o

References

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Bäuerle, G.G.A; de Kerf, E.A.; ten Kroode, A.P.E. (1997). A. van Groesen; E.M. de Jager, eds. Finite and infinite dimensional Lie algebras and their application in physics. Studies in mathematical physics. 7. North-Holland. ISBN 978-0-444-82836-1 – via ScienceDirect. (Subscription required (help)).
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D. Gaitsgory, Geometric Representation theory, Math 267y, Fall 2005
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