Applied Representation Theory : 007

Representations of Lie algebras and groups

We define modules for and representations of Lie algebras and groups. We then describe some important classifying details of those representations. 

In this section we define representations of both Lie algebras and groups. The groups need not be Lie groups, although typically are. Both kinds of representations are very similar, but have slight structural differences. We will aim to be clear on which of the two kinds of representations we are talking about, although mindfulness about that context is recommended in this piece. The easiest tell involves certain kinds of linear maps which we will discuss in what follows;

Endomorphisms $\leftrightarrow$ Lie algebras

Automorphisms $\leftrightarrow$ groups

This section will also deal with a lot of abstract ideas. Next time, we shall revisit them with concrete examples.

Representations of Lie Algebras

We begin by discussing Lie algebra modules and representations. Let $V$ be a vector space. The endomorphisms of $V$, denoted $\mathsf{End}\,V$ are the set of linear transformations from $V$ into itself. The endomorphisms of a vector space are also a vector space. We can also make them into an algebra by assigning the composition of maps as the vector multiplication.

A representation of a Lie algebra $\mathfrak{g}$ on some vector space $V$ is a linear map:

$$\pi : \mathfrak{g}\rightarrow \mathsf{End}\,V,$$

such that for $a,b$ in $\mathfrak{g}$, and $v$ in $V$,

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

In other words, they're a subspace of the linear maps on $V$ that furnish the same Lie algebra structure found in $\mathfrak{g}$. The vector space $V$ associated to such a representation $\pi$ is typically called a called a $\mathbf{\mathfrak{g}}$-module.

Given that Lie algebras are also vector spaces, any Lie algebra $\mathfrak{g}$ has a natural representation on itself, the adjoint representation:

$$\mathsf{ad}: \mathfrak{g}\rightarrow \mathsf{End}\,\mathfrak{g},$$

where for an $x$ in $\mathfrak{g}$, the given endomorphism is furnished by the Lie bracket:

$$\mathsf{ad} : x \mapsto [x,\cdot],$$

so that for $y$ also in $\mathfrak{g}$:

$$\mathsf{ad}_{x}(y) = [x,y].$$

Physicists often like to concretely represent the Lie bracket in terms of the structure constants of the Lie algebra, $f_{abc}$. To that end, let a collection $\{T_{a}\}_{a\in[1,2,\cdots \dim \mathfrak{g}]}$ be a basis for a Lie algebra $\mathfrak{g}$.

$$[T_{a},T_{b}] = i\sum_{c} f_{abc} T_{c}.$$

Let $n = \dim\mathfrak{g}$. We see then that the matrix

$$\sum_{c}f_{abc}T_{c} = \left(\sum_{c}f_{c}T_{c}\right)_{ab} \in \mathsf{Hom}_{n,n}\mathbb{C},$$

is an endomorphism of $\mathfrak{g}$. Such concrete notation is probably best reserved for concrete examples, as we’ll see in later lectures.

Representations of a Group

Let $\mathfrak{g}$ be a Lie algebra. Given that all Lie algebras are associated with a Lie group, it is perhaps unsurprising that $\mathfrak{g}$-modules may possess a natural action of the group $G$ associated to $\mathfrak{g}$. While we shall often confuse these concepts notationally, it is best not to confuse their definitions - or the students! To that end, a group representation of some group $G$ on a vector space $V$ is a group homomorphism $\rho$, from $G$ into the automorphisms of $V$

$$\rho : G \rightarrow \mathsf{Aut}\,V.$$

An automorphism of a vector space is a linear map that is also isomorphism. In particular, the linear dimension of the image of an automorphism is the same as the dimension of its domain. Note that all automorphisms are a special subclass of endomorphisms: they are invertible\(^{1}\).

Reducibility and Irreducibility

There are a few properties enjoyed by some representations that are worth remarking on, as they play important roles in our study of the representations of symmetry groups. In our discussions, let $G$ be a group, $\mathfrak{g}$ be a Lie algebra and $V$ can be a module for either, depending upon context. $\phi$ will denote the associated representation. 

A submodule $W$ of a group or algebra module is a linear subspace of $V$ that is closed under the action of the representation. By closed we mean that if $w$ in $W$, then $\phi(x) \cdot w$ is also in $W$ for all $x$ in the group or algebra. We can also say that $W$ is invariant under that group or algebra. Some folks might also call it an invariant subspace.

An irreducible module has no nontrivial submodules. These concepts are functionally the same for both Lie algebra and group modules.

Direct Sums and Tensor Products of Modules

For a Lie algebra $\mathfrak{g}$, let $V$ and $W$ be $\mathfrak{g}$-modules. The direct sum of $V$ and $W$ is the vector space of dimension $\dim V + \dim W$ that has two submodules, $V$ and $W$.

A module is completely reducible if it can be represented as a vector space direct sum of irreducible modules. A module that is not completely reducible is said to be indecomposable. Evidently every irreducible module is indecomposable, although the converse is not true. Such beasts may feel pathological from a physical perspective, and fortunately they aren't commonly dealt with.

The vector space direct product of $V$ and $W$ is the vector space of dimension $\dim V \times \dim W$, composed of the span over of all pairs of vectors $v$ and $w$, written as $v\otimes w$:

$$V\times W = \mathsf{span}\left\{ v\otimes w \;\Big|\; v\in V,\; w\in W\right\}.$$

Up until this point, these concepts are functionally the same for both Lie algebra and group modules. We will now discuss the tensor product of two representations, whose structure depends on whether they are Lie algebra or group modules.

For finite-dimensional $V$ and $W$, it’s easy to check that $V\times W$ is a vector space. We can promote $V\times W$ to a $\mathfrak{g}$-module, called the tensor product of Lie algebra modules $V$ and $W$ by demanding that for all $x$ in $\mathfrak{g}$, $\pi(x)$ acts via the product rule:

$$\pi(x)\cdot( v\otimes w) = (\pi(x)\cdot v)\otimes w + v\otimes(\pi(x)\cdot w ).$$

Let $G$ be a group and now let $V$ and $W$ be $G$-modules. A tensor product of group modules is a vector space direct product of $V$ and $W$ together with the $G$-action of a representation $\pi$:

$$\pi(x)\cdot( v\otimes w) = (\pi(x)\cdot v)\otimes (\pi(x) \cdot w), $$

for all $x$ in $G$.

It doesn’t typically make sense to discuss the tensor product of a group module with a Lie algebra module. One specific class of examples where they are the same will be investigated in the exercises.

A Sketch of the Applications to Particle Spin in Quantum Mechanics

One major task in the application of representation theory to physics is to convert tensor products of modules into a form that is explicitly completely reducible. In quantum mechanics this process is often called the addition of angular momentum. For example, if $\mathbf{2}$ is the doublet associated to the spin $\frac{1}{2}$ Pauli spinors representing electrons in the Hydrogen atom, then the two electrons of the Helium atom might be best represented as

$$\mathbf{2}\otimes \mathbf{2} = \mathbf{3}\oplus \mathbf{1},$$

where the $\mathbf{3}$ is the spin $1$ triplet or “symmetric” product and $\mathbf{1}$ is the spin $0$ “antisymmetric“ singlet. The three-electron system, we have

$$\mathbf{2}\otimes \mathbf{2}\otimes \mathbf{2} = \mathbf{4}\oplus \mathbf{2}\oplus \mathbf{2}.$$

In this language, ${\bf 1,2,3,4}$ are all irreducible representations of $\mathsf{SU}_{2}$ of eponymous dimension. The objective of the next lecture is to explicitly create these products of representations. Later we shall see similar examples of modules that are necessarily infinite dimensional\(^{2}\).

Exercises

Exercise 7.1 : Endomorphisms and Automorphisms

Verify that $\mathsf{End}\,V$ is a vector space. Is $\mathsf{Aut}\,V$ a vector space?

Exercise 7.2: Natural Associative Algebras

Argue that $\mathsf{End}\,V$ together with composition - $(\mathsf{End}\,V,\circ)$ - always forms an associative algebra.

Exercise 7.3 : The Automorphism Group

For a vector space $V$, prove that $\mathsf{Aut}\,V$ is a group whose multiplication is given by composition. Is $\mathsf{End}\,V$ similarly a group?

Exercise 7.4 : Modules for Associative Algebras

Use the definition of a Lie algebra representation to define a representation of an associative algebra. Of course, your definition should include the notion of an associative algebra module.

Exercise 7.5 : More on Tensor Products

Suppose $\pi_{1}$ and $\pi_{2}$ are two distinct representations of a Lie algebra $\mathfrak{g}$, whose actions don’t interact. That is, they commute:

$$[\pi_{1},\pi_{2}] = 0.$$

Argue that the linear combination $\pi_{1}$ + $\pi_{2}$ is a representation of $\mathfrak{g}$. Use this fact to argue that the definition of the tensor product given above is indeed a representation of $\mathfrak{g}$. Iterate the definition of the tensor product to define the tensor product of finitely many such $\mathfrak{g}$-modules.

Exercise 7.6 : Group Algebras

Let $G$ be a group. Let $\mathbb{C}[G]$ be the space of formal, complex linear combinations of elements of $G$.In other words, each distinct $g$ in $G$ can be thought of as a basis vector. Show that $\mathbb{C}[G]$ is an associative algebra using group multiplication. For any $G$-module $V$, argue that $V$ is also a $\mathbb{C}[G]$-module. 

Exercise 7.7: Induced Modules

This one is a little tricky, so let's go slowly. Let $A$ be an associative algebra with a subalgebra $B$. Let the vector space $V$ be an $B$-module, with representation $\pi$. Show that the tensor product vector space 

$$A\otimes V$$

is an $A$-module. Argue that it is not irreducible. Hint: there are two distinct actions of $B$ on $A\otimes V$

Next, we aim to convert $A\otimes V$ into something as close to an irreducible representation as possible. Let $a$ be an element of $A$, $b$ be an element of $B$ and $v$ be a vector in  $V$. As per our hint above, notice that  the object

$$ab \otimes v$$

is distinct from 

$$a\otimes(\pi(b)\cdot v).$$

Let us form a quotient vector space of $A\otimes V$ by formally identifying the vectors  $ab \otimes v $ and $a\otimes(\pi(b)\cdot v)$:

$$ab \otimes v \sim  a\otimes(\pi(b)\cdot v).$$

This defines the induced ${\bf A}$-module denoted by

$$A\otimes_{B}V = A\otimes V \Big/ \left\{ ab \otimes v\sim a\otimes(\pi(b)\cdot v)\right\}.$$

Argue that $A\otimes_{B}V$ has dimension $\dim A + \dim V - \dim B$. Under what conditions is $A\otimes_{B}V$ irreducible as an $A$-module?

Exercise 7.7: Induced Group Modules

Let $G$ be a group with a subgroup $H$, and let $V$ be an $H$-module. Use the results of the last exercise to build the $G$-module induced by $V$. Hint: you’ll need to first construct vector spaces to take a quotient.

\(^{1}\) An example of a non-invertible endomorphism is a projection. For example. if $\overline{r}$ is a position vector in $\mathbb{R}^{3}$, the restriction to the $\overline{x}$ direction:

$$P_{x} : \mathbb{R}^{3} \rightarrow \mathbb{R}$$

$$P_{x} : \overline{x}\mapsto (\overline{r}\cdot \overline{x})\overline{x}$$

is a linear map. But it cannot be inverted since it “forgets” information about the $\overline{y}$ and $\overline{z}$ directions. Similarly, taking the real part of a complex number is a projection. Also, multiplying by zero is a linear map to the trivial, zero-dimensional vector space.

\(^{2}\) The Lie group $\mathsf{SL}_{2}\mathbb{R}$ is one such example.

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