Representation theory of sl₂ℂ

The Lie algebra ${\displaystyle {\mathfrak {sl}}_{2}\mathbb {C} }$ is a three-dimensional complex Lie algebra. Its defining feature is that it contains a basis ${\displaystyle e,h,f}$ satisfying the commutation relations

${\displaystyle [e,f]=h}$, ${\displaystyle [h,f]=-2f}$, and ${\displaystyle [h,e]=2e}$.

This is a Cartan-Weyl basis for ${\displaystyle {\mathfrak {sl}}_{2}\mathbb {C} }$. It has an explicit realization in terms of two-by-two complex matrices with zero trace:

${\displaystyle E={\begin{bmatrix}0&1\\0&0\end{bmatrix}}}$, ${\displaystyle F={\begin{bmatrix}0&0\\1&0\end{bmatrix}}}$, ${\displaystyle H={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}$.

This is the fundamental or defining representation for ${\displaystyle {\mathfrak {sl}}_{2}\mathbb {C} }$.

The Lie algebra ${\displaystyle {\mathfrak {sl}}_{2}\mathbb {C} }$ can be viewed as a subspace of its universal enveloping algebra ${\displaystyle U=U({\mathfrak {sl}}_{2}\mathbb {C} )}$ and, in ${\displaystyle U}$, there are the following commutator relations shown by induction:^[1]

${\displaystyle [h,f^{k}]=-2kf^{k},\,[h,e^{k}]=2ke^{k}}$,

${\displaystyle [e,f^{k}]=-k(k-1)f^{k-1}+kf^{k-1}h}$.

Note that, here, the powers ${\displaystyle f^{k}}$, etc. refer to powers as elements of the algebra U and not matrix powers. The first basic fact (that follows from the above commutator relations) is:^[1]

From this lemma, one deduces the following fundamental result:^[2]

The first statement is true since either ${\displaystyle v_{j}}$ is zero or has ${\displaystyle h}$-eigenvalue distinct from the eigenvalues of the others that are nonzero. Saying ${\displaystyle v}$ is a ${\displaystyle {\mathfrak {b}}}$-weight vector is equivalent to saying that it is simultaneously an eigenvector of ${\displaystyle h,e}$; a short calculation then shows that, in that case, the ${\displaystyle e}$-eigenvalue of ${\displaystyle v}$ is zero: ${\displaystyle e\cdot v=0}$. Thus, for some integer ${\displaystyle N\geq 0}$, ${\displaystyle v_{N}\neq 0,v_{N+1}=v_{N+2}=\cdots =0}$ and in particular, by the early lemma,

${\displaystyle 0=e\cdot v_{N+1}=(\lambda -(N+1)+1)v_{N},}$

which implies that ${\displaystyle \lambda =N}$. It remains to show ${\displaystyle W=\operatorname {span} \{v_{j}|j\geq 0\}}$ is irreducible. If ${\displaystyle 0\neq W'\subset W}$ is a subrepresentation, then it admits an eigenvector, which must have eigenvalue of the form ${\displaystyle N-2j}$; thus is proportional to ${\displaystyle v_{j}}$. By the preceding lemma, we have ${\displaystyle v=v_{0}}$ is in ${\displaystyle W}$ and thus ${\displaystyle W'=W}$. ${\displaystyle \square }$

As a corollary, one deduces:

• If ${\displaystyle V}$ has finite dimension and is irreducible, then ${\displaystyle h}$-eigenvalue of v is a nonnegative integer ${\displaystyle N}$ and ${\displaystyle V}$ has a basis ${\displaystyle v,fv,f^{2}v,\cdots ,f^{N}v}$.
• Conversely, if the ${\displaystyle h}$-eigenvalue of ${\displaystyle v}$ is a nonnegative integer and ${\displaystyle V}$ is irreducible, then ${\displaystyle V}$ has a basis ${\displaystyle v,fv,f^{2}v,\cdots ,f^{N}v}$; in particular has finite dimension.

The beautiful special case of ${\displaystyle {\mathfrak {sl}}_{2}}$ shows a general way to find irreducible representations of Lie algebras. Namely, we divide the algebra to three subalgebras "h" (the Cartan Subalgebra), "e", and "f", which behave approximately like their namesakes in ${\displaystyle {\mathfrak {sl}}_{2}}$. Namely, in an irreducible representation, we have a "highest" eigenvector of "h", on which "e" acts by zero. The basis of the irreducible representation is generated by the action of "f" on the highest eigenvectors of "h". See the theorem of the highest weight.

Representation theory of sl_nℂ

When ${\displaystyle {\mathfrak {g}}={\mathfrak {sl}}_{n}\mathbb {C} ={\mathfrak {sl}}(V)}$ for a complex vector space ${\displaystyle V}$ of dimension ${\displaystyle n}$, each finite-dimensional irreducible representation of ${\displaystyle {\mathfrak {g}}}$ can be found as a subrepresentation of a tensor power of ${\displaystyle V}$.^[4]

The Lie algebra can be explicitly realized as a matrix Lie algebra of traceless ${\displaystyle n\times n}$ matrices. This is the fundamental representation for ${\displaystyle {\mathfrak {sl}}_{n}\mathbb {C} }$.

Set ${\displaystyle M_{i,j}}$ to be the matrix with one in the ${\displaystyle i,j}$ entry and zeroes everywhere else. Then

${\displaystyle H_{i}:=M_{i,i}-M_{i+1,i+1},{\text{ with }}1\leq i\leq n-1}$

${\displaystyle M_{i,j},{\text{ with }}i\neq j}$

Form a basis for ${\displaystyle {\mathfrak {sl}}_{n}\mathbb {C} }$. This is technically an abuse of notation, and these are really the image of the basis of ${\displaystyle {\mathfrak {sl}}_{n}\mathbb {C} }$ in the fundamental representation.

Furthermore, this is in fact a Cartan–Weyl basis, with the ${\displaystyle H_{i}}$ spanning the Cartan subalgebra. Introducing notation ${\displaystyle E_{i,j}=M_{i,j}}$ if ${\displaystyle j>i}$, and ${\displaystyle F_{i,j}=M_{i,j}^{T}=M_{j,i}}$, also if ${\displaystyle j>i}$, the ${\displaystyle E_{i,j}}$ are positive roots and ${\displaystyle F_{i,j}}$ are corresponding negative roots.

A basis of simple roots is given by ${\displaystyle E_{i,i+1}}$ for ${\displaystyle 1\leq i\leq n-1}$.