representation theory of $\mathfrak{sl}_{2}\mathbb{C}$

The special linear Lie algebra of $2\times 2$ matricies, denoted by $\mathfrak{sl}_{2}\mathbb{C}$ , is defined to be the span (over $\mathbb{C}$ ) of the matricies

$E=\left(\begin{array}[]{cc}0&1\\ 0&0\end{array}\right),H=\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right),F=\left(\begin{array}[]{cc}0&0\\ -1&0\end{array}\right)$

with Lie bracket given by the commutator of matricies: $[X,Y]:=X\cdot Y-Y\cdot X$ . The matricies $E, F, H$ satisfy the commutation relations: $[E,F]=H,[H,E]=2E,[H,F]=-2F$ .

The representation theory of $\mathfrak{sl}_{2}\mathbb{C}$ is a very important tool for understanding the structure theory and representation theory of other Lie algebras (semi-simple finite dimensional Lie algebras, as well as infinite dimensional Kac-Moody Lie algebras).

The finite dimensional, irreducible, representations of $\mathfrak{sl}_{2}\mathbb{C}$ are in bijection with the non-negative integers $\mathbb{Z}_{\geq 0}$ as follows. Let $k\in\mathbb{Z}_{\geq 0}$ , $V$ be a $\mathbb{C}$ -vector space spanned by vectors $v_{0},\ldots,v_{k}$ . The following action of $E, H, F$ on $V$ define the unique (up to isomorphism) irreducible representation of $\mathfrak{sl}_{2}\mathbb{C}$ of dimension $k+1$ (or of highest weight $k$ ):

$\begin{array}[]{ll}E.v_{0}&=0\\ E.v_{i}&=(i-1)(k-i+1)v_{i-1}\quad\forall\quad 1\leq i\leq k\\ H.v_{i}&=(k-2i)v_{i}\quad\forall\quad 1\leq i\leq k\\ F.v_{i}&=v_{i+1}\quad\forall\quad 0\leq i<k\\ F.v_{k}&=0\end{array}$

The main points are that the one dimensional spaces $\mathbb{C}\cdot v_{i}$ are eigenspaces for $H$ with eigenvalue $k-2i$ , the operator corresponding to $E$ kills $v_{0}$ and otherwise sends $\mathbb{C}\cdot v_{i}\to\mathbb{C}\cdot v_{i-1}$ , while $F$ kills $v_{k}$ and otherwise sends $\mathbb{C}\cdot v_{i}\to\mathbb{C}\cdot v_{i+1}$ . The operator corresponding to $E$ is often called a raising operator since it raises the eigenvalue for $H$ , and that of $F$ is called a lowering operator since it lowers the eigenvalue for $H$ .

$\mathfrak{sl}_{2}\mathbb{C}$ is a simple Lie algebra, thus by Weyl’s Theorem all finite dimensional representations for $\mathfrak{sl}_{2}\mathbb{C}$ are completely reducible. So any finite dimensional representation of $\mathfrak{sl}_{2}\mathbb{C}$ splits into a direct sum of irreducible representations for various non-negative integers as described above.

Title	representation theory of $\mathfrak{sl}_{2}\mathbb{C}$
Canonical name	RepresentationTheoryOfmathfraksl2mathbbC
Date of creation	2013-03-22 15:30:29
Last modified on	2013-03-22 15:30:29
Owner	benjaminfjones (879)
Last modified by	benjaminfjones (879)
Numerical id	7
Author	benjaminfjones (879)
Entry type	Definition
Classification	msc 22E60
Classification	msc 22E47
Defines	sl_2
Defines	special linear Lie algebra of 2x2 matricies

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representation theory of $\mathfrak{sl}_{2}\mathbb{C}$