| Version 8 |
Version 7 |
| A representation of a Lie algebra $\lag$ is a Lie algebra homomorphism |
A representation of a Lie algebra $\lag$ is a Lie algebra homomorphism |
| $$\rho:\lag \rightarrow \End V,$$ |
$$\rho:\lag \rightarrow \End V,$$ |
| where $\End V$ is the commutator Lie |
where $\End V$ is the commutator Lie |
| algebra of some vector space $V$. In other words, $\rho$ is a linear |
algebra of some vector space $V$. In other words, $\rho$ is a linear |
| mapping that satisfies |
mapping that satisfies |
| $$\rho([a,b]) = \rho(a)\rho(b)-\rho(b)\rho(a),\quad a,b\in\lag$$ |
$$\rho([a,b]) = \rho(a)\rho(b)-\rho(b)\rho(a),\quad a,b\in\lag$$ |
| Alternatively, one calls $V$ a $\lag$-module, and calls $\rho(a),\, |
Alternatively, one calls $V$ a $\lag$-module, and calls $\rho(a),\, |
| a\in \lag$ the action of $a$ on $V$. |
a\in \lag$ the action of $a$ on $V$. |
| We call the representation {\em faithful} if \rho is injective. |
We call the representation {\em faithful} if \rho is injective. |
| A invariant subsspace or sub-module $W\subset V$ is a subspace of $V$ satisfying $\rho(a)(W)\subset W$ for all $a\in\lag$. A representation is |
A invariant subsspace or sub-module $W\subset V$ is a subspace of $V$ satisfying $\rho(a)(W)\subset W$ for all $a\in\lag$. A representation is |
| called {\em irreducible} or simple if its only invariant subspaces are $\{0\}$ |
called {\em irreducible} or simple if its only invariant subspaces are $\{0\}$ |
| and the whole representation. |
and the whole representation. |
| The dimension of $V$ is called the dimension of the representation. |
The dimension of $V$ is called the dimension of the representation. |
| If $V$ is infinite-dimensional, then one speaks of an |
If $V$ is infinite-dimensional, then one speaks of an |
| infinite-dimensional representation. |
infinite-dimensional representation. |
| Given a representation or pair of representation, there are a couple of operations which will produce other representations: |
Given a representation or pair of representation, there are a couple of operations which will produce other representations: |
| First there is direct sum. If $\rho:\lag\to\End(V)$ and |
First there is direct sum. If $\rho:\lag\to\End(V)$ and |
| $\sigma:\lag\to\End(W)$ are representations, then $V\oplus W$ has the obvious Lie algebra action, by the embedding $\End(V)\times\End(W)\hookrightarrow\End(V\oplus W)$. |
$\sigma:\lag\to\End(W)$ are representations, then $V\oplus W$ has the obvious Lie algebra action, by the embedding $\End(V)\times\End(W)\hookrightarrow\End(V\oplus W)$. |
| %There is also tensor product, which is defined in a somwhat less obvious |
%There is also tensor product, which is defined in a somwhat less obvious |
| %manner. |
%manner. |
