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'Lie algebra representation'
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| Title of object: |
Lie algebra representation |
| Canonical Name: |
RepresentationLieAlgebra |
| Type: |
Definition |
| Created on: |
2002-05-29 09:33:04-04 |
| Modified on: |
2002-12-30 21:26:13.901061-05 |
| Classification: |
msc:17B10 |
| Defines: |
irreducible, module, dimension, finite dimensional, finite-dimensional, infinite dimensional, infinite-dimensional, faithful, tensor product |
| Synonyms: |
Lie algebra representation=representation |
Revision comment (for changes between this and next version):
| Changes for correction #1423 ('faithful, tensor product, etc.'). |
Preamble:
\newcommand{\lag}{\mathfrak{g}}
\newcommand{\ad}{\mathop{\mathrm{ad}}\nolimits}
\newcommand{\End}{\mathop{\mathrm{End}}\nolimits}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\natnums}{\mathbb{N}}
\newcommand{\cnums}{\mathbb{C}}
\newcommand{\znums}{\mathbb{Z}}
\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\lb}{\left[}
\newcommand{\rb}{\right]}
\newcommand{\supth}{^{\text{th}}}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}[proposition]{Definition}
\newcommand{\nl}[1]{\PMlinkescapetext{{#1}}}
\newcommand{\pln}[2]{\PMlinkname{#1}{#2}} |
Content:
A representation of a Lie algebra $\lag$ is a Lie algebra homomorphism
$$\rho:\lag \rightarrow \End V,$$
where $\End V$ is the commutator Lie
algebra of some vector space $V$. In other words, $\rho$ is a linear
mapping that satisfies
$$\rho([a,b]) = \rho(a)\rho(b)-\rho(b)\rho(a),\quad a,b\in\lag$$
Alternatively, one calls $V$ a $\lag$-module, and calls $\rho(a),\,
a\in \lag$ the action of $a$ on $V$.
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
called {\em irreducible} or simple if its only invariant subspaces are $\{0\}$
and the whole representation.
The dimension of $V$ is called the dimension of the representation.
If $V$ is infinite-dimensional, then one speaks of an
infinite-dimensional representation.
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
$\sigma:\lag\to\End(W)$ 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
%manner. |
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