Lie algebra representation

A representation of a Lie algebra $𝔤$ is a Lie algebra homomorphism

$$\rho :𝔤\to EndV,$$

where $EndV$ is the commutator Lie algebra of some vector space $V$. In other words, $\rho$ is a linear mapping that satisfies

$$\rho ([a,b])=\rho (a)\rho (b)-\rho (b)\rho (a),a,b\in 𝔤$$

Alternatively, one calls $V$ a $𝔤$-module, and calls $\rho (a),a\in 𝔤$ the action of $a$ on $V$.

We call the representation faithful if $\rho$ is injective.

A invariant subspace or sub-module $W\subset V$ is a subspace of $V$ satisfying $\rho (a)(W)\subset W$ for all $a\in 𝔤$. A representation is called 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 pair of representations, we can define a new representation, called the direct sum of the two given representations:

If $\rho :𝔤\to End(V)$ and $\sigma :𝔤\to End(W)$ are representations, then $V\oplus W$ has the obvious Lie algebra action, by the embedding $End(V)\times End(W)\hookrightarrow End(V\oplus W)$.

Title	Lie algebra representation
Canonical name	LieAlgebraRepresentation
Date of creation	2013-03-22 12:41:13
Last modified on	2013-03-22 12:41:13
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	16
Author	mathcam (2727)
Entry type	Definition
Classification	msc 17B10
Synonym	representation
Related topic	Dimension3
Defines	irreducible
Defines	module
Defines	dimension
Defines	finite dimensional
Defines	finite-dimensional
Defines	infinite dimensional
Defines	infinite-dimensional
Defines	faithful
Defines	direct sum of representations