📂Representation Theory

Representations of Lie Algebras

Definition¹

Let $\mathfrak{g}$ be a Lie algebra and $V$ a vector space. A Lie algebra homomorphism $\pi$ from $\mathfrak{g}$ to the general linear Lie algebra $\mathfrak{gl}(V)$ is called a representation of $\mathfrak{g}$.

$$ \pi : \mathfrak{g} \to \mathfrak{gl}(V) $$

• Such a $V$ is called a representation space.

Explanation

Since a representation is defined as a homomorphism, the commutation relations of $\mathfrak{g}$ are preserved in $\mathfrak{gl}(V)$.

$$ \pi \left( \left[ X, Y \right]_{\mathfrak{g}} \right) = \left[ \pi(X), \pi(Y) \right]_{\mathfrak{gl}(V)} $$

This also means that the structure constants are preserved. If $[X_{i}, X_{j}] = c_{ij}^{k}X_{k}$, then the following holds.

$$ [\pi(X_{i}), \pi(X_{j})] = c_{ij}^{k}\pi(X_{k}) $$

Invariant Representations

A subspace $W$ of $V$ is said to be $\pi$-invariant, or invariant under $\pi$, if it satisfies the following.

$$ \pi(X) (W) \subset W, \quad \forall X \in \mathfrak{g} $$

This means that no vector in $W$ is mapped outside of $W$ by any transformation $\pi(X)$.

Irreducible Representations

The subspaces $\left\{ \mathbf{0} \right\}$ and $V$ of a vector space $V$ are always invariant under any transformation, so they are called the trivial invariant subspaces. If a representation $\pi : \mathfrak{g} \to \mathfrak{gl}(V)$ has only the trivial invariant subspaces, it is said to be irreducible.