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I know that irreducible representations of associative $*$-algebras are fairly restricted: any $*$-algebra $A$ is isomorphic to a finite sum of simple algebras

$A\cong\oplus_{i=0}^{N}M_{n_i}(\mathbb{C})$

What's the cardinality of the irreps of a Lie algebra?

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    It depends on what you mean by *irrep*... and it depends on the Lie algebra. Usually, though, there are many, many irreps.2012-06-05

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The smallest non-zero Lie algebra of all is the one-dimensional Lie algebra.

If you only look at finite dimensional representations, its irreps are in bijection with pairs $(\lambda,n)$ with $\lambda\in\mathbb C$ and $n\in\mathbb N$: this fact is an immediate consequence of the theorem of Jordan Canonical forms.

If you want to consider infinite dimensional representations, then things are much more complicated. A representation of this algebra is roughly the same as an endomorphism of a vector space, so in a sense the study of its representions is the same thing as operator theory: whole books have been written about this.

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    As for the bijection: just write down what exactly a module over the $1$-dimensional Lie algebra is, and notice that it is just the same thing as an endomorphism of a vector space. If you cannot figure it out, ask this as a separate question. It is quite a different thing as the subject of *this* question which, strictly, has nothing to do with it.2012-06-06