# Algebra out of a set of modules of a Lie algebra? Fusion

The problem I faced is how to organize a set of finite-dimensional irreducible representations $U_\alpha$ of some simple Lie algebra $g$ into an Lie algebra $A$ that contains $g$ as a Lie subalgebra and such that $A$ as a $g$-module decomposes into $\bigoplus_\alpha U_\alpha$. The set of irreps $U_\alpha$ of $g$ is the input data (the adjoint of $g$ can be thought of as one of $U_\alpha$).

A simple positive answer is given by the universal enveloping algebra evaluated at some other irreducible representation, say $V$, then $A=U(g)|_{V}\sim V\otimes V^*$ is an algebra and $A$ as a $g$-module is given simply by decomposing $V\otimes V^*=\bigoplus_\alpha U_\alpha$ into irreps $U_\alpha$. So given $U_\alpha$ that come out of $V\otimes V^*$ there exists an algebra $A$, whose 'generators' are given by $U_\alpha$. This gives a lot of algebras of this type. I wonder if this is the complete answer. This is reminiscent of the fusion problem in conformal field theory.

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I don't understand precisely enough the mathematical question here, but your reference to fusion reminds me that there is substantial mathematical literature inspired by the Verlinde work. Probably the best explanation is found in the framework of quantum groups at a root of unity, but there is also an interesting Lie algebra formulation by H.H. Andersen and J. Paradowski, Fusion categories arising from semisimple Lie algebras, Comm. Math. Physics 169 (1995), 563-588 (see their section 1 and references). Maybe that's too far from your question; it's also technical. – Jim Humphreys Aug 16 '12 at 0:43
Thank you for the reference, I will have a look at it! May I also ask what is not clear in the question? (it is my fault, of course) – Eugene Starling Aug 16 '12 at 7:33
What I find not clear is the following: do you really mean that $A$ should be a Lie algebra? In the example you're giving which is the operation? The fusion algebras I know are associative algebras built from a set of representations $U_\alpha$, the product being given by decomposition of tensor products (which, of course, imposes restrictions on the family $U_\alpha$). – Nicola Ciccoli Aug 27 '13 at 7:42
I need $A$ to be just a Lie algebra. In the example I gave $A$ is generously provided by the universal enveloping algebra in any representation, say $V$, it is associative by construction, in fact just $gl(V)$, but it can be turned into a Lie and contains $\mathfrak{g}$ as required. The question is whether all such Lie algerbas are in fact associative and originate from $U(g)_{V}$ for some $V$. – Eugene Starling Aug 27 '13 at 18:25