Tensor factorization and spin construction for Kac-Moody algebras (Q734826)

When a representation \(V\) of a Lie algebra \(\tilde{\mathfrak g}\) is restricted to a subalgebra \(\mathfrak g\), \(V\) decomposes into the tensor product of representations for \(\mathfrak g\). For a semisimple finite dimensional Lie algebra \(\mathfrak g\) let \(V\) be a \(\mathfrak g\)-representation that has a nondegenerate symmetric bilinear form preserved by \(\mathfrak g\) (an orthogonal \(\mathfrak g\)-representation), then \(\mathfrak g\) acts on \(\text{Spin}(V)\) via the embedding of \(\mathfrak g\) in \(\mathfrak{so}(V)\). It turns out that \(\text{Spin}(V)\) decomposes as direct sum of copies of a smaller representation called \(\text{Spin}_0(V)\). If \(\mathfrak g\) is infinite dimensional this can be extended with some care, by considering a central extension of an image of \(\mathfrak g\) in a Clifford algebra. If \(\mathfrak g\) is a symmetrizable Kac-Moody algebra with a distinguished element \(d\), then if \(V\) is integrable then so is \(\text{Spin}(V)\) and it is still true that \(\text{Spin}(V)\) decomposes into copies of \(\text{Spin}_0(V)\). The character of \(\text{Spin}_0(V)\) is also computed in terms of the character of \(V\). Some interesting special cases are given.