## Paths and root operators in representation theory.(English)Zbl 0858.17023

Let $$X$$ be the weight lattice of a complex symmetrizable Kac-Moody algebra $${\mathfrak g}$$, and $$\pi$$ the set of all piecewise linear paths $$\pi: [0,1] \to X_\mathbb{Q}$$ starting at 0. For a simple root $$\alpha$$, the author defines linear operators $$e_\alpha$$ and $$f_\alpha$$ on the $$\mathbb{Z}$$-module $$\mathbb{Z}\pi$$ spanned by $$\pi$$. Let $$A\subset \text{End}_\mathbb{Z} \mathbb{Z} \pi$$ be the subalgebra generated by these operators. Let $$P^+$$ be the set of all paths $$\pi$$ such that the image is contained in the dominant Weyl chamber and $$\pi(1) \in X$$. Let $$\pi \in P^+$$ and $$M_\pi$$ be the $$A$$-module $$A\pi$$ and $$B_\pi$$ the set of paths contained in $$M_\pi$$. Let $$\pi(1) = \lambda$$, and $$V(\lambda)$$ the irreducible integrable $${\mathfrak g}$$-module with highest weight $$\lambda$$. The author shows that $$\text{char} V(\lambda) = \sum_{\eta \in B_\pi} e^{\eta(1)}$$. Denoting by $$*$$ the “concatariation” of paths, the author shows that $$M_{\pi_1}* M_{\pi_2} = \oplus_\pi M_\pi$$, where $$\pi_1, \pi_2 \in P^+$$ and $$\pi$$ runs over all paths in $$P^+$$ of the form $$\pi = \pi_1* \eta$$, for some $$\eta \in B_{\pi_2}$$. For a Levi subalgebra $${\mathfrak p}$$ of $${\mathfrak g}$$, denoting by $$A_{\mathfrak p}$$ the subalgebra generated by those $$e_\alpha$$, $$f_\alpha$$, such that $$\alpha$$ is a simple root of $${\mathfrak p}$$, the author shows that $$M_\pi = \oplus_\eta N_\eta$$ (as $$A_{\mathfrak p}$$-modules) where $$\pi \in P^+$$, and $$\eta$$ runs over all paths in $$B_\pi$$ which are contained in $$P^+_{\mathfrak p}$$. (Here $$P^+_{\mathfrak p}$$ denotes the set of paths contained in the Weyl chamber corresponding to $${\mathfrak p}$$, and for $$\eta \in P^+_{\mathfrak p}$$, $$N_\eta$$ denotes the $$A_{\mathfrak p}$$-module $$A_{\mathfrak p} \eta$$.
As an application the author obtains a tensor product decomposition rule and restriction rule for Kac-Moody algebras.
This paper makes an important contribution to representation theory.

### MSC:

 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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