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fusionProduct -- computes the multiplicities of irreducibles in the decomposition of the fusion product of U and V

Synopsis

Description

This function implements the Kac-Walton algorithm; see Di Francesco, Mathieu, and Senechal, Conformal Field Theory, Springer Graduate Texts in Theoretical Physics, Section 16.2.2.

Given two irreducible Lie algebra modules $U$ and $V$, the function returns the fusion product of $U$ and $V$ at level $l$. (We are abusing notation and terminology a little here; the fusion product is really a product for modules over an affine Lie algebra. However, since the Kac-Walton algorithm is defined entirely using the combinatorics of the root system of the underlying finite-dimensional Lie algebra, we may therefore use the Kac-Walton algorithm to define a product on Lie algebra modules as well.)

The example below shows that for $g=sl_3$ and $\lambda=2 \omega_1 + \omega_2 = (2,1)$, $\mu= \omega_1 + 2 \omega_2 = (1,2)$, the level 3 fusion product $V_{(2,1)} \otimes_3 V_{(1,2)}$ contains one copy of $V_{(0,0)}$ and one copy of $V_{(1,1)}$.

i1 : g=simpleLieAlgebra("A",2);
i2 : U=irreducibleLieAlgebraModule({2,1},g);
i3 : V=irreducibleLieAlgebraModule({1,2},g);
i4 : fusionProduct(U,V,3)

o4 = LL   (g) ++ LL   (g)
       0,0         1,1

o4 : LieAlgebraModule over g

Ways to use fusionProduct :

For the programmer

The object fusionProduct is a method function.