Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1993-07-29
Int.J.Mod.Phys. A9 (1994) 4619-4636
Physics
High Energy Physics
High Energy Physics - Theory
21 pages, Latex, DAMTP-93-38
Scientific paper
Following a recent proposal of Richard Borcherds to regard fusion as the ring-like tensor product of modules of a {\em quantum ring}, a generalization of rings and vertex operators, we define fusion as a certain quotient of the (vector space) tensor product of representations of the symmetry algebra ${\cal A}$. We prove that this tensor product is associative and symmetric up to equivalence. We also determine explicitly the action of ${\cal A}$ on it, under which the central extension is preserved. \\ Having given a precise meaning to fusion, determining the fusion rules is now a well-posed algebraic problem, namely to decompose the tensor product into irreducible representations. We demonstrate how to solve it for the case of the WZW- and the minimal models and recover thereby the well-known fusion rules. \\ The action of the symmetry algebra on the tensor product is given in terms of a comultiplication. We calculate the $R$-matrix of this comultiplication and find that it is triangular. This seems to shed some new light on the possible r\^{o}le of the quantum group in conformal field theory.
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