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Strongly multiplicity free modules for Lie algebras and quantum groups. (English) Zbl 1169.17003

Summary: Let \(\mathcal U\) be either the universal enveloping algebra of a complex semisimple Lie algebra \(\mathfrak g\) or its Drinfel’d-Jimbo quantisation over the field \(\mathbb C(z)\) of rational functions in the indeterminate \(z\). We define the notion of “strongly multiplicity free” (smf) for a finite-dimensional \(\mathcal U\)-module \(V\), and prove that for such modules the endomorphism algebras \(\mathrm{End}_\mathcal U(V^{\otimes r})\) are “generic” in the sense that in the classical (unquantised) case, they are quotients of Kohno’s infinitesimal braid algebra \(T_r\) while in the quantum case they are quotients of the group ring \(\mathbb C(z)B_r\) of the \(r\)-string braid group \(B_r\). In the classical case, the generators are generalisations of the quadratic Casimir operator \(C\) of \(\mathcal U\), while in the quantum case, they arise from \(R\)-matrices, which may be thought of as square roots of a quantum analogue of \(C\) in a completion of \(\mathcal U^{\otimes r}\). This unifies many known results and brings some new cases into their context. These include the irreducible 7-dimensional module in type \(G_2\) and arbitrary irreducibles for \(\mathfrak{sl}_2\). The work leads naturally to questions concerning non-semisimple deformations of the relevant endomorphism algebras, which arise when the ground rings are varied.

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
20G42 Quantum groups (quantized function algebras) and their representations
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