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\(W_n^{(n)}\) algebras. (English) Zbl 1123.17302

Summary: We construct \(W\)-algebra generalizations of the \(\widehat{sl}\) source algebra – \(W\) algebras \({\mathcal W}_n^{(2)}\) generated by two currents \(E\) and \(F\) with the highest pole of order \(n\) in their OPE. The \(n=3\) term in this series is the Bershadsky-Polyakov \({\mathcal W}_3^{(2)}\) source algebra. We define these algebras as a centralizer (commutant) of the \({\mathcal U}_q \text{sl}(n| 1)\) quantum supergroup and explicitly find the generators in a factored, “Miura-like” form. Another construction of the \({\mathcal W}_n^{(2)}\) source algebras is in terms of the coset \(\widehat{sl}(n| 1)\widehat{sl}(n)\). The relation between the two constructions involves the “duality” \((k+n-1)(k'+n-1)=1\) between levels \(k\) and \(k'\) of two \(\widehat{sl}(n)\) source algebras.

MSC:

17B68 Virasoro and related algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B69 Vertex operators; vertex operator algebras and related structures
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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References:

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