Feigin, B. L.; Semikhatov, A. M. \(W_n^{(n)}\) algebras. (English) Zbl 1123.17302 Nucl. Phys., B 698, No. 3, 409-449 (2004). 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. Cited in 1 ReviewCited in 36 Documents 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 PDFBibTeX XMLCite \textit{B. L. Feigin} and \textit{A. M. Semikhatov}, Nucl. Phys., B 698, No. 3, 409--449 (2004; Zbl 1123.17302) Full Text: DOI arXiv References: [1] Polyakov, A. M., Gauge transformations and diffeomorphisms, Int. J. Mod. Phys. A, 5, 833 (1990) · Zbl 0741.53057 [2] Bershadsky, M., Conformal field theories via Hamiltonian reduction, Commun. Math. Phys., 139, 71 (1991) · Zbl 0721.58046 [3] Feigin, B.; Jimbo, M.; Miwa, T., Vertex operator algebra arising from the minimal series \(M(3, p)\) and monomial basis · Zbl 1028.81025 [4] Blumenhagen, R.; Eholzer, W.; Honecker, A.; Hornfeck, K.; Huebel, R., Unifying \(W\)-algebras, Phys. Lett. B, 332, 51-60 (1994) · Zbl 1044.81594 [5] Feigin, B. L.; Stoianovsky, A. V., Functional models of representations of current algebras and semi-infinite Schubert cells, Funkcional. Anal. Prilozh., 28, 1, 68 (1994) [6] Semikhatov, A. M.; Yu Tipunin, I.; Feigin, B. L., Semi-infinite realization of unitary representations of the \(N = 2\) algebra and related constructions, Teor. Mat. Fiz.. Teor. Mat. Fiz., Theor. Math. Phys., 126, 1-47 (2001) · Zbl 0998.81032 [7] Gorbounov, V.; Malikov, F., The chiral de Rham complex and positivity of the equivariant signature of the loop space · Zbl 1063.14061 [8] Borisov, L. A., Vertex algebras and mirror symmetry, Commun. Math. Phys., 215, 517-557 (2001) · Zbl 0990.17023 [9] de Boer, J.; Tjin, T., The relation between quantum \(W\)-algebras and Lie algebras, Commun. Math. Phys., 160, 317-332 (1994) · Zbl 0796.17027 [10] Feigin, B. L.; Frenkel, E., Quantization of the Drinfeld-Sokolov reduction, Phys. Lett. B, 246, 75 (1990) · Zbl 1242.17023 [11] Bouwknegt, P.; Schoutens, K., \(W\)-symmetry in conformal field theory, Phys. Rep., 223, 183-276 (1993) [12] Feigin, B. L.; Semikhatov, A. M., The \(\hat{s \ell}(2) \oplus \hat{s \ell}(2) / \hat{s \ell}(2)\) coset theory as a Hamiltonian reduction of \(\hat{D}(2 | 1 \text{;} \alpha)\), Nucl. Phys. B, 610, 489-530 (2001) · Zbl 0971.81044 [13] Thielemans, K., An algorithmic approach to operator product expansions, \(W\)-algebras and \(W\)-strings, KUL, Leuven [14] Bowcock, P.; Feigin, B. L.; Semikhatov, A. M.; Taormina, A., \( \hat{s \ell}(2 | 1)\) and \(\hat{D}(2 | 1 \text{;} \alpha)\) as vertex operator extensions of dual affine \(s \ell(2)\) algebras, Commun. Math. Phys., 214, 495-545 (2000) · Zbl 0980.17016 [15] Feigin, B. L.; Frenkel, E. V., Lett. Math. Phys., 19, 307 (1990) [16] Petersen, J. L.; Rasmussen, J.; Yu, M., Free field realizations of 2D current algebras, screening currents and primary fields, Nucl. Phys. B, 502, 649-670 (1997) · Zbl 0934.81013 [17] Feigin, B. L.; Semikhatov, A. M., Free-field resolutions of the unitary representations of the \(N = 2\) Virasoro algebra, II: the butterfly resolution, Teor. Mat. Fiz.. Teor. Mat. Fiz., Theor. Math. Phys., 121, 1462-1472 (1999) · Zbl 1025.17008 [18] Feigin, B.; Jimbo, M.; Miwa, T.; Mukhin, E., A differential ideal of symmetric polynomials spanned by Jack polynomials at \(\beta = -(r - 1) /(k + 1)\) · Zbl 1012.05153 [19] Feigin, B.; Jimbo, M.; Miwa, T.; Mukhin, E., Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials · Zbl 1069.33019 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.