Abstract
An introduction is given to the basic concepts, construction and representation theory of W-algebras.
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H. Awata, M. Fukuma, Y. Matsuo, and S. Odake, Character and determinant formulae of quasifinite representation of the 1+∞ algebra, Commun. Math. Phys. 172 (1995) 377, hep-th/9405093; V. Kac and A. Radul, Representation theory of the vertex algebra W 1+∞, hep-th/9512150.
F.A. Bais, P. Bouwknegt, K. Schoutens and M. Surridge, Extensions of the Virasoro algebra constructed from Kac-Moody algebras using higher order Casimir invariants, Nucl. Phys. B304 (1988) 348.
F.A. Bais, P. Bouwknegt, K. Schoutens and M. Surridge, Coset construction for extended Virasoro algebras, Nucl. Phys. B304 (1988) 371.
Z. Bajnok, Singular vectors of the WA 2 algebra, Phys. Lett. B329 (1994) 225, hep-th/9403032.
Z. Bajnok, L. Palla and G. Takács, A 2 Toda theory in reduced WZNW framework and the representations of the W algebra, Nucl. Phys. B385 (1992) 329, hep-th/9206075.
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B241 (1984) 333.
R. Blumenhagen, M. Flohr, A. Kliem, W. Nahm, A. Recknagel, and R. Varnhagen. W-algebras with two and three generators, Nucl. Phys. B361 (1991) 255.
R. Blumenhagen, W. Eholzer, A. Honecker, K. Hornfeck, and R. Hübel, Unifying W-algebras, Phys. Lett. B322 (1994) 51, hep-th/9404113
R. Blumenhagen, W. Eholzer, A. Honecker, K. Hornfeck, and R. Hübel, Coset realisation of unifying W-algebras, Int. J. Mod. Phys. A10 (1995) 2367, hep-th/9406203
J. de Boer, L. Feher and A. Honecker, A class of W algebras with infinitely generated classical limit, Nucl. Phys. B420 (1993) 409, hep-th/9312049.
J. de Boer and T. Tjin, The relation between quantum W algebras and Lie algebras, Commun. Math. Phys. 160 (1994) 317, hep-th/9302006
P. Bouwknegt, Extended Conformal algebras, Phys. Lett. 207B (1988) 295.
P. Bouwknegt and K. Schoutens, W symmetry in conformal field theory, Phys. Rept. 223 (1993) 183.
P. Bowcock, Representation theory of a W algebra from generalised DS reduction, Durham preprint DTP-94-5, hep-th/9403157
P. Bowcock and G.M.T. Watts, On the classification of quantum W-algebras, Nucl. Phys. B379 (1992) 63, hep-th/9111062.
A. Cappelli, C. Itzykson and J.-B. Zuber, The A-D-E classification of minimal and a (1)1 conformal field theories, Commun. Math. Phys. 113 (1987) 1.
A. Cappelli, C.A. Trugenberger and G.R. Zemba, Stable hierarchical quantum Hall fluids as W 1+∞ minimal models, Nucl. Phys. B448 (1995) 470, hep-th/9502021; W 1+∞ minimal models and the hierarchy of the quantum Hall effect, Nucl. Phys. Proc. Suppl. 45A (1996) 112.
E.B. Dynkin, Transl. Am. Math. Soc. Series 2, 6 (1957) 112.
P. van Driel and K. de Vos, The Kazhdan-Lusztig conjecture for W-algebras, Bonn preprint BONN-TH-95-14, hep-th/9508020.
V.G. Drinfel'd and V.V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, J. Sov. Math. 30 (1985) 1975.
W. Eholzer, M. Flohr, A. Honecker, R. Hübel, W. Nahm and R. Varnhagen, Representations of W-algebras with two-generators and new rational models, Nucl. Phys. B383 (1992) 249.
W. Eholzer, A. Honecker and R. Hübel, How complete is the classification of W-symmetries?, Phys. Lett. B308 (1993) 42, hep-th/9302124.
V.A. Fateev and S.L. Luk'yanov, The models of two-dimensional conformal quantum field theory with Z n symmetry, Int. J. Mod. Phys. A3 (1988) 507.
V.A. Fateev and S.L. Luk'yanov, Additional symmetries and exactly-soluble models in two-dimensional conformal field theory, Sov. Sci. Rev. A15 (1990) 1.
V.A. Fateev and A.B. Zamolodchikov, Conformal quantum field theory models in two dimensions having Z 3 symmetry, Nucl. Phys. B280 [FS18] (1987) 644.
L. Feher, L. O'Raifeartaigh, P. Ruelle, I. Tsutsui and A. Wipf, On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories, Phys. Rep. 222 (1992) 1.
L. Feher, L O'Raifeartaigh and I. Tsutsui, The vacuum preserving Lie algebra of a classical W algebra, Phys. Lett. B316 (1993) 275, hep-th/9307190
B. Feigin and E. Frenkel, Quantization of the Drinfeld-Sokolov reduction, Phys. Lett. B246 (1990) 75.
B.L. Feigin and D.B. Fuchs, On the cohomology of some nilpotent subalgebras of Kac-Moody and the Virasoro algebras, J. Geom. Phys. 5 (1988) 209.
J.M. Figueroa-O'Farrill and S. Schrans, The spin 6 extended conformal algebra, Phys. Lett. B245 (1990) 471.
E.V. Frenkel, V. Kac and M. Wakimoto, Characters and fusion rules for W algebras via quantized Drinfeld-Sokolov reductions, Commun. Math. Phys. 147 (1992) 295.
P. Furlan, A.C. Ganchev, R. Paunov and V.B. Petkova, Solutions of the Knizhnik-Zamolodchikov equation with rational isospins and the reduction to the minimal models, Nucl. Phys. B394 (1993) 665, hep-th/9201080.
P. Furlan, A.C. Ganchev and V.B. Petkova, Singular vectors of W algebras via DS reduction of A (1)2 Nucl. Phys. B431 (1994) 622, hep-th/9403075.
M.R. Gaberdiel, Fusion rules of chiral algebras, Nucl. Phys. B417 (1994) 130, hep-th/9309105; Fusion in conformal field theory as the tensor product of the symmetry algebra, Int. J. Mod. Phys. A9 (1994) 4619, hep-th/9307183.
M.R. Gaberdiel and H.G. Kausch, A rational logarithmic conformal field theory, Phys. Lett. B386 (1996) 131, hep-th/9606050; Indecomposable fusion products, Nucl. Phys. B477 (1996) 293, hep-th/9604026.
T. Gannon, The classification of affine su(3) modular invariant partition functions, Commun. Math. Phys 161 (1994) 233, hep-th/9212060.
T. Gannon and M.A. Walton, On the classification of diagonal coset modular invariants, Commun. Math. Phys 173 (1995) 175, hep-th/9407055
P. Goddard, Meromorphic Conformal Field Theory, in: Infinite Dimensional Lie Algebras and Lie Groups, ed. V. G. Kac, World Scientific, 1989, CIRM-Luminy July conference on Infinite dimensional Lie algebras and Lie Groups, Marseilles 1988.
K. Hornfeck, W-algebras with set of primary fields of dimensions (3,4,5) and (3,4,5,6), Nucl. Phys. B407 (1993) 237, hep-th/9212104.
K. Hornfeck, Classification of structure constants for W-algebras from highest weights, Nucl. Phys. B411 (1994) 307, hep-th/9307170.
K. Hornfeck, W-algebras of negative rank, Phys. Lett. B343 (1995) 94, hep-th/9410013.
Y.-Z. Huang and J. Lepowsky, On the D-module and formal variable approaches to vertex algebras, in: ‘Topics in Geometry: In Memory of Joseph D'Atri', ed. S. Gindikin, Progress in Nonlinear Differential Equations, Vol. 20, Birkhauser, Boston, (1996) 175, q-alg/9603020.
H.G. Kausch, Extended conformal algebras generated by a multiplet of primary fields, Phys. Lett. B259 (1991) 448.
H.G. Kausch and G.M.T. Watts, A study of W-algebras using Jacobi identities, Nucl. Phys. B354 (1991) 740.
W. Nahm, Chiral algebras of two-dimensional chiral field theories and their normal ordered products, in ‘Recent developments in conformal field theories’ S. Randjbar-Daemi et al. eds, (World Scientific 1990) 81.
W. Nahm, Quasi-rational fusion products, Int. J. Mod. Phys. B8 (1994) 3693, hep-th/9402039.
E. Verlinde, Fusion rules and modular transformations in 2d conformal field theory, Nucl. Phys. B300 [FS22] (1988) 360.
G.M.T. Watts, Fusion in the W 3 algebra, Commun. Math. Phys. 171 (1995) 87, hep-th/9403163.
A.B. Zamolodchikov, Infinite additional symmetries in two-dimensional conformal quantum field theory, Theor. Mat. Fiz. 65 (1985) 347.
Y.-C. Zhu, Vertex operator algebras, elliptic functions and modular forms, Ph.D. thesis, Yale University, 1990
I.B. Frenkel and Y.-C. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), 123.
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© 1997 Springer-Verlag
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Watts, G.M.T. (1997). W-algebras and their representations. In: Horváth, Z., Palla, L. (eds) Conformal Field Theories and Integrable Models. Lecture Notes in Physics, vol 498. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0105278
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DOI: https://doi.org/10.1007/BFb0105278
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