196
D. Valeri
Acknowledgments This review is based on the talk I gave at the XIth International Symposium
Quantum Theory and Symmetry in Montréal. I wish to thank the organizers for the invitation and
the hospitality.
References
1. L.F. Alday, D. Gaiotto, Y. Tachikawa, Liouville correlation functions from four-dimensional
gauge theories. Lett. Math. Phys. 91(2), 167–197 (2010)
2. T. Arakawa, Representation theory of W -algebras and Higgs branch conjecture, in Proceedings of the International Congress of Mathematicians - Rio de Janeiro 2018. Invited Lectures,
vol. II (World Scientific Publishing, Hackensack), pp. 1263–1281
3. Bakalov B., Milanov T., W-constraints for the total descendant potential of a simple
singularity. Compos. Math. 149(5), 840–888 (2013)
4. A. Barakat, A. De Sole, V.G. Kac, Poisson vertex algebras in the theory of Hamiltonian
equations. Jpn. J. Math. 4(2), 141–252 (2009)
5. C. Beem, L. Rastelli, Vertex operator algebras, Higgs branches, and modular differential
equations. J. High Energ. Phys. 2018, 114 (2018)
6. R. Borcherds, Vertex algebras, Kac-Moody algebras and the Monster. Proc. Natl. Acad. Sci.
U. S. A. 83, 3068–3071 (1986)
7. P. Bouwknegt, K. Schoutens, W-symmetry. Advanced Series in Mathematical Physics, vol. 22
(World Scientific, Singapore, 1995)
8. A. Braverman, M. Finkelberg, H. Nakajima, Instanton moduli spaces and W -algebras.
Astérisque 385, pp. vii+128, (2016)
9. J. Brundan, A. Kleshchev, Shifted Yangians and finite W-algebras Adv. Math. 200(1), 136–195
(2006)
10. N. Burruoughs, M. de Groot, T. Hollowood, L. Miramontes, Generalized Drinfeld-Sokolov
hierarchies II: the Hamiltonian structures. Commun. Math. Phys. 153, 187–215 (1993)
11. S. Carpentier, A. De Sole, V.G. Kac, D. Valeri, J. van de Leur, p-reduced multicomponent KP
hierarchy and classical W-algebras W(gl N , p) (2019). arXiv:1909.03301
12. M. de Groot, T. Hollowood, L. Miramontes, Generalized Drinfeld-Sokolov hierarchies.
Commun. Math. Phys. 145, 57–84 (1992)
13. A. De Sole, On classical finite and affine W -algebras, in Advances in Lie Superalgebras.
Springer INdAM Series, vol. 7 (Springer, Berlin, 2014), pp. 51–66
14. A. De Sole, V.G. Kac, Finite vs. affine W-algebras. Jpn. J. Math. 1(1), 137–261 (2006)
15. A. De Sole, V.G. Kac, D. Valeri, Classical W-algebras and generalized Drinfeld-Sokolov biHamiltonian systems within the theory of Poisson vertex algebras. Commun. Math. Phys.
323(2), 663–711 (2013)
16. A. De Sole, V.G. Kac, D. Valeri, Classical W-algebras and generalized Drinfeld-Sokolov
hierarchies for minimal and short nilpotents. Commun. Math. Phys. 331(2), 623–676 (2014)
17. A. De Sole, V.G. Kac, D. Valeri, Structure of classical (finite and affine) W-algebras. J. Eur.
Math. Soc. 18(9), 1873–1908 (2016)
18. A. De Sole, V.G. Kac, D. Valeri, Classical affine W-algebras for gl N and associated integrable
Hamiltonian hierarchies. Commun. Math. Phys. 348(1), 265–319 (2016)
19. A. De Sole, V.G. Kac, D. Valeri, Finite W-algebras for gl N . Adv. Math. 327, 173–224 (2018)
20. A. De Sole, V.G. Kac, D. Valeri, Classical affine W -algebras and the associated integrable
hierarchies for classical Lie algebras. Commun. Math. Phys. 360(3), 851–918 (2018)
21. A. De Sole, V.G. Kac, D. Valeri, A Lax type operator for quantum finite W-algebras. Sel. Math.
New Ser. 24(5), 4617–4657 (2018)
22. A. De Sole, L. Fedele, D. Valeri, Generators of the quantum finite W -algebras in type A. J.
Algebra Appl. 19(9), 2050175, p. 76
D. Valeri
Acknowledgments This review is based on the talk I gave at the XIth International Symposium
Quantum Theory and Symmetry in Montréal. I wish to thank the organizers for the invitation and
the hospitality.
References
1. L.F. Alday, D. Gaiotto, Y. Tachikawa, Liouville correlation functions from four-dimensional
gauge theories. Lett. Math. Phys. 91(2), 167–197 (2010)
2. T. Arakawa, Representation theory of W -algebras and Higgs branch conjecture, in Proceedings of the International Congress of Mathematicians - Rio de Janeiro 2018. Invited Lectures,
vol. II (World Scientific Publishing, Hackensack), pp. 1263–1281
3. Bakalov B., Milanov T., W-constraints for the total descendant potential of a simple
singularity. Compos. Math. 149(5), 840–888 (2013)
4. A. Barakat, A. De Sole, V.G. Kac, Poisson vertex algebras in the theory of Hamiltonian
equations. Jpn. J. Math. 4(2), 141–252 (2009)
5. C. Beem, L. Rastelli, Vertex operator algebras, Higgs branches, and modular differential
equations. J. High Energ. Phys. 2018, 114 (2018)
6. R. Borcherds, Vertex algebras, Kac-Moody algebras and the Monster. Proc. Natl. Acad. Sci.
U. S. A. 83, 3068–3071 (1986)
7. P. Bouwknegt, K. Schoutens, W-symmetry. Advanced Series in Mathematical Physics, vol. 22
(World Scientific, Singapore, 1995)
8. A. Braverman, M. Finkelberg, H. Nakajima, Instanton moduli spaces and W -algebras.
Astérisque 385, pp. vii+128, (2016)
9. J. Brundan, A. Kleshchev, Shifted Yangians and finite W-algebras Adv. Math. 200(1), 136–195
(2006)
10. N. Burruoughs, M. de Groot, T. Hollowood, L. Miramontes, Generalized Drinfeld-Sokolov
hierarchies II: the Hamiltonian structures. Commun. Math. Phys. 153, 187–215 (1993)
11. S. Carpentier, A. De Sole, V.G. Kac, D. Valeri, J. van de Leur, p-reduced multicomponent KP
hierarchy and classical W-algebras W(gl N , p) (2019). arXiv:1909.03301
12. M. de Groot, T. Hollowood, L. Miramontes, Generalized Drinfeld-Sokolov hierarchies.
Commun. Math. Phys. 145, 57–84 (1992)
13. A. De Sole, On classical finite and affine W -algebras, in Advances in Lie Superalgebras.
Springer INdAM Series, vol. 7 (Springer, Berlin, 2014), pp. 51–66
14. A. De Sole, V.G. Kac, Finite vs. affine W-algebras. Jpn. J. Math. 1(1), 137–261 (2006)
15. A. De Sole, V.G. Kac, D. Valeri, Classical W-algebras and generalized Drinfeld-Sokolov biHamiltonian systems within the theory of Poisson vertex algebras. Commun. Math. Phys.
323(2), 663–711 (2013)
16. A. De Sole, V.G. Kac, D. Valeri, Classical W-algebras and generalized Drinfeld-Sokolov
hierarchies for minimal and short nilpotents. Commun. Math. Phys. 331(2), 623–676 (2014)
17. A. De Sole, V.G. Kac, D. Valeri, Structure of classical (finite and affine) W-algebras. J. Eur.
Math. Soc. 18(9), 1873–1908 (2016)
18. A. De Sole, V.G. Kac, D. Valeri, Classical affine W-algebras for gl N and associated integrable
Hamiltonian hierarchies. Commun. Math. Phys. 348(1), 265–319 (2016)
19. A. De Sole, V.G. Kac, D. Valeri, Finite W-algebras for gl N . Adv. Math. 327, 173–224 (2018)
20. A. De Sole, V.G. Kac, D. Valeri, Classical affine W -algebras and the associated integrable
hierarchies for classical Lie algebras. Commun. Math. Phys. 360(3), 851–918 (2018)
21. A. De Sole, V.G. Kac, D. Valeri, A Lax type operator for quantum finite W-algebras. Sel. Math.
New Ser. 24(5), 4617–4657 (2018)
22. A. De Sole, L. Fedele, D. Valeri, Generators of the quantum finite W -algebras in type A. J.
Algebra Appl. 19(9), 2050175, p. 76