### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 072, 36 pages      arXiv:2108.13883      https://doi.org/10.3842/SIGMA.2022.072

### Quadratic Relations of the Deformed $W$-Algebra for the Twisted Affine Lie Algebra of Type $A_{2N}^{(2)}$

Takeo Kojima
Department of Mathematics and Physics, Faculty of Engineering, Yamagata University, Jonan 4-3-16, Yonezawa 992-8510, Japan

Received December 15, 2021, in final form September 09, 2022; Published online October 04, 2022

Abstract
We revisit the free field construction of the deformed $W$-algebra by Frenkel and Reshetikhin [Comm. Math. Phys. 197 (1998), 1-32], where the basic $W$-current has been identified. Herein, we establish a free field construction of higher $W$-currents of the deformed $W$-algebra associated with the twisted affine Lie algebra $A_{2N}^{(2)}$. We obtain a closed set of quadratic relations and duality, which allows us to define deformed $W$-algebra ${\mathcal W}_{x,r}\big(A_{2N}^{(2)}\big)$ using generators and relations.

Key words: deformed $W$-algebra; twisted affine algebra; quadratic relation; free field construction; exactly solvable model.

pdf (892 kb)   tex (93 kb)

References

1. Awata H., Kubo H., Odake S., Shiraishi J., Quantum deformation of the ${\mathcal W}_N$ algebras, arXiv:q-alg/9612001.
2. Awata H., Kubo H., Odake S., Shiraishi J., Quantum ${\mathcal W}_N$ algebras and Macdonald polynomials, Comm. Math. Phys. 179 (1996), 401-416, arXiv:q-alg/9508011.
3. Brazhnikov V., Lukyanov S., Angular quantization and form factors in massive integrable models, Nuclear Phys. B 512 (1998), 616-636, arXiv:hep-th/9707091.
4. Ding J., Feigin B., Quantized $W$-algebra of $\mathfrak{sl}(2,1)$: a construction from the quantization of screening operators, in Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Contemp. Math., Vol. 248, Amer. Math. Soc., Providence, RI, 1999, 83-108, arXiv:math.QA/9801084.
5. Feigin B., Frenkel E., Quantum $\mathcal W$-algebras and elliptic algebras, Comm. Math. Phys. 178 (1996), 653-678, arXiv:q-alg/9508009.
6. Feigin B., Jimbo M., Mukhin E., Vilkoviskiy I., Deformations of $\mathcal {W}$ algebras via quantum toroidal algebras, Selecta Math. (N.S.) 27 (2021), 52, 62 pages, arXiv:2003.04234.
7. Frenkel E., Reshetikhin N., Quantum affine algebras and deformations of the Virasoro and ${\mathcal W}$-algebras, Comm. Math. Phys. 178 (1996), 237-264, arXiv:q-alg/9505025.
8. Frenkel E., Reshetikhin N., Deformations of $\mathcal W$-algebras associated to simple Lie algebras, Comm. Math. Phys. 197 (1998), 1-32, arXiv:q-alg/9708006.
9. Frenkel E., Reshetikhin N., Semenov-Tian-Shansky M.A., Drinfeld-Sokolov reduction for difference operators and deformations of ${\mathcal W}$-algebras. I. The case of Virasoro algebra, Comm. Math. Phys. 192 (1998), 605-629, arXiv:q-alg/9704011.
10. Harada K., Matsuo Y., Noshita G., Watanabe A., $q$-Deformation of corner vertex operator algebras by Miura transformation, J. High Energy Phys. 2021 (2021), no. 4, 202, 49 pages, arXiv:2101.03953.
11. Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
12. Kojima T., Quadratic relations of the deformed $W$-superalgebra $\mathcal W_{q,t} (A(M,N))$, J. Phys. A 54 (2021), 335201, 37 pages, arXiv:2101.01110.
13. Kojima T., Quadratic relations of the deformed $W$-superalgebra $\mathcal{W}_{q,t}(\mathfrak{sl}(2|1))$, J. Math. Phys. 62 (2021), 051702, 19 pages, arXiv:1912.03096.
14. Odake S., Comments on the deformed $W_N$ algebra, Internat. J. Modern Phys. B 16 (2002), 2055-2064, arXiv:math.QA/0111230.
15. Semenov-Tian-Shansky M.A., Sevostyanov A.V., Drinfeld-Sokolov reduction for difference operators and deformations of ${\mathcal W}$-algebras. II. The general semisimple case, Comm. Math. Phys. 192 (1998), 631-647, arXiv:q-alg/9702016.
16. Sevostyanov A., Drinfeld-Sokolov reduction for quantum groups and deformations of $W$-algebras, Selecta Math. (N.S.) 8 (2002), 637-703, arXiv:math.QA/0107215.
17. Shiraishi J., Kubo H., Awata H., Odake S., A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions, Lett. Math. Phys. 38 (1996), 33-51, arXiv:q-alg/9507034.
18. van de Leur J.W., Contragredient Lie superalgebras of finite growth, Ph.D. Thesis, Utrecht University, 1986.
19. van de Leur J.W., A classification of contragredient Lie superalgebras of finite growth, Comm. Algebra 17 (1989), 1815-1841.