2000 character limit reached
The coproduct for the affine Yangian and the parabolic induction for non-rectangular $W$-algebras (2404.14096v5)
Published 22 Apr 2024 in math.RA, math-ph, math.MP, and math.QA
Abstract: By using the coproduct and evaluation map for the affine Yangian and the Miura map for non-rectangular $W$-algebras, we construct a homomorphism from the affine Yangian associated with $\widehat{\mathfrak{sl}}(n)$ to the universal enveloping algebra of a non-rectangular $W$-algebra of type $A$, which is an affine analogue of the one given in De Sole-Kac-Valeri. As a consequence, we find that the coproduct for the affine Yangian is compatible with some of the parabolic induction for non-rectangular $W$-algebras via this homomorphism. We also show that the image of this homomorphism is contained in the affine coset of the $W$-algebra in the special case that the $W$-algebra is associated with a nilpotent element of type $(1{m-n},2n)$.
- T. Arakawa. Representation theory of W𝑊Witalic_W-algebras. Invent. Math., 169(2):219–320, 2007, https://doi.org/10.1007/s00222-007-0046-1.
- J. Brundan and A. Kleshchev. Shifted Yangians and finite W𝑊Witalic_W-algebras. Adv. Math., 200(1):136–195, 2006, https://doi.org/10.1016/j.aim.2004.11.004.
- Affine Laumon spaces and iterated 𝒲𝒲{\cal W}caligraphic_W-algebras. Comm. Math. Phys., 402(3):2133–2168, 2023.
- T. Creutzig and A. R. Linshaw. Trialities of 𝒲𝒲\mathcal{W}caligraphic_W-algebras. Camb. J. Math., 10(1):69–194, 2022.
- A. De Sole and V. G. Kac. Finite vs affine W𝑊Witalic_W-algebras. Jpn. J. Math., 1(1):137–261, 2006, https://doi.org/10.1007/s11537-006-0505-2.
- A Lax type operator for quantum finite W𝑊Witalic_W-algebras. Selecta Math. (N.S.), 24(5):4617–4657, 2018.
- V. G. Drinfeld. Hopf algebras and the quantum Yang-Baxter equation. Dokl. Akad. Nauk SSSR, 283(5):1060–1064, 1985, https://doi.org/10.1142/9789812798336.0013.
- V. G. Drinfeld. A new realization of Yangians and of quantum affine algebras. Dokl. Akad. Nauk SSSR, 296(1):13–17, 1987.
- M. Finkelberg and A. Tsymbaliuk. Multiplicative slices, relativistic Toda and shifted quantum affine algebras. In Representations and nilpotent orbits of Lie algebraic systems, volume 330 of Progr. Math., pages 133–304. Birkhäuser/Springer, Cham, 2019.
- I. B. Frenkel and Y. Zhu. Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J., 66(1):123–168, 1992, https://doi.org/10.1215/S0012-7094-92-06604-X.
- D. Gaiotto and M. Rapčák. Vertex algebras at the corner. J. High Energy Phys., (1):160, front matter+85, 2019.
- N. Genra. Screening operators for 𝒲𝒲\mathcal{W}caligraphic_W-algebras. Selecta Math. (N.S.), 23(3):2157–2202, 2017, https://doi.org/10.1007/s00029-017-0315-9.
- N. Genra. Screening operators and parabolic inductions for affine 𝒲𝒲{\cal W}caligraphic_W-algebras. Adv. Math., 369:107179, 62, 2020, https://doi.org/10.1016/j.aim.2020.107179.
- N. Guay. Cherednik algebras and Yangians. Int. Math. Res. Not., (57):3551–3593, 2005, https://doi.org/10.1155/IMRN.2005.3551.
- N. Guay. Affine Yangians and deformed double current algebras in type A. Adv. Math., 211(2):436–484, https://doi.org/10.1016/j.aim.2006.08.007, 2007.
- Coproduct for Yangians of affine Kac-Moody algebras. Adv. Math., 338:865–911, 2018, https://doi.org/10.1016/j.aim.2018.09.013.
- V. G. Kac and M. Wakimoto. Quantum reduction and representation theory of superconformal algebras. Adv. Math., 185(2):400–458, 2004, https://doi.org/10.1016/j.aim.2003.12.005.
- R. Kodera. Braid group action on affine Yangian. SIGMA Symmetry Integrability Geom. Methods Appl., 15:Paper No. 020, 28, 2019, https://doi.org/10.3842/SIGMA.2019.020.
- R. Kodera. On Guay’s evaluation map for affine Yangians. Algebr. Represent. Theory, 24(1):253–267, 2021, https://doi.org/10.1007/s10468-019-09945-w.
- R. Kodera and M. Ueda. Coproduct for affine Yangians and parabolic induction for rectangular W𝑊Witalic_W-algebras. Lett. Math. Phys., 112(1):Paper No. 3, 37, 2022.
- Quasi-finite algebras graded by Hamiltonian and vertex operator algebras. London Math. Soc. Lecture Note Ser., 372:282–329, 2010, https://doi.org/10.1017/CBO9780511730054.015.
- A. Premet. Special transverse slices and their enveloping algebras. Adv. Math., 170(1):1–55, 2002, https://doi.org/10.1006/aima.2001.2063. With an appendix by Serge Skryabin.
- O. Schiffmann and E. Vasserot. Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on 𝔸2superscript𝔸2\mathbb{A}^{2}blackboard_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. Publ. Math. Inst. Hautes Études Sci., 118:213–342, 2013, https://doi.org/10.1007/s10240-013-0052-3.
- M. Ueda. An Example of Homomorphisms from the Guay’s affine Yangians to non-rectangular W𝑊Witalic_W-algebras. Transformation Groups(2023).
- M. Ueda. A homomorphism from the affine Yangian Yℏ,ε(𝔰𝔩^(n))subscript𝑌Planck-constant-over-2-pi𝜀^𝔰𝔩𝑛Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n))italic_Y start_POSTSUBSCRIPT roman_ℏ , italic_ε end_POSTSUBSCRIPT ( over^ start_ARG fraktur_s fraktur_l end_ARG ( italic_n ) ) to the affine Yangian Yℏ,ε(𝔰𝔩^(n+1))subscript𝑌Planck-constant-over-2-pi𝜀^𝔰𝔩𝑛1Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n+1))italic_Y start_POSTSUBSCRIPT roman_ℏ , italic_ε end_POSTSUBSCRIPT ( over^ start_ARG fraktur_s fraktur_l end_ARG ( italic_n + 1 ) ). arXiv:2312.09933.
- M. Ueda. Notes on a homomorphism from the affine Yangian associated with 𝔰𝔩^(n)^𝔰𝔩𝑛\widehat{\mathfrak{sl}}(n)over^ start_ARG fraktur_s fraktur_l end_ARG ( italic_n ) to the affine Yangian associated with 𝔰𝔩^(n)^𝔰𝔩𝑛\widehat{\mathfrak{sl}}(n)over^ start_ARG fraktur_s fraktur_l end_ARG ( italic_n ). arXiv:2402.01870.
- M. Ueda. Two homomorphisms from the affine Yangian associated with 𝔰𝔩^(n)^𝔰𝔩𝑛\widehat{\mathfrak{sl}}(n)over^ start_ARG fraktur_s fraktur_l end_ARG ( italic_n ) to the affine Yangian associated with 𝔰𝔩^(n)^𝔰𝔩𝑛\widehat{\mathfrak{sl}}(n)over^ start_ARG fraktur_s fraktur_l end_ARG ( italic_n ). arXiv:2404.10923.
- M. Ueda. Affine super Yangians and rectangular W𝑊Witalic_W-superalgebras. J. Math. Phys., 63(5):Paper No. 051701, 34, 2022.
- M. Ueda. Guay’s affine Yangians and non-rectangular W𝑊Witalic_W-algebras. Adv. Math., 438:Paper No. 109468, 44, 2024.
Collections
Sign up for free to add this paper to one or more collections.
Paper Prompts
Sign up for free to create and run paper prompts using GPT-5.