Restricted shifted Yangians and restricted finite 𝑊-algebras

Goodwin, Simon; Topley, Lewis

doi:10.1090/btran/63

Restricted shifted Yangians and restricted finite $W$-algebras
HTML articles powered by AMS MathViewer

by Simon M. Goodwin and Lewis Topley HTML | PDF

Trans. Amer. Math. Soc. Ser. B 8 (2021), 190-228

Abstract:

We study the truncated shifted Yangian $Y_{n,l}(\sigma )$ over an algebraically closed field $\Bbbk$ of characteristic $p >0$, which is known to be isomorphic to the finite $W$-algebra $U(\mathfrak {g},e)$ associated to a corresponding nilpotent element $e \in \mathfrak {g} = \mathfrak {gl}_N(\Bbbk )$. We obtain an explicit description of the centre of $Y_{n,l}(\sigma )$, showing that it is generated by its Harish-Chandra centre and its $p$-centre. We define $Y_{n,l}^{[p]}(\sigma )$ to be the quotient of $Y_{n,l}(\sigma )$ by the ideal generated by the kernel of trivial character of its $p$-centre. Our main theorem states that $Y_{n,l}^{[p]}(\sigma )$ is isomorphic to the restricted finite $W$-algebra $U^{[p]}(\mathfrak {g},e)$. As a consequence we obtain an explicit presentation of this restricted $W$-algebra.

References

Roman Bezrukavnikov and Ivan Mirković, Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution, Ann. of Math. (2) 178 (2013), no. 3, 835–919. With an appendix by Eric Sommers. MR 3092472, DOI 10.4007/annals.2013.178.3.2
N. Bourbaki, Algebra. II. Chapters 4–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1990. Translated from the French by P. M. Cohn and J. Howie. MR 1080964
Jonathan Brown and Jonathan Brundan, Elementary invariants for centralizers of nilpotent matrices, J. Aust. Math. Soc. 86 (2009), no. 1, 1–15. MR 2495993, DOI 10.1017/S1446788708000608
Kenneth A. Brown and Iain Gordon, The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras, Math. Z. 238 (2001), no. 4, 733–779. MR 1872572, DOI 10.1007/s002090100274
Jonathan Brundan, Mœglin’s theorem and Goldie rank polynomials in Cartan type $A$, Compos. Math. 147 (2011), no. 6, 1741–1771. MR 2862061, DOI 10.1112/S0010437X11005653
Jonathan Brundan and Alexander Kleshchev, Shifted Yangians and finite $W$-algebras, Adv. Math. 200 (2006), no. 1, 136–195. MR 2199632, DOI 10.1016/j.aim.2004.11.004
J. Brundan and A. Kleschev, Representations of shifted Yangians and finite $W$-algebras, Mem. Amer. Math. Soc. 196 (2008).
Jonathan Brundan and Lewis Topley, The $p$-centre of Yangians and shifted Yangians, Mosc. Math. J. 18 (2018), no. 4, 617–657. MR 3914107, DOI 10.17323/1609-4514-2018-18-4-617-657
Simon M. Goodwin and Lewis W. Topley, Modular finite $W$-algebras, Int. Math. Res. Not. IMRN 18 (2019), 5811–5853. MR 4012128, DOI 10.1093/imrn/rnx295
Simon M. Goodwin and Lewis Topley, Minimal-dimensional representations of reduced enveloping algebras for $\frak {gl}_n$, Compos. Math. 155 (2019), no. 8, 1594–1617. MR 3984027, DOI 10.1112/s0010437x19007474
Jens Carsten Jantzen, Representations of Lie algebras in positive characteristic, Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math., vol. 40, Math. Soc. Japan, Tokyo, 2004, pp. 175–218. MR 2074594, DOI 10.2969/aspm/04010175
B. Ju. Veĭsfeĭler and V. G. Kac, The irreducible representations of Lie $p$-algebras, Funkcional. Anal. i Priložen. 5 (1971), no. 2, 28–36 (Russian). MR 0285575
Ivan Losev, Finite W-algebras, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1281–1307. MR 2827841
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
C. Mœglin, Idéaux complètement premiers de l’algèbre enveloppante de $\textrm {gl}_n(\textbf {C})$, J. Algebra 106 (1987), no. 2, 287–366 (French). MR 880962, DOI 10.1016/0021-8693(87)90001-9
Alexander Premet, Special transverse slices and their enveloping algebras, Adv. Math. 170 (2002), no. 1, 1–55. With an appendix by Serge Skryabin. MR 1929302, DOI 10.1006/aima.2001.2063
A. Premet, Primitive ideals, non-restricted representations and finite $W$-algebras, Mosc. Math. J. 7, 743–762.
A. Premet, Modular Representations and Finite $W$-algebras, MIMS Eprint 2010.38, (2010).
Alexander Premet, Commutative quotients of finite $W$-algebras, Adv. Math. 225 (2010), no. 1, 269–306. MR 2669353, DOI 10.1016/j.aim.2010.02.020
B. Shu and Y. Zeng, Centers and Azumaya loci of finite W-algebras, arXiv:1710.05514 (2017).
Lewis W. Topley, Invariants of centralisers in positive characteristic, J. Algebra 399 (2014), 1021–1050. MR 3144625, DOI 10.1016/j.jalgebra.2013.11.003