On the functor of Arakawa, Suzuki and Tsuchiya

S. M. Khoroshkin; Nazarov M.

doi:10.2969/aspm/07610000

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On the functor of Arakawa, Suzuki and Tsuchiya

Ch. 8. P. 275–302.

Khoroshkin S. M., Nazarov M.

Arakawa, Suzuki and Tsuchiya defined a correspondence between certain modules of the trigonometric Cherednik algebra CN depending on a parameter κ∈C, and certain modules of the affine Lie algebra slˆm of level κ−m. We give a detailed proof of this correspondence by working with the affine Lie algebra glˆm alongside of slˆm. We also relate this construction to a correspondence between certain modules of the degenerate affine Hecke algebra HN and all modules of sl^m or gl^m. The latter correspondence was constructed earlier by Cherednik

Language: English

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Keywords: cherednik algebras Affine Lie algebra

In book

Advanced Studies in Pure Mathematics (т.76 Representations Theory, Special Functions and Painleve Equations - RIMS 2015)

Vol. 76. , Tokyo: Mathematical Society of Japan, 2018.

Stochastic local volatility models and the Wei-Norman factorization method

Guerrero J., Orlando G., Discrete and Continuous Dynamical Systems - Series S 2022 Vol. 15 No. 12 P. 3699–3722

In this paper, we show that a time-dependent local stochastic volatility (SLV) model can be reduced to a system of autonomous PDEs that can be solved using the heat kernel, by means of the Wei-Norman factorization method and Lie algebraic techniques. Then, we compare the results of traditional Monte Carlo simulations with the explicit solutions ...

Added: February 23, 2024

Cyclotomic double affine Hecke algebras

Braverman A., Etingof P., Michael Finkelberg, Annales Scientifiques de l'Ecole Normale Superieure 2020 Vol. 53 No. 5 P. 1249–1312

We show that the partially spherical cyclotomic rational Cherednik algebra (obtained from the full rational Cherednik algebra by averaging out the cyclotomic part of the underlying reflection group) has four other descriptions: (1) as a subalgebra of the degenerate DAHA of type A given by generators; (2) as an algebra given by generators and relations; (3) as ...

Added: December 8, 2020

Beilinson-Drinfeld Schubert varieties and global Demazure modules

Dumanski I., Feigin E., Finkelberg M. V., / Series math "arxiv.org". 2020. No. 2003.12930.

We compute the spaces of sections of powers of the determinant line bundle on the spherical Schubert subvarieties of the Beilinson- Drinfeld affine Grassmannians. The answer is given in terms of global Demazure modules of the current Lie algebra. ...

Added: April 2, 2020

Cherednik algebras and Zhelobenko operators

Khoroshkin S. M., Nazarov M., Transformation Groups 2018 Vol. 23 No. 1 P. 119–147

We study canonical intertwining operators between induced modules of the trigonometric Cherednik algebra. We demonstrate that these operators correspond to the Zhelobenko operators for the affine Lie algebra . To establish the correspondence, we use the functor of Arakawa, Suzuki and Tsuchiya which maps certain modules to modules of the Cherednik algebra. ...

Added: August 1, 2018

Conjecture on theta-blocks of order 1

Valery Gritsenko, Wang H., Russian Mathematical Surveys 2017 Vol. 72 No. 5 P. 968–970

In this paper we prove the indicated conjecture in the last case of known infinite series of theta-blocks of weight 2. ...

Added: January 29, 2018

Гипотеза о тэта-блоках первого порядка

Gritsenko V., Ванг Х., Успехи математических наук 2017 Т. 72 № 5 С. 191–192

In this paper we prove the conjecture above in the last case of known theta-blocks of weight 2. This gives a new intereting series of Borcherds products of weight 2. ...

Added: October 11, 2017

A Combinatorial Formula for Affine Hall-Littlewood Functions via a Weighted Brion Theorem

Feigin B. L., Makhlin I., / Series math "arxiv.org". 2015.

We present a new combinatorial formula for Hall-Littlewood functions associated with the affine root system of type A~n−1, i.e. corresponding to the affine Lie algebra slˆn. Our formula has the form of a sum over the elements of a basis constructed by Feigin, Jimbo, Loktev, Miwa and Mukhin in the corresponding irreducible representation.Our formula can ...

Added: August 7, 2015

Characters of Feigin-Stoyanovsky subspaces and Brion's theorem

Makhlin I., Functional Analysis and Its Applications 2015 Vol. 49 No. 1 P. 15–24

We give an alternative proof of the main result of [1]; the proof relies on Brion’s theorem about convex polyhedra. The result itself can be viewed as a formula for the character of the Feigin-Stoyanovsky subspace of an integrable irreducible representation of the affine Lie algebra widehatsln(C). Our approach is to assign integer points of a ...

Added: August 7, 2015

24 faces of the Borcherds modular form Phi_{12}

Gritsenko V., / Series math "arxiv.org". 2012. No. 6503.

The fake monster Lie algebra is determined by the Borcherds function Phi_{12} which is the reflective modular form of the minimal possible weight with respect to O(II_{2,26}). We prove that the first non-zero Fourier-Jacobi coefficient of Phi_{12} in any of 23 Niemeier cusps is equal to the Weyl-Kac denominator function of the affine Lie algebra ...

Added: March 3, 2015

Degenerate affine Grassmannians and loop quivers

Feigin E., Finkelberg M. V., Reineke M., / Series math "arxiv.org". 2014. No. 1410.0777.

We study the connection between the affine degenerate Grassmannians in type $A$, quiver Grassmannians for one vertex loop quivers and affine Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type $GL_n$ and identify it with semi-infinite orbit closure of type $A_{2n-1}$. We show that principal quiver Grassmannians for the one vertex loop ...

Added: October 6, 2014

Finite-dimensional subalgebras in polynomial Lie algebras of rank one.

Arzhantsev I., Makedonskii E. A., Petravchuk A. P., Ukrainian Mathematical Journal 2011 Vol. 63 No. 5 P. 708–712

Added: July 10, 2014

Коммутативные вертексные алгебры и их вырождения

Feigin B. L., Функциональный анализ и его приложения 2014 № 3

We study commutative vertex operator algebras. These algebras are isomorphic to certain subalgebras in Kac-Moody vertex operator algebras. We describe systems of relations and degenerations to quadratic algebras. Our approach leads to the fermionic formulas for characters. ...

Added: April 14, 2014