Evgeny Mukhin
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Research Interests
Evgeny Mukhin has a total of 47 co-authors. Here are the top co-authors by number of shared publications:
- Alexander Varchenko
Department of Mathematics University of North Carolina at Chapel Hill 27599-3250 Chapel Hill NC USA
26 shared publications - Boris Feigin
Hebrew University of Jerusalem, Einstein Institute of Mathematics, Jerusalem, Israel
23 shared publications - Michio Jimbo
Department of Mathematics, Rikkyo University, Toshima-ku, Japan
20 shared publications - Michio Jimbo
Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan
14 shared publications - B. Feigin
Landau Institute for Theoretical Physics
10 shared publications - Vitaly Tarasov
Department of Mathematical Sciences Indiana University -Purdue University Indianapolis 402 North Blackford St 46202-3216 Indianapolis IN USA
10 shared publications
Latest Publications
- 28 October 2026
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 407 (12) , 253
Affinization of Shifted Quantum Affine $\mathfrak {gl}_2$- 28 October 2026
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 407 (12) , 253
Affinization of Shifted Quantum Affine $\mathfrak {gl}_2$- 11 May 2026
- Published by Springer Nature in Annales Henri Poincare
- p. 1-43
Extensions of a Commuting Pair of Quantum Toroidal $\mathfrak {gl}_1$- 11 May 2026
- Published by Springer Nature in Annales Henri Poincare
- p. 1-43
Extensions of a Commuting Pair of Quantum Toroidal $\mathfrak {gl}_1$- 2 January 2026
- Published by SIGMA (Symmetry, Integrability and Geometry: Methods and Application) in Symmetry, Integrability and Geometry: Methods and Applications
Finding All Solutions of qKZ Equations in Characteristic $p$In [J. Lond. Math. Soc. 109 (2024), e12884, 22 pages], the difference qKZ equations were considered modulo a prime number $p$ and a family of polynomial solutions of the qKZ equations modulo $p$ was constructed by an elementary procedure as suitable $p$-approximations of the hypergeometric integrals. In this paper, we study in detail the first family of nontrivial examples of the qKZ equations in characteristic $p$. We describe all solutions of these qKZ equations in characteristic $p$ by demonstrating that they all stem from the $p$-hypergeometric solutions. We also prove a Lagrangian property (called the orthogonality property) of the subbundle of the qKZ bundle spanned by the $p$-hypergeometric sections. This paper extends the results of [arXiv:2405.05159] on the differential KZ equations to the difference qKZ equations.- 2 January 2026
- Published by SIGMA (Symmetry, Integrability and Geometry: Methods and Application) in Symmetry, Integrability and Geometry: Methods and Applications
Finding All Solutions of qKZ Equations in Characteristic $p$In [J. Lond. Math. Soc. 109 (2024), e12884, 22 pages], the difference qKZ equations were considered modulo a prime number $p$ and a family of polynomial solutions of the qKZ equations modulo $p$ was constructed by an elementary procedure as suitable $p$-approximations of the hypergeometric integrals. In this paper, we study in detail the first family of nontrivial examples of the qKZ equations in characteristic $p$. We describe all solutions of these qKZ equations in characteristic $p$ by demonstrating that they all stem from the $p$-hypergeometric solutions. We also prove a Lagrangian property (called the orthogonality property) of the subbundle of the qKZ bundle spanned by the $p$-hypergeometric sections. This paper extends the results of [arXiv:2405.05159] on the differential KZ equations to the difference qKZ equations.- Open AccessAffinization of shifted quantum affine $\mathfrak{gl}_2$We give a realization $\mathcal{A}_0$ of quantum toroidal algebra associated to $\mathfrak{gl}_2$ which can be viewed as an affinization of the Drinfeld new realization of quantum affine $\mathfrak{gl}_2$. We use this realization to define an affinization $\mathcal{A}_N$, $N\in{\mathbb Z}$, of shifted quantum affine $\mathfrak{gl}_2$. We construct a large family of representations of dominantly shifted algebra $\mathcal A_N$, $N>0$. The examples of representations with even positive $N$ appear in the study of extensions of deformed $W$-algebras of type $\mathfrak{gl}(N+2|1)$.Open AccessAffinization of shifted quantum affine $\mathfrak{gl}_2$We give a realization $\mathcal{A}_0$ of quantum toroidal algebra associated to $\mathfrak{gl}_2$ which can be viewed as an affinization of the Drinfeld new realization of quantum affine $\mathfrak{gl}_2$. We use this realization to define an affinization $\mathcal{A}_N$, $N\in{\mathbb Z}$, of shifted quantum affine $\mathfrak{gl}_2$. We construct a large family of representations of dominantly shifted algebra $\mathcal A_N$, $N>0$. The examples of representations with even positive $N$ appear in the study of extensions of deformed $W$-algebras of type $\mathfrak{gl}(N+2|1)$.
- 17 September 2025
- Published by Springer Nature in Letters in Mathematical Physics
- Vol. 115 (5) , 1-87
Q-functions for lambda opers
- 17 September 2025
- Published by Springer Nature in Letters in Mathematical Physics
- Vol. 115 (5) , 1-87
Q-functions for lambda opers
- Open Access$p$-curvature operators and Satake-type phenomenon for $\frak{sl}_2$ KZ equations with $κ=\pm 2$The $\frak{sl}_2$ KZ differential equations with values in the tensor power of the fundamental representation with parameter $κ=\pm 2$ are considered. A Satake-type correspondence is established over complex numbers and subsequently reduced to finite characteristic. This correspondence enables the study of the KZ equations on the lower weight subspaces of the tensor power in terms of the wedge powers of the weight subspace of the weight just below the highest weight. We apply this approach to analyze the $p$-curvature operators associated with our KZ equations, evaluate the dimension of the solution space in characteristic $p$, and determine whether all solutions are generated by the so-called $p$-hypergeometric solutions. In particular, we show that not all solutions of the KZ equations with $κ=2$ in characteristic $p$ are generated by $p$-hypergeometric solutions. Previously, no such examples were known.Open Access$p$-curvature operators and Satake-type phenomenon for $\frak{sl}_2$ KZ equations with $κ=\pm 2$The $\frak{sl}_2$ KZ differential equations with values in the tensor power of the fundamental representation with parameter $κ=\pm 2$ are considered. A Satake-type correspondence is established over complex numbers and subsequently reduced to finite characteristic. This correspondence enables the study of the KZ equations on the lower weight subspaces of the tensor power in terms of the wedge powers of the weight subspace of the weight just below the highest weight. We apply this approach to analyze the $p$-curvature operators associated with our KZ equations, evaluate the dimension of the solution space in characteristic $p$, and determine whether all solutions are generated by the so-called $p$-hypergeometric solutions. In particular, we show that not all solutions of the KZ equations with $κ=2$ in characteristic $p$ are generated by $p$-hypergeometric solutions. Previously, no such examples were known.
- 1 May 2025
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 66 (5)
Intertwiners of representations of twisted quantum affine algebras
We use the q-characters to compute explicit expressions of the R-matrices for first fundamental representations of all types of twisted quantum affine algebras.
- 1 May 2025
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 66 (5)
Intertwiners of representations of twisted quantum affine algebras
We use the q-characters to compute explicit expressions of the R-matrices for first fundamental representations of all types of twisted quantum affine algebras.
- 1 May 2025
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 66 (5)
Intertwiners of representations of untwisted quantum affine algebras and Yangians revisited
We discuss applications of the q-characters to the computation of the R-matrices. In particular, we describe the R-matrix acting in the tensor square of the first fundamental representation of E8 and in a number of other cases, where the decomposition of the tensor squares with respect to non-affine quantum algebra has non-trivial multiplicities. As an illustration, we also recover R-matrices acting in the multiplicity free-case on the tensor squares of the first fundamental representations of all other types of untwisted quantum affine algebras. The answer is written in terms of projectors related to the decomposition of the tensor squares with respect to non-affine quantum algebras. Then we give explicit expressions for the R-matrices in terms of matrix units with respect to a natural basis (except for the case of E8). We give similar formulas for the Yangian R-matrices.
- 1 May 2025
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 66 (5)
Intertwiners of representations of untwisted quantum affine algebras and Yangians revisited
We discuss applications of the q-characters to the computation of the R-matrices. In particular, we describe the R-matrix acting in the tensor square of the first fundamental representation of E8 and in a number of other cases, where the decomposition of the tensor squares with respect to non-affine quantum algebra has non-trivial multiplicities. As an illustration, we also recover R-matrices acting in the multiplicity free-case on the tensor squares of the first fundamental representations of all other types of untwisted quantum affine algebras. The answer is written in terms of projectors related to the decomposition of the tensor squares with respect to non-affine quantum algebras. Then we give explicit expressions for the R-matrices in terms of matrix units with respect to a natural basis (except for the case of E8). We give similar formulas for the Yangian R-matrices.
- Open Access
- 22 November 2024
- Published by IOP Publishing in Journal of Physics A: Mathematical and Theoretical
- Vol. 57 (48) , 485201
Remarks on q-difference opers arising from quantum toroidal algebras
We propose a conjectural correspondence between the spectra of the Bethe algebra for the quantum toroidal gl 2 algebra on relaxed Verma modules, and q-hypergeometric opers with apparent singularities. We introduce alongside the notion of apparent singularities for linear q-difference operators and discuss some of their properties. We also touch on a generalization to gl n .
- 22 November 2024
- Published by IOP Publishing in Journal of Physics A: Mathematical and Theoretical
- Vol. 57 (48) , 485201
Open AccessRemarks on q-difference opers arising from quantum toroidal algebras
We propose a conjectural correspondence between the spectra of the Bethe algebra for the quantum toroidal gl 2 algebra on relaxed Verma modules, and q-hypergeometric opers with apparent singularities. We introduce alongside the notion of apparent singularities for linear q-difference operators and discuss some of their properties. We also touch on a generalization to gl n .
- 20 September 2024
- Published by Springer Nature in Algebras and Representation Theory
- Vol. 27 (6) , 2019-2044
The Deformed Tanisaki-Garsia-Procesi Modules
- 20 September 2024
- Published by Springer Nature in Algebras and Representation Theory
- Vol. 27 (6) , 2019-2044
The Deformed Tanisaki-Garsia-Procesi Modules