About

Igor Frenkel is the Sol Goldman Family Professor of Mathematics at Yale University. His research focuses on infinite-dimensional algebras, representation theory, and mathematical physics. He is a member of the American Academy of Arts and Sciences and the National Academy of Sciences. Professor Frenkel received his Ph.D. from Yale in 1980 and is recognized for his significant contributions to his fields of study.

Research topics

• Mathematics
• Physics
• Pure mathematics
• Mathematical physics
• Combinatorics
• Geometry
• Theoretical physics
• Quantum mechanics

Selected publications

• Quasi regular functions in quaternionic analysis

  Journal of Functional Analysis · 2026-02-10

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• Perspectives on the past, present, and future of representation theory and mathematical physics: An interview of Igor Frenkel by Joshua Sussan

  Contemporary mathematics - American Mathematical Society · 2025-01-01

  other1st authorCorresponding
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• Quasi Regular Functions in Quaternionic Analysis

  arXiv (Cornell University) · 2024-02-11

  preprintOpen access1st authorCorresponding

  We study a new class of functions that arise naturally in quaternionic analysis, we call them "quasi regular functions". Like the well-known quaternionic regular functions, these functions provide representations of the quaternionic conformal group. However, unlike the regular functions, the quasi regular ones do not admit an invariant unitary structure but rather a pseudounitary equivalent. The reproducing kernels of these functions have an especially simple form: (Z-W)^{-1}. We describe the K-type bases of quasi regular functions and derive the reproducing kernel expansions. We also show that the restrictions of the irreducible representations formed from the quasi regular functions to the Poincare group have three irreducible components. Our interest in the quasi regular functions arises from an application to the study of conformal-invariant algebras of quaternionic functions. We also introduce a factorization of certain intertwining operators between tensor products of spaces of quaternionic functions. This factorization is obtained using fermionic Fock spaces constructed from the quasi regular functions.

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• Sketch of a Program for Automorphic Functions from Universal Teichmüller Theory to Capture Monstrous Moonshine

  Communications in Mathematical Physics · 2022 · 5 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics
  • Combinatorics
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• Three dimensional construction of the Virasoro-Bott group

  arXiv (Cornell University) · 2021-07-24

  preprintOpen access1st authorCorresponding

  We present a three-dimensional geometric construction of the Virasoro-Bott group, which is a central extension of the group of diffeomorphisms of the circle. Our approach is analogous to the well-known construction of a central extension of the loop group by means of the Wess-Zumino topological term. In particular, the Virasoro-Bott group is realized as a quotient group of diffeomorphisms of the disc with special boundary conditions. We identify the Lie algebra corresponding to our group with the Virasoro algebra. We also show that for generalized boundary conditions the Virasoro algebra is extended to a semidirect product with the Heisenberg algebra. We discuss the relation between our construction, the Chern-Simons theory, and the three-dimensional gravity.

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• Canonical basis and homology of local systems

  UNC Libraries · 2021-08-14

  articleOpen access1st authorCorresponding

  Let Uqsl2 be the quantum group corresponding to the Lie algebra sl2, and let Mλ be the Verma module over the Uqsl2 with the highest weight λ. It follows from the results of [V1] that one can identify the weight space M[μ] = (Mλ1 ⊗ . . . ⊗ Mλn)[μ], μ = Σλi − 2l with a suitable homology space of the configuration space Xn,l = {x ∈ Cl; xi ≠ xj , xi ≠ 0, xi ≠ zk, k = 1, . . . , n} with coefficients in a certain one-dimensional local system. This result plays a crucial role in the construction of quantum group symmetries in conformal field theory, since this homology space naturally appears in the integral formulas for the solutions of Knizhnik-Zamolodchikov equations (see [SV1, SV2]).

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• Quaternionic analysis, representation theory and physics II

  Advances in Theoretical and Mathematical Physics · 2021 · 18 citations

  1st authorCorresponding
  • Physics
  • Mathematical physics
  • Theoretical physics

  We develop further quaternionic analysis introducing left and right doubly regular functions. We derive Cauchy-Fueter type formulas for these doubly regular functions that can be regarded as another counterpart of Cauchy's integral formula for the second order pole, in addition to the one studied in the first paper with the same title. We also realize the doubly regular functions as a subspace of the quaternionic-valued functions satisfying a Euclidean version of Maxwell's equations for the electromagnetic field. Then we return to the study of the original quaternionic analogue of Cauchy's second order pole formula and its relation to the polarization of vacuum. We find the decomposition of the space of quaternionic-valued functions into irreducible components that include the spaces of doubly left and right regular functions. Using this decomposition, we show that a regularization of the vacuum polarization diagram is achieved by subtracting the component corresponding to the one-dimensional subrepresentation of the conformal group. After the regularization, the vacuum polarization diagram is identified with a certain second order differential operator which yields a quaternionic version of Maxwell equations. Next, we introduce two types of quaternionic algebras consisting of spaces of scalar-valued and quaternionic-valued functions. We emphasize that these algebra structures are invariant under the action of the conformal Lie algebra. This uses techniques from our study of the vacuum polarization diagram. These algebras are not associative, but we can define an infinite family of n-multiplications, and we conjecture that they have structures of weak cyclic A-infinity algebras. We also conjecture the relation between the multiplication operations of the scalar and non-scalar quaternionic algebras with the n-photon Feynman diagrams in the scalar and ordinary conformal QED.

  PublisherDOI

• Sketch of a Program for Universal Automorphic Functions to Capture Monstrous Moonshine

  arXiv (Cornell University) · 2020 · 3 citations

  1st authorCorresponding
  • Mathematics
  • Combinatorics
  • Pure mathematics

  We review and reformulate old and prove new results about the triad $ {\rm PPSL}_2({\mathbb Z})\subseteq{\rm PPSL}_2({\mathbb R})\circlearrowright ppsl_2({\mathbb R}) $, which provides a universal generalization of the classical automorphic triad ${\rm PSL}_2({\mathbb Z})\subseteq{\rm PSL}_2({\mathbb R})\circlearrowright psl_2({\mathbb R})$. The leading P or $p$ in the universal setting stands for $piecewise$, and the group ${\rm PPSL}_2({\mathbb Z})$ plays at once the role of universal modular group, universal mapping class group, Thompson group $T$ and Ptolemy group. We produce a new basis of the Lie algebra $ppsl_2({\mathbb R})$, compute its structure constants, define a central extension which is compared with the Weil-Petersson 2-form, and discuss its representation theory. We construct and study new framed holographic coordinates on the universal Teichmüller space and its symmetry group ${\rm PPSL}_2({\mathbb R})$, and construct an invariant 1-form as its Maurer-Cartan form analogous to the invariant Eisenstein 1-form $E_2(z)dz$, which gives rise to the spin 1 representation of $psl_2({\mathbb R})$ extended by the trivial representation. This suggests the full program for developing the theory of universal automorphic functions conjectured to yield the bosonic CFT$_2$. Relaxing the automorphic condition to the commutant leads to our ultimate conjecture on realizing the Monster CFT$_2$ via the automorphic representation for the universal triad. This conjecture is also bolstered by the links of both the universal Teichmüller and the Monster CFT$_2$ theories to the three-dimensional quantum gravity.

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• n-Regular functions in quaternionic analysis

  International Journal of Mathematics · 2020-12-19

  preprintOpen access1st authorCorresponding

  In this paper, we study left and right [Formula: see text]-regular functions that originally were introduced in [I. Frenkel and M. Libine, Quaternionic analysis, representation theory and physics II, accepted in Adv. Theor. Math. Phys]. When [Formula: see text], these functions are the usual quaternionic left and right regular functions. We show that [Formula: see text]-regular functions satisfy most of the properties of the usual regular functions, including the conformal invariance under the fractional linear transformations by the conformal group and the Cauchy–Fueter type reproducing formulas. Arguably, these Cauchy–Fueter type reproducing formulas for [Formula: see text]-regular functions are quaternionic analogues of Cauchy’s integral formula for the [Formula: see text]th-order pole [Formula: see text] We also find two expansions of the Cauchy–Fueter kernel for [Formula: see text]-regular functions in terms of certain basis functions, we give an analogue of Laurent series expansion for [Formula: see text]-regular functions, we construct an invariant pairing between left and right [Formula: see text]-regular functions and we describe the irreducible representations associated to the spaces of left and right [Formula: see text]-regular functions of the conformal group and its Lie algebra.

  PublisherOA PDFDOI

• Quaternionic Analysis, Representation Theory and Physics II

  arXiv (Cornell University) · 2019-07-02

  preprintOpen access1st authorCorresponding

  We develop further quaternionic analysis introducing left and right doubly regular functions. We derive Cauchy-Fueter type formulas for these doubly regular functions that can be regarded as another counterpart of Cauchy's integral formula for the second order pole, in addition to the one studied in the first paper with the same title. We also realize the doubly regular functions as a subspace of the quaternionic-valued functions satisfying a Euclidean version of Maxwell's equations for the electromagnetic field. Then we return to the study of the original quaternionic analogue of Cauchy's second order pole formula and its relation to the polarization of vacuum. We find the decomposition of the space of quaternionic-valued functions into irreducible components that include the spaces of doubly left and right regular functions. Using this decomposition, we show that a regularization of the vacuum polarization diagram is achieved by subtracting the component corresponding to the one-dimensional subrepresentation of the conformal group. After the regularization, the vacuum polarization diagram is identified with a certain second order differential operator which yields a quaternionic version of Maxwell equations. Next, we introduce two types of quaternionic algebras consisting of spaces of scalar-valued and quaternionic-valued functions. We emphasize that these algebra structures are invariant under the action of the conformal Lie algebra. This uses techniques from our study of the vacuum polarization diagram. These algebras are not associative, but we can define an infinite family of n-multiplications, and we conjecture that they have structures of weak cyclic A-infinity algebras. We also conjecture the relation between the multiplication operations of the scalar and non-scalar quaternionic algebras with the n-photon Feynman diagrams in the scalar and ordinary conformal QED.

  PublisherOA PDFDOI

Recent grants

• Representation Theory of Infinite Dimensional Lie Aglebras and Quantum Field Theory

  NSF · $429k · 2005–2011

• Representation Theory and Physical Models

  NSF · $429k · 2010–2015

Frequent coauthors

• Pavel Etingof

  17 shared

• Matvei Libine

  16 shared

• Alex J. Feingold

  Binghamton University

  15 shared

• Alexander Kirillov

  14 shared

• Naihuan Jing

  University of Sydney

  13 shared

• John F. X. Ries

  13 shared

• James Lepowsky

  12 shared

• Mikhail Khovanov

  10 shared

Labs

• Igor Frenkel LabPI

Awards & honors

• Sloan Fellowship
• Guggenheim Fellowship
• Weyl-Wigner Award