Publications

  • Frenkel, I. B. Spinor representations of affine Lie algebras. Proc. Nat. Acad. Sci. U.S.A. 77 (1980), no. 11, part 1, 6303–6306.
  • Frenkel, I. B.; Kac, V. G. Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62 (1980/81), no. 1, 23–66.
  • Frenkel, Igor Borisovich ORBITAL THEORY FOR AFFINE LIE ALGEBRAS. Thesis (Ph.D.)–Yale University. 1980. 98 pp, ProQuest LLC
  • Frenkel, I. B. Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory. J. Functional Analysis 44 (1981), no. 3, 259–327.
  • Frenkel, I. B. Representations of affine Lie algebras, Hecke modular forms and Korteweg-de Vries type equations. Lie algebras and related topics (New Brunswick, N.J., 1981), pp. 71–110, Lecture Notes in Math., 933, Springer, Berlin-New York, 1982.
  • Feingold, Alex J.; Frenkel, Igor B. A hyperbolic Kac-Moody algebra and the theory of Siegel modular forms of genus 2. Math. Ann. 263 (1983), no. 1, 87–144.
  • Frenkel, I. B.; Lepowsky, J.; Meurman, A. A natural representation of the Fischer-Griess Monster with the modular function J as character. Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 10, , Phys. Sci., 3256–3260.
  • Frenkel, I. B. Orbital theory for affine Lie algebras. Invent. Math. 77 (1984), no. 2, 301–352. (Reviewer: Alex Jay Feingold)
  • Frenkel, Igor B.; Lepowsky, James; Meurman, Arne A moonshine module for the Monster. Vertex operators in mathematics and physics (Berkeley, Calif., 1983), 231–273, Math. Sci. Res. Inst. Publ., 3, Springer, New York, 1985.
  • Feingold, Alex J.; Frenkel, Igor B. Classical affine algebras. Adv. in Math. 56 (1985), no. 2, 117–172.
  • (pdf) Frenkel, I. B. Representations of Kac-Moody algebras and dual resonance models. Applications of group theory in physics and mathematical physics (Chicago, 1982), 325–353, Lectures in Appl. Math., 21, Amer. Math. Soc., Providence, RI, 1985.
  • Frenkel, I. B.; Lepowsky, J.; Meurman, A. An E8-approach to F1. Finite groups—coming of age (Montreal, Que., 1982), 99–120, Contemp. Math., 45, Amer. Math. Soc., Providence, RI, 1985.
  • Frenkel, I. B. Representations of affine Lie algebras and soliton equations (abstract only). New developments in the theory and application of solitons. Philos. Trans. Roy. Soc. London Ser. A 315 (1985), no. 1533, 391.
  • Frenkel, I. B.; Lepowsky, J.; Meurman, A. An introduction to the Monster. Workshop on unified string theories (Santa Barbara, Calif., 1985), 533–546, World Sci. Publishing, Singapore, 1986.
  • Frenkel, I. B.; Garland, H.; Zuckerman, G. J. Semi-infinite cohomology and string theory. Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 22, 8442–8446.
  • Frenkel, I. B.; Lepowsky, J.; Meurman, A. Vertex operator calculus. Mathematical aspects of string theory (San Diego, Calif., 1986), 150–188, Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987.
  • Frenkel, I. B. Beyond affine Lie algebras. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 821–839, Amer. Math. Soc., Providence, RI, 1987.
  • Frenkel, Igor B.; Jing, Nai Huan Vertex representations of quantum affine algebras. Proc. Nat. Acad. Sci. U.S.A. 85 (1988), no. 24, 9373–9377.
  • Frenkel, Igor; Lepowsky, James; Meurman, Arne Vertex operator algebras and the Monster. Pure and Applied Mathematics, 134. Academic Press, Inc., Boston, MA, 1988. liv+508 pp.
  • Feingold, Alex J.; Ries, John F. X.; Frenkel, Igor B. The exceptional affine algebra E(1)8, triality and chiral algebras. Lie algebra and related topics (Madison, WI, 1988), 67–73, Contemp. Math., 110, Amer. Math. Soc., Providence, RI, 1990.
  • Frenkel, Igor; Moore, Gregory Simplex equations and their solutions. Comm. Math. Phys. 138 (1991), no. 2, 259–271.
  • Feingold, Alex J.; Frenkel, Igor B.; Ries, John F. X. Spinor construction of vertex operator algebras, triality, and E(1)8. Contemporary Mathematics, 121. American Mathematical Society, Providence, RI, 1991. x+146 pp.
  • Frenkelʹ, I.; Lepovski, Ĭ.; Merman, A. Introduction from the book Vertex operator algebras and the Monster [pp. xv–l, Academic Press, Boston, MA, 1988; MR0996026]. (Russian) Funktsional. Anal. i Prilozhen. 25 (1991), no. 4, 36–52.
  • Frenkel, Igor B.; Zhu, Yongchang Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66 (1992), no. 1, 123–168.
  • Frenkel, I. B.; Reshetikhin, N. Yu. Quantum affine algebras and holonomic difference equations. Comm. Math. Phys. 146 (1992), no. 1, 1–60.
  • Frenkel, I. B.; Reshetikhin, N. Yu. Quantum affine algebras, commutative systems of difference equations and elliptic solutions to the Yang-Baxter equations. Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, Vol. 1, 2 (New York, 1991), 46–107, World Sci. Publ., River Edge, NJ, 1992.
  • Frenkel, Igor B.; Huang, Yi-Zhi; Lepowsky, James On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104 (1993), no. 494, viii+64 pp.
  • Feingold, Alex J.; Frenkel, Igor B.; Ries, John F. X. Representations of hyperbolic Kac-Moody algebras. J. Algebra 156 (1993), no. 2, 433–453.
  • Ding, Jin Tai; Frenkel, Igor B. Isomorphism of two realizations of quantum affine algebra Uq(gl(n)). Comm. Math. Phys. 156 (1993), no. 2, 277–300.
  • Crane, Louis; Frenkel, Igor B. Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. Topology and physics. J. Math. Phys. 35 (1994), no. 10, 5136–5154.
  • Etingof, Pavel I.; Frenkel, Igor B. Central extensions of current groups in two dimensions. Comm. Math. Phys. 165 (1994), no. 3, 429–444. (Reviewer: C. J. Atkin)
  • Ding, Jin Tai; Frenkel, Igor B. Spinor and oscillator representations of quantum groups. Lie theory and geometry, 127–165, Progr. Math., 123, Birkhäuser Boston, Boston, MA, 1994.
  • Etingof, Pavel I.; Frenkel, Igor B.; Kirillov, Alexander A., Jr. Spherical functions on affine Lie groups. Duke Math. J. 80 (1995), no. 1, 59–90.
  • Frenkel, Igor B.; Turaev, Vladimir G. Trigonometric solutions of the Yang-Baxter equation, nets, and hypergeometric functions. Functional analysis on the eve of the 21st century, Vol. 1 (New Brunswick, NJ, 1993), 65–118, Progr. Math., 131, Birkhäuser Boston, Boston, MA, 1995. (Reviewer: Louis H. Kauffman)
  • Frenkel, Igor B.; Khesin, Boris A. Four-dimensional realization of two-dimensional current groups. Comm. Math. Phys. 178 (1996), no. 3, 541–562.
  • Frenkel, Igor B.; Turaev, Vladimir G. Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions. The Arnold-Gelfand mathematical seminars, 171–204, Birkhäuser Boston, Boston, MA, 1997.
  • Frenkel, Igor B.; Khovanov, Mikhail G. Canonical bases in tensor products and graphical calculus for Uq(sl2). Duke Math. J. 87 (1997), no. 3, 409–480.
  • Frenkel, Igor; Kirillov, Alexander, Jr.; Varchenko, Alexander Canonical basis and homology of local systems. Internat. Math. Res. Notices 1997, no. 16, 783–806.
  • Etingof, Pavel I.; Frenkel, Igor B.; Kirillov, Alexander A., Jr. Lectures on representation theory and Knizhnik-Zamolodchikov equations. Mathematical Surveys and Monographs, 58. American Mathematical Society, Providence, RI, 1998. xiv+198 pp.
  • Beck, Jonathan; Frenkel, Igor B.; Jing, Naihuan Canonical basis and Macdonald polynomials. Adv. Math. 140 (1998), no. 1, 95–127.
  • Frenkel, I. B.; Khovanov, M. G.; Kirillov, A. A., Jr. Kazhdan-Lusztig polynomials and canonical basis. Transform. Groups 3 (1998), no. 4, 321–336.
  • Bernstein, Joseph; Frenkel, Igor; Khovanov, Mikhail A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl2) via projective and Zuckerman functors. Selecta Math. (N.S.) 5 (1999), no. 2, 199–241.
  • Malikov, F. G.; Frenkelʹ, I. B. Annihilating ideals and tilting functors. (Russian) Funktsional. Anal. i Prilozhen. 33 (1999), no. 2, 31–42, 95; translation in Funct. Anal. Appl. 33 (1999), no. 2, 106–115
  • Frenkel, Igor B.; Jing, Naihuan; Wang, Weiqiang Vertex representations via finite groups and the McKay correspondence. Internat. Math. Res. Notices 2000, no. 4, 195–222.
  • Frenkel, Igor B.; Jing, Naihuan; Wang, Weiqiang Quantum vertex representations via finite groups and the McKay correspondence. Comm. Math. Phys. 211 (2000), no. 2, 365–393.
  • Frenkel, Igor B.; Wang, Weiqiang Virasoro algebra and wreath product convolution. J. Algebra 242 (2001), no. 2, 656–671.
  • Frenkel, Igor; Malkin, Anton; Vybornov, Maxim Affine Lie algebras and tame quivers. Selecta Math. (N.S.) 7 (2001), no. 1, 1–56.
  • Frenkel, Igor B.; Jing, Naihuan; Wang, Weiqiang Twisted vertex representations via spin groups and the McKay correspondence. Duke Math. J. 111 (2002), no. 1, 51–96.
  • Frenkel, Igor B.; Savage, Alistair Bases of representations of type A affine Lie algebras via quiver varieties and statistical mechanics. Int. Math. Res. Not. 2003, no. 28, 1521–1547.
  • Frenkel, Igor B.; Jardim, Marcos Quantum instantons with classical moduli spaces. Comm. Math. Phys. 237 (2003), no. 3, 471–505.
  • Frenkel, Igor; Malkin, Anton; Vybornov, Maxim Quiver varieties, affine Lie algebras, algebras of BPS states, and semicanonical basis. Algebraic combinatorics and quantum groups, 11–29, World Sci. Publ., River Edge, NJ, 2003.
  • Frenkel, Igor; Khovanov, Mikhail; Schiffmann, Olivier Homological realization of Nakajima varieties and Weyl group actions. Compos. Math. 141 (2005), no. 6, 1479–1503.
  • Frenkel, Igor B.; Styrkas, Konstantin Modified regular representations of affine and Virasoro algebras, VOA structure and semi-infinite cohomology. Adv. Math. 206 (2006), no. 1, 57–111.
  • Benkart, Georgia; Frenkel, Igor; Kang, Seok-Jin; Lee, Hyeonmi Level 1 perfect crystals and path realizations of basic representations at q=0. Int. Math. Res. Not. 2006, Art. ID 10312, 28 pp.
  • Frenkel, Igor; Khovanov, Mikhail; Stroppel, Catharina A categorification of finite-dimensional irreducible representations of quantum sl2 and their tensor products. Selecta Math. (N.S.) 12 (2006), no. 3-4, 379–431.
  • Frenkel, Igor B.; Todorov, Andrey N. Complex counterpart of Chern-Simons-Witten theory and holomorphic linking. Adv. Theor. Math. Phys. 11 (2007), no. 4, 531–590.
  • Frenkel, Igor B.; Jardim, Marcos Complex ADHM equations and sheaves on P3. J. Algebra 319 (2008), no. 7, 2913–2937.
  • Frenkel, Igor; Libine, Matvei Quaternionic analysis, representation theory and physics. Adv. Math. 218 (2008), no. 6, 1806–1877.
  • Frenkel, Igor B.; Zeitlin, Anton M. Quantum groups as semi-infinite cohomology. Comm. Math. Phys. 297 (2010), no. 3, 687–732.
  • Feigin, Boris; Finkelberg, Michael; Frenkel, Igor; Rybnikov, Leonid Gelfand-Tsetlin algebras and cohomology rings of Laumon spaces. Selecta Math. (N.S.) 17 (2011), no. 2, 337–361.
  • Frenkel, Igor; Libine, Matvei Split quaternionic analysis and separation of the series for SL(2,R) and SL(2,C)/SL(2,R). Adv. Math. 228 (2011), no. 2, 678–763.
  • Duncan, John F. R.; Frenkel, Igor B. Rademacher sums, moonshine and gravity. Commun. Number Theory Phys. 5 (2011), no. 4, 849–976.
  • Frenkel, Igor B.; Kim, Hyun Kyu Quantum Teichmüller space from the quantum plane. Duke Math. J. 161 (2012), no. 2, 305–366.
  • Frenkel, Igor; Zhu, Minxian Vertex algebras associated to modified regular representations of the Virasoro algebra. Adv. Math. 229 (2012), no. 6, 3468–3507.
  • Frenkel, Igor; Stroppel, Catharina; Sussan, Joshua Categorifying fractional Euler characteristics, Jones-Wenzl projectors and 3j-symbols. Quantum Topol. 3 (2012), no. 2, 181–253.
  • Frenkel, Igor; Libine, Matvei Quaternionic analysis and the Schrödinger model for the minimal representation of O(3,3). Int. Math. Res. Not. IMRN 2012, no. 21, 4904–4923.
  • Frenkel, Igor B.; Zeitlin, Anton M. Quantum group GLq(2) and quantum Laplace operator via semi-infinite cohomology. J. Noncommut. Geom. 7 (2013), no. 4, 1007–1026.
  • Frenkel, Igor B.; Zeitlin, Anton M. On the continuous series for sl(2,R)ˆ. Comm. Math. Phys. 326 (2014), no. 1, 145–165.
  • Frenkel, Igor B.; Ip, Ivan C. H. Positive representations of split real quantum groups and future perspectives. Int. Math. Res. Not. IMRN 2014, no. 8, 2126–2164.
  • Frenkel, Igor; Libine, Matvei Anti de Sitter deformation of quaternionic analysis and the second-order pole. Int. Math. Res. Not. IMRN 2015, no. 13, 4840–4900.
  • Frenkel, Igor; Penkov, Ivan; Serganova, Vera A categorification of the boson-fermion correspondence via representation theory of sl(∞). Comm. Math. Phys. 341 (2016), no. 3, 911–931.
  • Frenkel, Igor; Linine, Matvei, Quaternionic Analysis, Representation Theory and Physics II, arXiv:1907.01594
  • Frenkel, Igor; Libine, Matvei n-regular functions in quaternionic analysis. Internat. J. Math. 32 (2021), no. 2, 2150008, 30 pp.
  • Frenkel, Igor; Penner, Robert, Sketch of a Program for Universal Automorphic Functions to Capture Monstrous Moonshine, arXiv:2012.14220.