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In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups, compact matrix quantum groups, and bicrossproduct quantum groups.
Vladimir Gershonovich Drinfeld, surname also romanized as Drinfel'd, is a renowned mathematician from the former USSR, who emigrated to the United States and is currently working at the University of Chicago.
In arithmetic geometry, the Weil–Châtelet group or WC-group of an algebraic group such as an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, defined over K. John Tate (1958) named it for François Châtelet (1946) who introduced it for elliptic curves, and André Weil (1955), who introduced it for more general groups. It plays a basic role in the arithmetic of abelian varieties, in particular for elliptic curves, because of its connection with infinite descent.
A quasi-Hopf algebra is a generalization of a Hopf algebra, which was defined by the Russian mathematician Vladimir Drinfeld in 1989.
Rodney James Baxter FRS FAA is an Australian physicist, specializing in statistical mechanics. He is well known for his work in exactly solved models, in particular vertex models such as the six-vertex model and eight-vertex model, and the chiral Potts model and hard hexagon model. A recurring theme in the solution of such models, the Yang–Baxter equation, also known as the "star–triangle relation", is named in his honour.
In representation theory, a branch of mathematics, the Gelfand–Graev representation is a representation of a reductive group over a finite field introduced by Gelfand & Graev (1962), induced from a non-degenerate character of a Sylow subgroup.
The Proceedings of the USSR Academy of Sciences was a Soviet journal that was dedicated to publishing original, academic research papers in physics, mathematics, chemistry, geology, and biology. It was first published in 1933 and ended in 1992 with volume 322, issue 3.
In representation theory, a Yangian is an infinite-dimensional Hopf algebra, a type of a quantum group. Yangians first appeared in physics in the work of Ludvig Faddeev and his school in the late 1970s and early 1980s concerning the quantum inverse scattering method. The name Yangian was introduced by Vladimir Drinfeld in 1985 in honor of C.N. Yang.
Michio Jimbo is a Japanese mathematician working in mathematical physics and is a professor of mathematics at Rikkyo University. He is a grandson of the linguist Kaku Jimbo.
In mathematical physics, the quantum KZ equations or quantum Knizhnik–Zamolodchikov equations or qKZ equations are the analogue for quantum affine algebras of the Knizhnik–Zamolodchikov equations for affine Kac–Moody algebras. They are a consistent system of difference equations satisfied by the N-point functions, the vacuum expectations of products of primary fields. In the limit as the deformation parameter q approaches 1, the N-point functions of the quantum affine algebra tend to those of the affine Kac–Moody algebra and the difference equations become partial differential equations. The quantum KZ equations have been used to study exactly solved models in quantum statistical mechanics.
In arithmetic geometry, the Tate–Shafarevich groupШ(A/K), introduced by Serge Lang and John Tate (1958) and Igor Shafarevich (1959), of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group WC(A/K) = H1(GK, A) that become trivial in all of the completions of K (i.e. the p-adic fields obtained from K, as well as its real and complex completions). Thus, in terms of Galois cohomology, it can be written as
In mathematics, a grope is a construction used in 4-dimensional topology, introduced by Štan'ko (1971) and named by Cannon (1978) "because of its multitudinous fingers". Capped gropes were used by Freedman (1984) as a substitute for Casson handles, that work better for non-simply-connected 4-manifolds.
In algebra, the Kantor–Koecher–Tits construction is a method of constructing a Lie algebra from a Jordan algebra, introduced by Jacques Tits (1962), Kantor (1964), and Koecher (1967).
In mathematics, a general hypergeometric function or Aomoto–Gelfand hypergeometric function is a generalization of the hypergeometric function that was introduced by Gelfand (1986). The general hypergeometric function is a function that is defined on a Grassmannian, and depends on a choice of some complex numbers and signs.
In mathematics, a Novikov–Shubin invariant, introduced by Sergei Novikov and Mikhail Shubin (1986), is an invariant of a compact Riemannian manifold related to the spectrum of the Laplace operator acting on square-integrable differential forms on its universal cover.
In mathematical analysis, the Agranovich–Dynin formula is a formula for the index of an elliptic system of differential operators, introduced by Agranovich and Dynin (1962).
In number theory, a Parshin chain is a higher-dimensional analogue of a place of an algebraic number field. They were introduced by Parshin (1978) in order to define an analogue of the idele class group for 2-dimensional schemes.
Alexander Nikolaevich Varchenko is a Soviet and Russian mathematician working in geometry, topology, combinatorics and mathematical physics.
In mathematics, a quantum or quantized enveloping algebra is a q-analog of a universal enveloping algebra. Given a Lie algebra , the quantum enveloping algebra is typically denoted as . Among the applications, studying the limit led to the discovery of crystal bases.
Tetsuji Miwa is a Japanese mathematician, specializing in mathematical physics.
References
- Drinfeld, V. G. (1985), "Hopf algebras and the quantum Yang–Baxter equation", Doklady Akademii Nauk SSSR, 283 (5): 1060–1064, ISSN 0002-3264, MR 0802128
- Drinfeld, V. G. (1987), "A new realization of Yangians and of quantum affine algebras", Doklady Akademii Nauk SSSR, 296 (1): 13–17, ISSN 0002-3264, MR 0914215
- Frenkel, Igor B.; Reshetikhin, N. Yu. (1992), "Quantum affine algebras and holonomic difference equations", Communications in Mathematical Physics , 146 (1): 1–60, Bibcode:1992CMaPh.146....1F, doi:10.1007/BF02099206, ISSN 0010-3616, MR 1163666, S2CID 119818318
- Jimbo, Michio (1985), "A q-difference analogue of U(g) and the Yang-Baxter equation", Letters in Mathematical Physics , 10 (1): 63–69, Bibcode:1985LMaPh..10...63J, doi:10.1007/BF00704588, ISSN 0377-9017, MR 0797001, S2CID 123313856
- Jimbo, Michio; Miwa, Tetsuji (1995), Algebraic analysis of solvable lattice models, CBMS Regional Conference Series in Mathematics, 85, Published for the Conference Board of the Mathematical Sciences, Washington, DC, ISBN 978-0-8218-0320-2, MR 1308712
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