124
124

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 124

#
favorite 1

#
comment 0

Fermionic quantum mechanics. Let us now pass from �nite dimensional fermionic integrals to quantum mechanics, i.e. integrals over fermionic functions of one (even) real variable t.

Topics: Physics, Quantum Physics, Quantum Field Theory, Physics

Source: http://www.flooved.com/reader/1523

154
154

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 154

#
favorite 0

#
comment 0

We will now prove our �rst result � Schur�s lemma. Although it is very easy to prove, it is fundamental in the whole subject of representation theory.

Topics: Maths, Algebra, Representation Theory, Mathematics

Source: http://www.flooved.com/reader/1654

62
62

Sep 23, 2013
09/13

by
Pavel Etingof

texts

#
eye 62

#
favorite 0

#
comment 0

Recently, motivated by supersymmetric gauge theory, Cachazo, Douglas, Seiberg, and Witten proposed a conjecture about finite dimensional simple Lie algebras, and checked it in the classical cases. Later V. Kac and the author proposed a uniform approach to this conjecture, based on the theory of abelian ideals in the Borel subalgebra; this allowed them to check the conjecture for type G_2. In this note we further develop this approach, and propose three natural conjectures which imply the three...

Source: http://arxiv.org/abs/math/0312452v1

181
181

Sep 17, 2013
09/13

by
Pavel Etingof

texts

#
eye 181

#
favorite 0

#
comment 0

This talk is inspired by two previous ICM talks, by V.Drinfeld (1986) and G.Felder (1994). Namely, one of the main ideas of Drinfeld's talk is that the quantum Yang-Baxter equation (QYBE), which is an important equation arising in quantum field theory and statistical mechanics, is best understood within the framework of Hopf algebras, or quantum groups. On the other hand, in Felder's talk, it is explained that another important equation of mathematical physics, the star-triangle relation, may...

Source: http://arxiv.org/abs/math/0207008v3

69
69

Jul 20, 2013
07/13

by
Pavel Etingof

texts

#
eye 69

#
favorite 0

#
comment 0

Let G be a finite group of linear transformations of a finite dimensional complex vector space V. To this data one can attach a family of algebras H_{t,c}(G,V), parametrized by complex numbers t and conjugation invariant functions c on the set of complex reflections in G, which are called rational Cherednik algebras. These algebras have been studied for several years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this...

Source: http://arxiv.org/abs/math/0406499v3

111
111

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 111

#
favorite 0

#
comment 0

One may consider systems of classical particles or �elds some of which are bosonic and some fermionic. In this case, the space of states will be a supervector space, i.e. the direct sum of an even and an odd space (or, more generally, a supermanifold � a notion we will de�ne below). When such a theory is quantized using the path integral approach, one has to integrate functions over supermanifolds. Thus, we should learn to integrate over supermanifolds and then generalize to this case our...

Topics: Physics, Quantum Physics, Quantum Field Theory, Physics

Source: http://www.flooved.com/reader/1532

49
49

Sep 19, 2013
09/13

by
Pavel Etingof

texts

#
eye 49

#
favorite 0

#
comment 0

The paper introduces a new geometric interpretation of the quantum Knizhnik-Zamolodchikov equations introduced in 1991 by I.Frenkel and N.Reshetikhin. It turns out that these equations can be linked to certain holomorphic vector bundles on the N-th Cartesian power of an elliptic curve. These bundles are naturally constructed by a gluing procedure from a system of trigonometric quantum affine $R$-matrices. Meromorphic solutions of the quantum KZ equations are interpreted as sections of such a...

Source: http://arxiv.org/abs/hep-th/9303066v1

77
77

Sep 21, 2013
09/13

by
Pavel Etingof

texts

#
eye 77

#
favorite 0

#
comment 0

In this paper the quantum integrals of the Hamiltonian of the quantum many-body problem with the interaction potential K/sinh^2(x) (Sutherland operator) are constructed as images of higher Casimirs of the Lie algebra gl(N) under a certain homomorphism from the center of U(gl(N)) to the algebra of differential operators in N variables. A similar construction applied to the affine gl(N) at the critical level k=-N defines a correspondence between higher Sugawara operators and quantum integrals of...

Source: http://arxiv.org/abs/hep-th/9311132v1

64
64

Sep 21, 2013
09/13

by
Pavel Etingof

texts

#
eye 64

#
favorite 0

#
comment 0

In this note we determine the values of parameters c for which the polynomial representation of the degenerate double affine Hecke algebra (DAHA), i.e. the trigonometric Cherednik algebra, is reducible. Namely, we show that c is a reducibility point for the polynomial representation of the trigonometric Cherednik algebra for a root system R if and only if it is a reducibility point for the rational Cherednik algebra for the Weyl group of some root subsystem R' of R of the same rank; such...

Source: http://arxiv.org/abs/0706.4308v1

65
65

Sep 21, 2013
09/13

by
Pavel Etingof

texts

#
eye 65

#
favorite 0

#
comment 0

In 1986 Goldman introduced a Lie algebra structure on the linear span L of conjugacy classes of the fundamental group of a closed oriented surface. It is easy to see that the class e_1 of the trivial loop is a central element in L. We prove that any central element of L is a multiple of e_1 (a conjecture communicated to the author by Chas and Sullivan), and moreover that any central element of the Poisson algebra SL is a polynomial of e_1.

Source: http://arxiv.org/abs/math/0407505v1

7
7.0

Jun 30, 2018
06/18

by
Pavel Etingof

texts

#
eye 7

#
favorite 0

#
comment 0

P. Deligne defined interpolations of the tensor category of representations of the symmetric group S_n to complex values of n. Namely, he defined tensor categories Rep(S_t) for any complex t. This construction was generalized by F. Knop to the case of wreath products of S_n with a finite group. Generalizing these results, we propose a method of interpolating representations categories of various algebras containing S_n (such as degenerate affine Hecke algebras, symplectic reflection algebras,...

Topics: Mathematics, Quantum Algebra, Representation Theory

Source: http://arxiv.org/abs/1401.6321

5
5.0

Jun 30, 2018
06/18

by
Pavel Etingof

texts

#
eye 5

#
favorite 1

#
comment 0

We use a version of Haboush's theorem over complete local Noetherian rings to prove faithfulness of the lifting for semisimple cosemisimple Hopf algebras and separable (braided, symmetric) fusion categories from characteristic $p$ to characteristic zero (arXiv/math:0203060, Section 9), showing that, moreover, any isomorphism between such structures can be reduced modulo $p$. This fills a gap in arXiv/math:0203060, Subsection 9.3. We also show that lifting of semisimple cosemisimple Hopf...

Topics: Quantum Algebra, Mathematics

Source: http://arxiv.org/abs/1704.07855

136
136

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 136

#
favorite 0

#
comment 0

We�ll see now how Schur�s lemma allows us to classify subrepresentations in �nite dimensional semisimple representations.

Topics: Maths, Algebra, Representation Theory, Mathematics

Source: http://www.flooved.com/reader/1655

146
146

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 146

#
favorite 0

#
comment 0

We have seen that a central question about representations of quivers is whether a certain connected quiver has only �nitely many indecomposable representations. In the previous subsection it is shown that only those quivers whose underlying undirected graph is a Dynkin diagram may have this property. To see if they actually do have this property, we �rst explicitly decompose representations of certain easy quivers.

Topics: Maths, Algebra, Representation Theory, Mathematics

Source: http://www.flooved.com/reader/1658

57
57

Sep 21, 2013
09/13

by
Pavel Etingof

texts

#
eye 57

#
favorite 0

#
comment 0

The goal of this paper is to present some results and (more importantly) state a number of conjectures suggesting that the representation theory of symplectic reflection algebras for wreath products categorifies certain structures in the representation theory of affine Lie algebras (namely, decompositions of the restriction of the basic representation to finite dimensional and affine subalgebras). These conjectures arose from the insight due to R. Bezrukavnikov and A. Okounkov on the link...

Source: http://arxiv.org/abs/1011.4584v4

65
65

Sep 17, 2013
09/13

by
Pavel Etingof

texts

#
eye 65

#
favorite 0

#
comment 0

In this note we prove two main results. 1. In a rigid braided finite tensor category over C (not necessarily semisimple), some power of the Casimir element and some even power of the braiding is unipotent. 2. In a (semisimple) modular category, the twists are roots of unity dividing the algebraic integer D^{5/2}, where D is the global dimension of the category (the sum of squares of dimensions of simple objects). Both results generalize Vafa's theorem, saying that in a modular category twists...

Source: http://arxiv.org/abs/math/0207007v1

146
146

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 146

#
favorite 0

#
comment 0

Topics: Physics, Quantum Physics, Quantum Field Theory, Physics

Source: http://www.flooved.com/reader/1527

98
98

Sep 21, 2013
09/13

by
Pavel Etingof

texts

#
eye 98

#
favorite 0

#
comment 0

The purpose of this paper is to introduce and study a q-analogue of the holonomic system of differential equations associated to the Belavin's classical r-matrix (elliptic r-matrix equations), or, equivalently, to define an elliptic deformation of the quantum Knizhnik-Zamolodchikov equations of Frenkel and Reshetikhin. In hep-th 9303018, it was shown that solutions of the elliptic r-matrix equations admit a representation as traces of products of intertwining operators between certain modules...

Source: http://arxiv.org/abs/hep-th/9312057v1

67
67

Sep 19, 2013
09/13

by
Pavel Etingof

texts

#
eye 67

#
favorite 0

#
comment 0

We derive explicit integral formulas for eigenfunctions of quantum integrals of the Calogero-Sutherland-Moser operator with trigonometric interaction potential. In particular, we derive explicit formulas for Jack's symmetric functions. To obtain such formulas, we use the representation of these eigenfunctions by means of traces of intertwining operators between certain modules over the Lie algebra $\frak gl_n$, and the realization of these modules on functions of many variables.

Source: http://arxiv.org/abs/hep-th/9405038v2

46
46

Sep 23, 2013
09/13

by
Pavel Etingof

texts

#
eye 46

#
favorite 0

#
comment 0

We give a new proof of the Macdonald-Mehta-Opdam integral identity for finite Coxeter groups. This identity was conjectured by Macdonald and proved by Opdam in 1993 using the theory of multivariable Bessel functions, but in non-crystallographic cases the proof relied on a computer calculation by F. Garvan. Our proof is somewhat more elementary (in particular, it does not use multivariable Bessel functions), and uniform (does not refer to the classification of finite Coxeter groups and does not...

Source: http://arxiv.org/abs/0903.5084v1

4
4.0

Jun 28, 2018
06/18

by
Pavel Etingof

texts

#
eye 4

#
favorite 0

#
comment 0

Let $G$ be a finite group. There is a standard theorem on the classification of $G$-equivariant finite dimensional simple commutative, associative, and Lie algebras (i.e., simple algebras of these types in the category of representations of $G$). Namely, such an algebra is of the form $A={\rm Fun}_H(G,B)$, where $H$ is a subgroup of $G$, and $B$ is a simple algebra of the corresponding type with an $H$-action. We explain that such a result holds in the generality of algebras over a linear...

Topics: Mathematics, Rings and Algebras

Source: http://arxiv.org/abs/1512.07658

122
122

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 122

#
favorite 0

#
comment 0

Topics: Physics, Quantum Physics, Quantum Field Theory, Physics

Source: http://www.flooved.com/reader/1529

120
120

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 120

#
favorite 0

#
comment 0

So far we have considered quantum �eld theory with 0-dimensional spacetime (to make a joke, one may say that the dimension of the space is _1). In this section, we will move closer to actual physics: we will consider 1-dimensional spacetime, i.e. the dimension of the space is 0. This does not mean that we will study motion in a 0-dimensional space (which would be really a pity) but just means that we will consider only point-like quantum objects (particles) and not extended quantum objects...

Topics: Physics, Quantum Physics, Quantum Field Theory, Physics

Source: http://www.flooved.com/reader/1530

47
47

Sep 20, 2013
09/13

by
Pavel Etingof

texts

#
eye 47

#
favorite 0

#
comment 0

This is the text of the lecture given by the author in Naples at "Giornata IndAM", June 7, 2005. The lecture is addressed at the general mathematical audience and reviews several topics in deformation theory of associative algebras.

Source: http://arxiv.org/abs/math/0506144v4

55
55

Sep 22, 2013
09/13

by
Pavel Etingof

texts

#
eye 55

#
favorite 0

#
comment 0

In this note we prove the Davies-Foda-Jimbo-Miwa-Nakayashiki conjecture on the asymptotics of the composition of n quantum vertex operators for the quantum affine algebra U_q(\hat sl_2), as n goes to infinity. For this purpose we define and study the leading eigenvalue and eigenvector of the product of two components of the quantum vertex operator. This eigenvector and the corresponding eigenvalue were recently computed by M.Jimbo. The results of his computation are given in Section 4.

Source: http://arxiv.org/abs/hep-th/9410208v1

64
64

Sep 19, 2013
09/13

by
Pavel Etingof

texts

#
eye 64

#
favorite 0

#
comment 0

There were some errors in paper hep-th/9303018 in formulas 6.1, 6.6, 6.8, 6.11. These errors have been corrected in the present version of this paper. There are also some minor changes in the introduction.

Source: http://arxiv.org/abs/hep-th/9303018v3

141
141

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 141

#
favorite 0

#
comment 0

In this section, we begin a systematic development of representation theory of �nite groups.

Topics: Maths, Algebra, Representation Theory, Mathematics

Source: http://www.flooved.com/reader/1656

10
10.0

Jun 29, 2018
06/18

by
Pavel Etingof

texts

#
eye 10

#
favorite 0

#
comment 0

In 1998 Brou\'e, Malle and Rouquier conjectured that the Hecke algebra of a finite complex reflection group W is a free module over the algebra of parameters of rank |W|. We give an exposition of a proof of this conjecture in characteristic zero (and sufficiently large positive characteristic), due to I. Losev and I. Marin - G. Pfeiffer.

Topics: Representation Theory, Mathematics

Source: http://arxiv.org/abs/1606.08456

5
5.0

Jun 30, 2018
06/18

by
Pavel Etingof

texts

#
eye 5

#
favorite 0

#
comment 0

This paper is a sequel to arXiv:1401.6321. We define and study representation categories based on Deligne categories Rep(GL_t), Rep(O_t), Rep(Sp_2t), where t is any (non-integer) complex number. Namely, we define complex rank analogs of the parabolic category O and the representation categories of real reductive Lie groups and supergroups, affine Lie algebras, and Yangians. We develop a framework and language for studying these categories, prove basic results about them, and outline a number of...

Topics: Mathematics, Representation Theory

Source: http://arxiv.org/abs/1407.0373

143
143

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 143

#
favorite 0

#
comment 0

Remark. This discussion assumes that the extremum of S at q is actually a minimum, which we know is not always the case.

Topics: Physics, Quantum Physics, Quantum Field Theory, Physics

Source: http://www.flooved.com/reader/1522

107
107

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 107

#
favorite 0

#
comment 0

Topics: Physics, Quantum Physics, Quantum Field Theory, Physics

Source: http://www.flooved.com/reader/1524

123
123

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 123

#
favorite 0

#
comment 0

Matrix integrals (in particular, computation of the polynomial Pm(x)) can be used to calculate the orbifold Euler characteristic of the moduli space of curves. This was done by Harer and Zagier in 1986. Here we will give a review of this result (with some omissions).

Topics: Physics, Quantum Physics, Quantum Field Theory, Physics

Source: http://www.flooved.com/reader/1528

8
8.0

Jun 29, 2018
06/18

by
Pavel Etingof

texts

#
eye 8

#
favorite 0

#
comment 0

We generalize the theory of Koszul complexes and Koszul algebras (in particular, Koszul duality between symmetric and exterior algebras) to symmetric tensor categories. In characteristic $p\ge 5$, this theory exhibits peculiar effects, not observed in the classical theory. In particular, we show that the symmetric and exterior algebras of a non-invertible simple object in the Verlinde category ${\rm Ver}_p$ are almost Koszul (although not Koszul), and show how this gives examples of...

Topics: Rings and Algebras, Category Theory, Quantum Algebra, Mathematics

Source: http://arxiv.org/abs/1603.08133

56
56

Sep 18, 2013
09/13

by
Pavel Etingof

texts

#
eye 56

#
favorite 0

#
comment 0

These are lecture notes of a course on Calogero-Moser systems and their connections with representation theory and geometry, given by the author in Zurich in May-June 2005.

Source: http://arxiv.org/abs/math/0606233v4

94
94

Sep 18, 2013
09/13

by
Pavel Etingof

texts

#
eye 94

#
favorite 0

#
comment 0

We describe a generalization of Drinfeld's description of the center of a quantum group to the case of quantum affine algebras. We use the obtained central elements to construct the affine analogue of Macdonald's difference operators.

Source: http://arxiv.org/abs/q-alg/9412007v1

125
125

Sep 18, 2013
09/13

by
Pavel Etingof

texts

#
eye 125

#
favorite 0

#
comment 0

In this paper we q-deform a construction of Kazhdan and Kostant from 1970's which produces quantum Toda Hamiltonians by considering the action of Casimirs of a simple Lie algebra on Whittaker functions on the corresponding Lie group. We also give the affine analog of this generalization. This is done by extending the notion of a Whittaker function to quantum groups and quantum affine algebras. We compute the q-deformed Toda Hamiltonians for Lie algebras of type A and show that they coincide...

Source: http://arxiv.org/abs/math/9901053v1

125
125

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 125

#
favorite 0

#
comment 0

We are now passing to deeper results in representation theory of �nite groups. These results require the theory of algebraic numbers, which we will now brie�y review.

Topics: Maths, Algebra, Representation Theory, Mathematics

Source: http://www.flooved.com/reader/1657

135
135

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 135

#
favorite 0

#
comment 0

We will now give a very short introduction to Category theory, highlighting its relevance to the topics in representation theory we have discussed. For a serious acquaintance with category theory, the reader should use the classical book [McL].

Topics: Maths, Algebra, Representation Theory, Mathematics

Source: http://www.flooved.com/reader/1659

115
115

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 115

#
favorite 0

#
comment 0

In mechanics and �eld theory (both classical and quantum), there are two main languages � La-grangian and Hamiltonian. In the classical setting, the Lagrangian language is the language of vari-ational calculus (i.e. one studies extremals of the action functional), while the Hamiltonian language is that of symplectic geometry and Hamilton equations. Correspondingly, in the quantum setting, the Lagrangian language is the language of path integrals, while the Hamiltonian language is the...

Topics: Physics, Quantum Physics, Quantum Field Theory, Physics

Source: http://www.flooved.com/reader/1531

118
118

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 118

#
favorite 0

#
comment 0

Topics: Physics, Quantum Physics, Quantum Field Theory, Physics

Source: http://www.flooved.com/reader/1526

8
8.0

Jun 29, 2018
06/18

by
Pavel Etingof

texts

#
eye 8

#
favorite 0

#
comment 0

Let $F$ be an algebraically closed field. We show that if a quantum formal deformation $A$ of a commutative domain $A_0$ over $F$ is a PI algebra, then $A$ is commutative if ${\rm char}(F)=0$, and has PI degree a power of $p$ if ${\rm char}(F)=p>0$. This implies the same result for filtered deformations (i.e., filtered algebras $A$ such that ${\rm gr}(A)=A_0$).

Topics: Rings and Algebras, Mathematics

Source: http://arxiv.org/abs/1602.06480

110
110

Nov 14, 2013
11/13

by
Pavel Etingof

texts

#
eye 110

#
favorite 0

#
comment 0

Topics: Physics, Quantum Physics, Quantum Field Theory, Physics

Source: http://www.flooved.com/reader/1525

5
5.0

Jun 29, 2018
06/18

by
Pavel Etingof; Chelsea Walton

texts

#
eye 5

#
favorite 0

#
comment 0

We study when a finite dimensional Hopf action on a quantum formal deformation A of a commutative domain A_0 (i.e., a deformation quantization) must factor through a group algebra. In particular, we show that this occurs when the Poisson center of the fraction field of A_0 is trivial.

Topics: Rings and Algebras, Quantum Algebra, Mathematics

Source: http://arxiv.org/abs/1602.00532

54
54

Sep 18, 2013
09/13

by
Pavel Etingof; Olivier Schiffmann

texts

#
eye 54

#
favorite 0

#
comment 0

The purpose of this note is to define and construct highest weight modules for Felder's elliptic quantum groups. This is done by using exchange matrices for intertwining operators between (not necessarily finite-dimensional) modules over quantum affine algebras.

Source: http://arxiv.org/abs/math/9809161v1

65
65

Sep 19, 2013
09/13

by
Pavel Etingof; Shlomo Gelaki

texts

#
eye 65

#
favorite 0

#
comment 0

In this paper we contribute to the classification of Hopf algebras of dimension pq, where p,q are distinct prime numbers. More precisely, we prove that if p and q are odd primes with p

Source: http://arxiv.org/abs/math/0303359v1

69
69

Sep 18, 2013
09/13

by
Pavel Etingof; Shlomo Gelaki

texts

#
eye 69

#
favorite 0

#
comment 0

We develop a theory of descent and forms of tensor categories over arbitrary fields. We describe the general scheme of classification of such forms using algebraic and homotopical language, and give examples of explicit classification of forms. We also discuss the problem of categorification of weak fusion rings, and for the simplest families of such rings, determine which ones are categorifiable.

Source: http://arxiv.org/abs/1102.0657v2

59
59

Sep 20, 2013
09/13

by
Pavel Etingof; Xiaoguang Ma

texts

#
eye 59

#
favorite 0

#
comment 0

Following the works by Wiegmann-Zabrodin, Elbau-Felder, Hedenmalm-Makarov, and others, we consider the normal matrix model with an arbitrary potential function, and explain how the problem of finding the support domain for the asymptotic eigenvalue density of such matrices (when the size of the matrices goes to infinity) is related to the problem of Hele-Shaw flows on curved surfaces, considered by Entov and the first author in 1990-s. In the case when the potential function is the sum of a...

Source: http://arxiv.org/abs/math/0612108v3

72
72

Sep 22, 2013
09/13

by
Alexander Braverman; Pavel Etingof

texts

#
eye 72

#
favorite 0

#
comment 0

We introduce the notion of the (instanton part of the) Seiberg-Witten prepotential for general Schrodinger operators with periodic potential. In the case when the operator in question is integrable we show how to compute the prepotential in terms of period integrals; this implies that in the integrable case our definition of the prepotential coincides with the one that has been extensively studied in both mathematical and physical literature. As an application we give a proof of Nekrasov's...

Source: http://arxiv.org/abs/math/0409441v3

57
57

Sep 19, 2013
09/13

by
Pavel Etingof; Alexander Varchenko

texts

#
eye 57

#
favorite 0

#
comment 0

For any simple Lie algebra g and any complex number q which is not zero or a nontrivial root of unity, we construct a dynamical quantum group (Hopf algebroid), whose representation theory is essentially the same as the representation theory of the quantum group U_q(g). This dynamical quantum group is obtained from the fusion and exchange relations between intertwining operators in representation theory of U_q(g), and is an algebraic structure standing behind these relations.

Source: http://arxiv.org/abs/math/9801135v2

45
45

Sep 18, 2013
09/13

by
Pavel Etingof; Alexandre Soloviev

texts

#
eye 45

#
favorite 0

#
comment 0

In this note we define geometric classical r-matrices and quantum R-matrices, and show how any geometric classical r-matrix can be quantized to a geometric quantum R-matrix. This is one of the simplest nontrivial examples of quantization of solutions of the classical Yang-Baxter equation, which can be explicitly computed.

Source: http://arxiv.org/abs/math/9811001v3