Hopf Algebras and the Quantum Yang-Baxter Equation

Drinfeld, Vladimir

doi:10.1142/9789812798336_0013

Cited by 526 publications

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“…Thus we have shown the classical analogue of quantum affine algebra in the sense of Drinfeld's first realization [42]. This algebra can be interpreted as the remnant of the relativistic case [29,30] after taking the fast-moving string limit, while the left Yangians are not realized any more.…”

Section: Jhep08(2014)110mentioning

confidence: 76%

“…In subsection 3.4, we will show that a (classical analogue of) quantum affine algebra is generated by Q 0 , Q ± and Q ± in the sense of Drinfeld's first realization [42].…”

Section: 2mentioning

confidence: 99%

“…Let us see the relation to a classical analogue of Drinfeld's first realization of quantum affine algebra [42]. It is convenient to rescale the charges Q 0 (0) , Q ± (1) and Q ± (1) as follows:…”

Section: Quantum Affine Algebramentioning

confidence: 99%

“…These are the defining relations of Yangian in the sense of Drinfeld's first realization [42]. But the check of the defining relations has not been completed yet.…”

Section: Jhep08(2014)110mentioning

confidence: 99%

“…Just after that, a standard q-deformed AdS 5 ×S 5 superstring action has been constructed with a linear R-operator of Drinfeld-Jimbo type [42][43][44] satisfying the modified classical Yang-Baxter equation [45]. Then the metric (in the string frame) and NS-NS two-form have been determined [46].…”

Section: Introductionmentioning

confidence: 99%

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Anisotropic Landau-Lifshitz sigma models from q-deformed AdS5×S5 superstrings

Kameyama

Yoshida

2014

J. High Energ. Phys.

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Abstract:We consider bosonic subsectors of the q-deformed AdS 5 ×S 5 superstring action and study the classical integrable structure of anisotropic Landau-Lifshitz sigma models (LLSMs) derived by taking fast-moving string limits. The subsectors are 1) deformed AdS 3 × S 1 and 2) R × deformed S 3 . The cases 1) and 2) lead to a time-like warped SL(2) LLSM and a squashed S 3 LLSM, respectively. For each of them, we construct an infinite number of non-local conserved charges and show a quantum affine algebra at the classical level. Furthermore, a pp-wave like limit is applied for the case 1). The resulting system is a null-like warped SL(2) LLSM and exhibits a couple of Yangians through non-local gauge transformations associated with Jordanian twists.

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Section: Jhep08(2014)110mentioning

confidence: 76%

“…In subsection 3.4, we will show that a (classical analogue of) quantum affine algebra is generated by Q 0 , Q ± and Q ± in the sense of Drinfeld's first realization [42].…”

Section: 2mentioning

confidence: 99%

Section: Quantum Affine Algebramentioning

confidence: 99%

“…These are the defining relations of Yangian in the sense of Drinfeld's first realization [42]. But the check of the defining relations has not been completed yet.…”

Section: Jhep08(2014)110mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Anisotropic Landau-Lifshitz sigma models from q-deformed AdS5×S5 superstrings

Kameyama

Yoshida

2014

J. High Energ. Phys.

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“Classical” Flag Varieties for Quantum Groups: The Standard Quantum SL(n,C)

Ohn

2002

Advances in Mathematics

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We suggest a possible programme to associate geometric ''flag-like'' data to an arbitrary simple quantum group, in the spirit of the noncommutative algebraic geometry developed by Artin, Tate, and Van den Bergh. We then carry out this programme for the standard quantum SLðnÞ of Drinfel'd and Jimbo, where the varieties involved are certain T-stable subvarieties of the (ordinary) flag variety. # 2002 Elsevier Science (USA) INTRODUCTIONThe study of quantum analogues of flag varieties, first suggested by Manin [31], has been undertaken during the past decade by several authors, from various points of view; see e.g. [1,8,13,16,17,23,26,29,35,38,39,40]. Around the same time, an approach to noncommutative projective algebraic geometry was initiated by Artin et al. [4,5] and Artin and Van den Bergh [6], and considerably developed since (see e.g. [3, 7, 9, 24, 28, 37, 41, 42, 44]). One attractive feature of their approach is the association of actual geometric data to certain classes of graded noncommutative algebras.The present work is an attempt to study quantum flag varieties from this point of view. As a consequence, our ''quantum flag varieties'' will be actual varieties (with some bells and whistles).Recall the original idea of [4][5][6]. If A is the homogeneous coordinate ring of a projective scheme E; then the points of E are in one-to-one correspondence with the isomorphism classes of the so-called point modules of A; i.e. N-graded cyclic A-modules P such that dim P n ¼ 1 for all n: Now if A is an N-graded noncommutative algebra, one may still try to parametrize its point modules by the points of some projective scheme E: Of course, one cannot hope to reconstruct A from E alone, but there is now an additional ingredient: the shift operation s : P/P½1; where P½1 is the N-graded A-module defined by P½1 n :¼ P nþ1 : (When A is commutative, this shift is trivial: P½1 ' P for every point module P:) Assume that s may be viewed as an automorphism of E: one may then hope, at least in ''good'' cases, to recover A from the triple ðE; s; LÞ; where L is the line bundle over E defined by its embedding into a projective space. The first step of this recovery is the construction of the twisted homogeneous coordinate ring BðE; s; LÞ of a triple ðE; s; LÞ; defined in [6] as follows:(where L s denotes the pullback of L along s), the multiplication being given by ab :¼ a b s m for all a 2 B m ; b 2 B n : (When s is the identity, this algebra coincides, in high degree, with the (commutative) homogeneous coordinate ring of E w.r.t. the polarization L:) If the triple ðE; s; LÞ comes from an algebra A as above, the second step then consists in analysing the canonical morphism A ! BðE; s; LÞ: The initial success of this method has been a complete study of all regular algebras of dimension three [4] (where the kernel of A ! BðE; s; LÞ turns out to be generated by a single element of degree three).The present paper is organized as follows.In Part I, we give a general outline of a possible theory of flag varieties for quantum groups, usi...

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