## Schur-Weyl Categories and non-quasiclassical Weyl type Formula (2008)

Citations: | 4 - 1 self |

### BibTeX

@MISC{Gurevich08schur-weylcategories,

author = {Dimitri Gurevich and Zakaria Mriss and Université De Valenciennes},

title = {Schur-Weyl Categories and non-quasiclassical Weyl type Formula},

year = {2008}

}

### OpenURL

### Abstract

To a vector space V equipped with a non-quasiclassical involutary solution of the quantum Yang-Baxter equation and a partition λ, we associate a vector space Vλ and compute its dimension. The functor

### Citations

988 |
Intersection Theory
- Fulton
- 1984
(Show Context)
Citation Context ...ay that a morphism is an isomorphism if its inverse exists and is a morphism as well. Remark that the map V ↦−→ Vλ is a twisted analogue of the well-known Schur functor defined in the case S = σ (cf. =-=[11]-=-). Let us also observe that for two different embeddings Vλ ֒→ T m (V ), there exists a morphism sending one of them to the other one. Remark 2.7 It is worth saying that the definitions of SW category... |

762 |
A Guide to Quantum Groups
- Chari, Pressley
- 1994
(Show Context)
Citation Context ...(D.G.) was supported by the grant CNRS PICS-608. 1 Even symmetries We begin with the following observation. Usually, we assume that a tensor category (or a quasitensor category, in the terminology of =-=[5]-=-) is given, for example the category of Uq(g)-modules), and we study the properties of the objects of such a category. Our approach will be completely opposite: We begin with a basic object VS and gen... |

364 | Differential Geometry and Symmetric Spaces - Helgason - 1962 |

113 |
Foundations of Quantum Group Theory, Cambridge Univ
- Majid
- 1995
(Show Context)
Citation Context ...cribe twisted orbits and define twisted Casimir operator. From this viewpoint, Hopf or twisted Hopf algebras are derived objects themselves and they can be found from the reconstruction theorems (cf. =-=[18]-=-), although their explicit description is not always easy (cf. [1] where an attempt is made to describe an analogue of the QG Uq(g) for some non-quasiclassical Hecke symmetries). Let us pass now to de... |

54 | Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type
- Braverman, Gaitsgory
- 1996
(Show Context)
Citation Context ... U(g). Proposition 3.2 There exists a natural isomorphism ∧+(g) ∼ = GrU(g) where GrU(g) is the graded quadratic algebra associated to the filtered algebra U(g). Proof The algebra ∧+(g) is Kozsul (cf. =-=[4]-=- for the definition). It follows from the exactness of the Koszul complex of the first kind from [12]. Then by [4] we have the result. 23We say that a linear space W is a g-module if there exists a t... |

48 |
Algebraic Aspects of the Quantum Yang-Baxter Equation
- Gurevich
- 1991
(Show Context)
Citation Context ...p to a factor to the Laplace-Beltrami one if the latter is g-invariant.) To explain the reason of this phenomenon let us recall first some aspects of “twisted linear algebra” developed essentially in =-=[12]-=- and [16]. Let us fix a symmetry (0.1) and associate to it symmetric and skew-symmetric algebras in a natural way Sym(V ) = ∧+(V ) = T(V )/{Im (Id − S)}, ∧−(V ) = T(V )/{Im (Id + S)} (0.5) where T(V )... |

18 |
The quantum group of a non-degenerated bilinear form
- Dubois-Violette, Launer
- 1990
(Show Context)
Citation Context ...= a and u22 = b, then we have u31 = −a/x, v 13 = x/(2a), v 22 = −1/2b, v 31 = −1/2a where x is a solution of the equation x + x −1 = 3. 3 Some symmetries of this type were discovered independently in =-=[10]-=-. 12Our next aim is to describe a way to construct even symmetries of rank greater then 2 with central determinant. First, assume that two symmetries S1 : V ⊗2 1 ⊗2 → V1 and S2 : V ⊗2 ⊗2 2 → V2 (1.20... |

12 | Poincaré Series of Quantum Spaces Associated to Hecke Operators
- Hai
- 1999
(Show Context)
Citation Context ...ic polynomial (i.e. a polynomial with leading coefficient 1). For an even symmetry S, we call the degree of the polynomial P−(t) the rank of V , and we denote this by rankV . 7Remark 1.3 As shown in =-=[19]-=-, the Poincaré series of a Hecke symmetry is a rational function (a proof in the case of symmetries also appeared in [6]). Let us assume that P−(t) is a rational function with monic numerator and deno... |

11 | Quantum Lie Algebras of Type An - Lyubashenko, Sudbery |

9 | Braiding of the Lie algebra sl(2 - Donin, Gurevich - 1995 |

9 |
Double quantization on CP n type orbits by generalized Verma modules
- Donin, Gurevich, et al.
- 1998
(Show Context)
Citation Context ...ifferent from the usual flip. Such algebras can be “twisted commutative” or “twisted non-commutative”. In the latter case they are realized as operator algebras in twisted categories in the spirit of =-=[8]-=-. 2following question appears from the very beginning: which algebras can be considered as twisted analogues of commutative algebras and, in particular, which system of equations compatible with the ... |

9 |
Quantization of Poisson pairs: R-matrix approach
- Gurevich, Rubtsov, et al.
- 1992
(Show Context)
Citation Context ...nalogue of the notion of volume. It is interesting to compare this result with the analysis of the spectrum of an “exotic harmonic oscillator” arising from non-quasiclassical symmetries introduced in =-=[14]-=-. Let us discuss now the case p = rankVS > 2. In this case it is not so easy to find a system of equations which would define a twisted nonquasiclassical variety. We restrict attention to the “twisted... |

9 |
Superanalysis and Solutions to the Triangles Equation
- Lyubashenko
- 1987
(Show Context)
Citation Context ...ctor to the Laplace-Beltrami one if the latter is g-invariant.) To explain the reason of this phenomenon let us recall first some aspects of “twisted linear algebra” developed essentially in [12] and =-=[16]-=-. Let us fix a symmetry (0.1) and associate to it symmetric and skew-symmetric algebras in a natural way Sym(V ) = ∧+(V ) = T(V )/{Im (Id − S)}, ∧−(V ) = T(V )/{Im (Id + S)} (0.5) where T(V ) is the f... |

6 |
Some aspects of braided geometry : differential calculus, tangent space, gauge theory
- Akueson, Gurevich
- 1999
(Show Context)
Citation Context ... vectors fields” which are completely different from those coming from Uq(sl(2)), in spite of a tradition assigning the meaning of vector fields to the images of the elements X, Y, H ∈ Uq(sl(2)) (see =-=[2]-=-). Now we assume that n = dim VS > 2 (the case n = 2 corresponds to the classical hyperboloid). Proposition 4.2 On the hyperboloid in question the eigenvalules λl of the Casimir operator Csl and their... |

6 | A reconstruction result for the R-matrix quantizations of SU(N). Arxiv preprint arxiv:9806063v3
- Banica
- 1998
(Show Context)
Citation Context ... previous section is central. Finally, we recover just the same fusion ring as in the classical case but with n replaced by p. This fact has been already mentioned in the mathematical literature (cf. =-=[3]-=-), even in more general situation related to Hecke symmetries. However, dimensions of the spaces Vλ and the corresponding ClebschGordan coefficients (which are defined if we fix some bases in the comp... |

6 | V.Rubtsov Quantum hyperboloid and braided modules, Actes du Septième Contact Franco-Belge, Juin - Donin - 1995 |

5 |
Noncommutative differential geometry and Yang–Baxter equation
- Gurevich, Radul, et al.
- 1988
(Show Context)
Citation Context ...s operator as a second order twisted differential operator. In order to do this we will say some words on twisted differential operators on the hyperboloid in question. Connected to this is the paper =-=[13]-=- where some aspects of differential calculus arising from symmetries are considered. Recall that a twisted vector field (or S-vector field) on a twisted commutative algebra A is an operator X : A → A ... |

1 |
Some algebraic structures related to Temperley-Lieb algebra
- Akueson, Gurevich
- 1988
(Show Context)
Citation Context ... viewpoint, Hopf or twisted Hopf algebras are derived objects themselves and they can be found from the reconstruction theorems (cf. [18]), although their explicit description is not always easy (cf. =-=[1]-=- where an attempt is made to describe an analogue of the QG Uq(g) for some non-quasiclassical Hecke symmetries). Let us pass now to describing a possible form of an even symmetry. Let us fix a space V... |