arXiv subjects

Yoshihisa Saito

Publications and source records attributed to Yoshihisa Saito.

10 recordsLinked to original sources

arXiv2014

Quantized coordinate rings, PBW-type bases and $q$-boson algebras

Recently, Kuniba, Okado and Yamada proved that the transition matrix of PBW-type bases of the positive-half of a quantized universal enveloping algebra $U_q(\mathfrak{g})$ coincides with a matrix coefficients of the intertwiner between certain irreducible modules over the corresponding quantized coordinate ring $A_q(\mathfrak{g})$, introduced by Soibelman. In the present article, we give a new proof of their result, by using representation theory of the $q$-boson algebra, and the Drinfeld paring of $U_q(\mathfrak{g})$.

math.QA↗

arXiv2012

Toward Berenstein-Zelevinsky data in affine type $A$, part III: Proof of the connectedness

We prove the connectedness of the crystal, which we introduced in our previous works.

math.QA↗

arXiv2011

Mirković-Vilonen Polytopes and a quiver construction of crystal basis in type $A$

In the current paper, we give a quiver theoretical interpretation of Mirković-Vilonen polytopes in type $A_n$. As a by-product, we give a new proof of the Anderson-Mirković conjecture which describes the explicit forms of the actions of lowering Kashiwara operators on the set of Mirković-Vilonen polytopes.

math.RT↗

arXiv2011

Toward Berenstein-Zelevinsky data in affine type $A$, part II: Explicit description

In the present paper, we give an explicit description of the affine analogs of Berenstein-Zelevinsky data constructed in our previous paper: Toward Berenstein-Zelevinsky data in affine type $A$, I: Construction of affine analogs (arXiv:1009.4526), in terms of certain collections of nonnegative integers, which we call Lusztig data of type $A_{l-1}^{(1)}$.

math.QA↗

arXiv2010

Indecomposable decomposition of tensor products of modules over the restricted quantum universal enveloping algebra associated to $\boldsymbol{\mathfrak{sl}_2}$

In this paper we study the tensor category structure of the module category of the restricted quantum enveloping algebra associated to $\mathfrak{sl}_2$. Indecomposable decomposition of all tensor products of modules over this algebra is completely determined in explicit formulas. As a by-product, we show that the module category of the restricted quantum enveloping algebra associated to $\mathfrak{sl}_2$ is not a braided tensor category.

math.QA↗

arXiv2010

Toward Berenstein-Zelevinsky data in affine type $A$, I: Construction of affine analogs

We give (conjectural) analogs of Berenstein-Zelevinsky data for affine type $A$. Moreover, by using these affine analogs of Berenstein-Zelevinsky data, we realize the crystal basis of the negative part of the quantized universal enveloping algebra of the (Langlands dual) Lie algebra of affine type $A$.

math.QA↗

arXiv2009

The Rational qKZ Equation and Shifted Non-Symmetric Jack Polynomials

We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated with the Lie algebra $gl_N$. The main ingredient is a special class of the shifted non-symmetric Jack polynomials. It may be regarded as a shifted version of the singular polynomials studied by Dunkl. We prove that our solutions contain those obtained as a scaling limit of matrix elements of the vertex operators of level one.

math.QA↗

arXiv2001

Crystal bases and quiver varieties (Geometric construction of crystal bases II)

We give a crystal structure on the set of all irreducible components of Lagrangian subvarieties of quiver varieties. One con show that, as a crystal, it is isomorphic to the crystal base of an irreducible highest weight representation of a quantized universal enveloping algebra.

math.QA↗

arXiv1997

Quantum toroidal algebras and their vertex representations

We change the definition of the vertex representations. As a result the vertex representations has one parameter.

q-alg↗

arXiv1996

Geometric Construction of Crystal Bases

We realize the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety. As a by-product, we give a counterexample to the conjecture of Kazhdan--Lusztig on the irreducibility of the characteristic variety of the intersection cohomology sheaves associated with the Schubert cells of type A and also to the similar problem asked by Lusztig on the characteristic variety of the perverse sheaves corresponding to canonical bases.

q-alg↗