Geometric Crystals on Schubert Varieties
We define geometric crystals and unipotent crystals for arbitrary Kac-Moody groups and describe geometric and unipotent crystal structures on the Schubert varieties.
math.QA↗
arXiv subjects
Toshiki Nakashima
Publications and source records attributed to Toshiki Nakashima.
On certain maximal cyclic modules for the quantized special linear algebra at a root of unity
By properly specializing the parameters irreducible modules of maximal dimension for the De Concini-Kac version of the Drinfeld-Jimbo quantum algebra in type $A$ may be transformed into modules over Lusztig's infinitesimal quantum algeba. Thus obtained modules have a simple socle and a simple head, and share the same dimension as the infinitesimal Verma modules. Despite these common features we find that they are never isomorphic to infinitesimal Verma modules unless they are irreducible. The same carry over to the modular setup for the special linear groups in positive characteristic.
math.QA↗
Extremal Projectors of q-Boson Algebras
We define the extremal projector of the $q$-boson Kashiwara algebra $B_q(\ge)$ and study their basic properties. Applying their proerties to the representation theory of the category ${\cal O}(B_q(\ge))$, whose objects are ''upper bounded '' $B_q(\ge)$-modules, we obtain its semi-simplicity and the classification of simple modules.
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Irreducible Modules of Finite Dimensional Quantum Algebras of type A at Roots of Unity
Specializing properly the parameters contained in the maximal cyclic representation of the non-restricted A-type quantum algebra at roots of unity, we find the unique primitive vector in it. We show that the submodule generated by the primitive vector can be identified with an irreducble highest weight module of the finite dimensional A-type quantum algebra which is defined as the subalgebra of the restricted quantum algebra at roots of unity.
math.QA↗
Polytopes for Crystallized Demazure Modules and Exremal Vectors
We give a parametrization for crystal bases of Demazure modules as a set of lattice points in some convex polytope and we also describe explicitly the extremal vectors as solutions of some system of linear equations.
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Polyhedral Realizations of Crystal Bases and Braid-type Isomorphisms
We review the polyhedral realizations of crystal bases in the former half and in the latter half, we introduce braid-type isomorphisms for some rank 2 finite type crystals. Using this isomorphisms, for semi-simple Lie algebra we can show that polyhedral realizations can be obtained in the lattice of finite rank which coincides with the length of the longest element in the associated Weyl group.
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Polyhedral Realizations of Crystal Bases for Integrable Highest Weight Modules
We give a general way of representing the crystal (base) corresponding to the intgrable highest weight modules of quantum Kac-Moody algebras, which is called polyhedral realizations. This is applied to describe explicitly the crystal bases of integrable highest weight modules for arbitrary rank 2 Kac-Moody algebra cases, the classical A_n-case and the affine A^{(1)}_{n-1}-case.
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Polyhedral Realizations of Crystal Bases for Quantized Kac-Moody Algebras
Let B(\infty) be the crystal corresponding to the nilpotent part of a quantized Kac-Moody algebra. We suggest a general way to represent B(\infty) as the set of integer solutions of a system of linear inequalities. As an application, we treat in a unified manner all Kac-Moody algebras of rank 2 (sharpening the result by Kashiwara), as well as the algebras of types A_n and A_{n-1}^{(1)}.
q-alg↗
Crystallized Peter-Weyl type decomposition for level 0 part of modified quantum algebra $\widetilde U_q(\widehat{sl_2})_0$
We shall give some criteria for the existence of the Peter-Weyl type decomposition for modified quantum algebra, which is motivated by the fact that modified quantum algebra has commutative two crystal structures. We shall apply the criteria to the crystal base of the level 0 part of the modified quantum affine algebra $\widetilde U_q(\widehat{sl_2})_0$ and give more precise description of Peter-Weyl type decomposition.
q-alg↗
Crystallized structure for level 0 part of modified quantum affine algebra
Crystal base of the level 0 part of the modified quantum affine algebra $\widetilde U_q(\widehat{sl_2})_0$ is given by path. Weyl group actions, extremal vectors and crystal structure of all irreducible components are described explicitly.
q-alg↗
Quantum R-matrix and Intertwiners for the Kashiwara Algebra
We study the algebra $B_q(\ge)$ presented by Kashiwara and introduce intertwiners similar to $q$-vertex operators. We show that a matrix determined by 2-point functions of the intertwiners coincides with a quantum R-matrix (up to a diagonal matrix) and give the commutation relations of the intertwiners. We also introduce an analogue of the universal R-matrix for the Kashiwara algebra.
hep-th↗
Quantum affine symmetry in vertex models
We study the higher spin anologs of the six vertex model on the basis of its symmetry under the quantum affine algebra $U_q(\slth)$. Using the method developed recently for the XXZ spin chain, we formulate the space of states, transfer matrix, vacuum, creation/annihilation operators of particles, and local operators, purely in the language of representation theory. We find that, regardless of the level of the representation involved, the particles have spin $1/2$, and that the $n$-particle space has an RSOS-type structure rather than a simple tensor product of the $1$-particle space. This agrees with the picture proposed earlier by Reshetikhin.
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