quantum 
We consider some remarkable central elements of the universal enveloping algebraU(gl(n)) which we call quantum immanants.


They result in many nontrivial properties of quantum immanants.


Quantum integrable systems and differential Galois theory


This paper is devoted to a systematic study of quantum completely integrable systems (i.e., complete systems of commuting differential operators) from the point of view of algebraic geometry.


We study the tensor category of tilting modules over a quantum groupUq with divided powers.


Recently Varagnolo and Vasserot established that theqdeformed Fock spaces due to Hayashi, and Kashiwara, Miwa and Stern, admit actions of the quantum toroidal algebra with the level (0,1).


The quantum toroidal action on the Fock space depends on a certain parameter κ.


We find that with a specific choice of this parameter, the action on the Fock spaces gives rise to the toroidal action on irreducible level1 highest weight modules of the affine quantum algebra.


Similarly, by a specific choice of the parameter, the level (1,0) vertex representation of the quantum totoidal algebra gives rise to a structure on irreducible level1 highest weightmodules.


Simple modules over the multiparameter quantum function algebra at roots of 1


We construct essentially all the irreducible modules for the multiparameter quantum function algebraF?φ[G], whereG is a simple simply connected complex algebraic group, and ? is a root of unity.


On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al.


Fr?nsdal [Fr1, Fr2] made a penetrating observation that both of them are quasiHopf algebras, obtained by twisting the standard quantum affine algebraUq(g).


Another proof of Joseph and Letzter's separation of variables theorem for quantum groups


Using the theory of crystal bases as the main tool, we prove a quantum analogue of Richardson's theorem.


From it, we recover Joseph and Letzter's result by a kind of "quantum duality principle".


We express the vanishing conditions satisfied by the correlation functions of Drinfeld currents of quantum affine algebras, imposed by the quantum Serre relations.


Geometric construction of the global base of the quantum modified algebra of


A geometric construction of the modified quantum algebra ofgln was given in [BLM].


The form of these generic polynomials is that of a Bethe eigenfunction and they imitate, on a more elementary level, the Rmatrix construction of quantum immanants.

