Yangian Construction of the Virasoro Algebra

We show that a Yangian construction based on the algebra of an infinite number of harmonic oscillators (i.e. a vibrating string) terminates after one step, yielding the Virasoro algebra.Comment: 5 pages, AMS-Latex 2

Convex Bases of PBW type for Quantum Affine Algebras

This note has two purposes. First we establish that the map defined in [L, $\S 40.2.5$ (a)] is an isomorphism for certain admissible sequences. Second we show the map gives rise to a convex basis of Poincar\'e--Birkhoff--Witt (PBW) type for \bup, an affine untwisted quantized enveloping algebra of Drinfel$'$d and Jimbo. The computations in this paper are made possible by extending the usual braid group action by certain outer automorphisms of the algebra.Comment: 7 pages, to appear in Comm. Math. Phy

q-deformation of z→az+bcz+dz\to {az+b\over cz+d}

We construct the action of the quantum double of \uq on the standard Podle\'s sphere and interpret it as the quantum projective formula generalizing to the q-deformed setting the action of the Lorentz group of global conformal transformations on the ordinary Riemann sphere.Comment: LaTeX, 16 pages, we add a reference where an alternative construction of the q-Lorentz group action on the Podles sphere is give

Braid Group Action and Quantum Affine Algebras

We lift the lattice of translations in the extended affine Weyl group to a braid group action on the quantum affine algebra. This action fixes the Heisenberg subalgebra pointwise. Loop like generators are found for the algebra which satisfy the relations of Drinfel$'$d's new realization. Coproduct formulas are given and a PBW type basis is constructed.Comment: 15 page

Search-Money-and-Barter Models of Financial Stabilization

A macroeconomic model based on search-theoretical foundations is built to show that in an economy with structural deficiencies of the Russian Virtual Economy, money substitutes appear as a result of optimizing behavior of agents. Moreover, the volume of money substitutes is typically large, and it is impossible to reduce their volume significantly by using standard instruments as an increase of the money supply or decreasing the tax level. The result obtains for an economy, where there are large natural monopolies and widespread informal networks.http://deepblue.lib.umich.edu/bitstream/2027.42/39716/3/wp332.pd

Intertwining operators and Hirota bilinear equations

An interpretation of Hirota bilinear relations for classical $\tau$ functions is given in terms of intertwining operators. Noncommutative example of $U_q(sl_2)$ is presented.Comment: Latex, 13 pages, no figures. Contribution to the Proceedings of Alushta Conference, June 199

Coherent States for Quantum Compact Groups

Coherent states are introduced and their properties are discussed for all simple quantum compact groups. The multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates on a general quantum dressing orbit and interpret the coherent state as a holomorphic function on this orbit with values in the carrier Hilbert space of an irreducible representation of the corresponding quantized enveloping algebra. Using Gauss decomposition, the commutation relations for the holomorphic coordinates on the dressing orbit are derived explicitly and given in a compact R--matrix formulation (generalizing this way the $q$--deformed Grassmann and flag manifolds). The antiholomorphic realization of the irreducible representations of a compact quantum group (the analogue of the Borel--Weil construction) are described using the concept of coherent state. The relation between representation theory and non--commutative differential geometry is suggested.}Comment: 25 page

Highest weight representations of the quantum algebra U_h(gl_\infty)

A class of highest weight irreducible representations of the quantum algebra U_h(gl_\infty) is constructed. Within each module a basis is introduced and the transformation relations of the basis under the action of the Chevalley generators are explicitly written.Comment: 7 pages, PlainTe

Twisted Classical Poincar\'{e} Algebras

We consider the twisting of Hopf structure for classical enveloping algebra $U(\hat{g})$, where $\hat{g}$ is the inhomogenous rotations algebra, with explicite formulae given for $D=4$ Poincar\'{e} algebra $(\hat{g}={\cal P}_4).$ The comultiplications of twisted $U^F({\cal P}_4)$ are obtained by conjugating primitive classical coproducts by $F\in U(\hat{c})\otimes U(\hat{c}),$ where $\hat{c}$ denotes any Abelian subalgebra of ${\cal P}_4$, and the universal $R-$matrices for $U^F({\cal P}_4)$ are triangular. As an example we show that the quantum deformation of Poincar\'{e} algebra recently proposed by Chaichian and Demiczev is a twisted classical Poincar\'{e} algebra. The interpretation of twisted Poincar\'{e} algebra as describing relativistic symmetries with clustered 2-particle states is proposed.Comment: \Large \bf 19 pages, Bonn University preprint, November 199

Categorical geometric skew Howe duality

We categorify the R-matrix isomorphism between tensor products of minuscule representations of U_q(sl(n)) by constructing an equivalence between the derived categories of coherent sheaves on the corresponding convolution products in the affine Grassmannian. The main step in the construction is a categorification of representations of U_q(sl(2)) which are related to representations of U_q(sl(n)) by quantum skew Howe duality. The resulting equivalence is part of the program of algebro-geometric categorification of Reshitikhin-Turaev tangle invariants developed by the first two authors.Comment: 31 page