Highest weight representations of quantum current algebras

We study the highest weight and continuous tensor product representations ofq-deformed Lie algebras through the mappings of a manifold into a locally compact group. As an example the highest weight representation of theq-deformed algebra sl_q(2,) is calculated in detail.Alexander von Humboldt-Stiftung fellow. On leave from Institute of Physics, Chinese Academy of Sciences, Beijing, P.R. China.     

Construction of cyclic representations of quantum algebras at q p =1 from their regular representations

Cyclic representations of maximal dimension of the quantum algebra U _q L associated with any finite-dimensional simple Lie algebra L are studied from its regular representation at q ^p =1, which is proved to be a quotient module of itself as a left module with respect to some submodules. The general theory is given after an instructive example U _q sl(2) is studied. Another explicit example U _q sl(3) is also presented.This work is supported in part by the National Natural Science Foundation of China. Author Fu is also supported by the Jilin Provincial Science and Technology Foundation of China     

Braided differential operators on quantum algebras

We propose a general scheme of constructing braided differential algebras via algebras of “quantum exponentiated vector fields” and those of “quantum functions”. We treat a reflection equation algebra as a quantum analog of the algebra of vector fields. The role of a quantum function algebra is played by a general quantum matrix algebra. As an example we mention the so-called RTT algebra of quantized functions on the linear matrix group GL(m)$GL\left(m\right)$. In this case our construction essentially coincides with the quantum differential algebra introduced by S. Woronowicz. If the role of a quantum function algebra is played by another copy of the reflection equation algebra we get two different braided differential algebras. One of them is defined via a quantum analog of (co)adjoint vector fields, the other algebra is defined via a quantum analog of right-invariant vector fields. We show that the former algebra can be identified with a subalgebra of the latter one. Also, we show that “quantum adjoint vector fields” can be restricted to the so-called “braided orbits” which are counterparts of generic GL(m)$GL\left(m\right)$-orbits in gl^∗(m)$g{l}^{\ast }\left(m\right)$. Such braided orbits endowed with these restricted vector fields constitute a new class of braided differential algebras.     

Casimir invariants for quantized affine Lie algebras

Casimir invariants for quantized affine Lie algebras are constructed and their eigenvalues computed in any irreducible highest-weight representation.     

Fundamental representations of quantum groups at roots of 1

To every finite-dimensional irreducible representation V of the quantum group U(g) where is a primitive lth root of unity (l odd) and g is a finite-dimensional complex simple Lie algebra, de Concini, Kac and Procesi have associated a conjugacy class C _V in the adjoint group G of g. We describe explicitly, when g is of type A _n , B _n , C _n , or D _n , the representations associated to the conjugacy classes of minimal positive dimension. We call such representations fundamental and prove that, for any conjugacy class, there is an associated representation which is contained in a tensor product of fundamental representations.     

Quantum affine algebras and universalR-matrix with spectral parameter

Using the previously obtained universalR-matrix for the quantized nontwisted affine Lie algebras U _q (A ₁ ⁽¹⁾ ) and U _q (A ₂ ⁽¹⁾ ), we determine the explicitly spectral dependent universalR-matrix for the corresponding quantum Lie algebras U _q (A ₁) and U _q (A ₂). As applications, we reproduce the well known results in the fundamental representations and we also derive an extremely explicit formula of the spectral-dependentR-matrix for the adjoint representation of U _q (A ₂), the simplest nontrivial case when the tensor product decomposition of the representation with itself has nontrivial multiplicity.     

Boson realizations of quantum algebras and some physical spaces with quantum algebra symmetry

The generators ofq-boson algebra are expressed in terms of those of boson algebra, and the relations among the representations of a quantum algebra onq-Fock space, on Fock space, and on coherent state space are discussed in a general way. Two examples are also given to present concrete physical spaces with quantum algebra symmetry. Finally, a new homomorphic mapping from a Lie algebra to boson algebra is presented.This work is supported by the National Foundation of Natural Science of China.     

Quantized affine Lie algebras and diagonalization of braid generators

Let U _q be a quantized affine Lie algebra. It is proven that the universal R-matrix R of U _q satisfies the celebrated conjugation relationR ⁺ =TR withT the usual twist map. As applications, the braid generator is shown to be diagonalizable on arbitrary tensor product modules of integrable irreducible highest weight U _q -module and a spectral decomposition formula for the braid generator is obtained which is the generalization of Reshetikhin and Gould forms to the present affine case. Casimir invariants are constructed and their eigenvalues computed by means of the spectral decomposition formula. As a by-product, an interesting identity is found.     

Minimal affinizations of representations of quantum groups: the irregular case

Let be a finite-dimensional complex simple Lie algebra and U_q( ) the associated quantum group (q is a nonzero complex number which we assume is transcendental). IfV is a finitedimensional irreducible representation of U_q( ), an affinization ofV is an irreducible representationVV of the quantum affine algebra U_q( ) which containsV with multiplicity one and is such that all other irreducible U_q( )-components ofV have highest weight strictly smaller than the highest weight ofV. There is a natural partial order on the set of U_q( ) classes of affinizations, and we look for the minimal one(s). In earlier papers, we showed that (i) if is of typeA, B, C, F orG, the minimal affinization is unique up to U_q( )-isomorphism; (ii) if is of typeD orE and is not orthogonal to the triple node of the Dynkin diagram of , there are either one or three minimal affinizations (depending on ). In this paper, we show, in contrast to the regular case, that if U_q( ) is of typeD ₄ and is orthogonal to the triple node, the number of minimal affinizations has no upper bound independent of .As a by-product of our methods, we disprove a conjecture according to which, if is of typeA _n,every affinization is isomorphic to a tensor product of representations of U_q( ) which are irreducible under U_q( ) (in an earlier paper, we proved this conjecture whenn=1).Both authors were partially supported by the NSF, DMS-9207701.     

On universalR-matrix for quantized nontwisted rank 3 affine KM algebras

Explicit formulas of the universalR-matrix are given for all quantized nontwisted rank 3 affine KM algebras U _q (A ₂ ⁽¹⁾ ), U _q (C ₂ ⁽¹⁾ ) and U _q (G ₂ ⁽¹⁾ ).     

Singular vectors of quantum group representations for straight Lie algebra roots

We give explicit formulae for singular vectors of Verma modules over U_q(G), where G is any complex simple Lie algebra. The vectors we present correspond exhaustively to a class of positive roots of G which we call straight roots. In some special cases, we give singular vectors corresponding to arbitrary positive roots. For our vectors we use a special basis of U_q(G ^-), where G ^- is the negative roots subalgebra of G, which was introducted in our earlier work in the case q=1. This basis seems more economical than the Poincaré-Birkhoff-Witt type of basis used by Malikov, Feigin, and Fuchs for the construction of singular vectors of Verma modules in the case q=1. Furthermore, this basis turns out to be part of a general basis recently introduced for other reasons by Lusztig for U_q(^-), where ^- is a Borel subalgebra of G.A. v. Humboldt-Stiftung fellow, permanent address and after 22 September 1991: Bulgarian Academy of Sciences, Institute of Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria.     

Differential geometry on linear quantum groups

An exterior derivative, inner derivation, and Lie derivative are introduced on the quantum group GL _q (N). SL _q (N) is then found by constructing matrices with determnant unity, and the induced calculus is found.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY90-21139.     

On affine yangians

A Yangian , a deformation of the universal enveloping algebra of the two-dimensional loop algebra sl(2) C [t ^–1,t;u], is constructed. This deformation is an analogue of a Yangian which was constructed by V. Drinfeld for any simple Lie algebra. The PBW theorem for is proved and some representations are constructed. Like usual Yangians, possesses a one-dimensional group of auto- morphisms and at zero level - a two-dimensional group of automorphisms. This observation allows one to conjecture that the representation theory of should give rise to new solutions of QYBE.Yangians of other affine algebras can be constructed similarly and they enjoy similar properties.     

Braided Hopf algebras and differential calculus

We show that the algebra of the bicovariant differential calculus on a quantum group can be understood as a projection of the cross-product between a braided Hopf algebra and the quantum double of the quantum group. The resulting super-Hopf algebra can be reproduced by extending the exterior derivative to tensor products.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-90-21139.Supported in part by a Feodor-Lynen Fellowship.     

Finite-dimensional representations of U q (osp(1/2n)) and its connection with quantum so(2n+1)

It is shown that the quantum supergroup U _q (osp(1/2_n)) is essentially isomorphic to the quantum group U _-q (so(2n+1)) restricted to tensorial representations. This renders it straightforward to classify all the finite-dimensional irreducible representations of U _q (osp(1/2n)) at generic q. In particular, it is proved that at generic q, every-dimensional irrep of this quantum supergroup is a deformation of an osp(1/2n) irrep, and all the finite-dimensional representations are completely reducible.     

Linear connections on the quantum plane

A general definition has been proposed recently of a linear connection and a metric in noncommutative geometry. It is shown that to within normalization there is a unique linear connection on the quantum plane and there is no metric.     

Noncommutative symmetric functions and laplace operators for classical Lie algebras

New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the use of some properties of the noncommutative symmetric functions associated with a matrix. The decomposition of the Sklyanin determinant into a product of quasi-determinants play the main role in the construction. Analogous decomposition for the quantum determinant provides an alternative proof of the known construction for the Lie algebra gl(N).