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1.
Using previous results we construct theq-analogues of the left invariant vector fields of the quantum enveloping algebra corresponding to the complex Lie algebras of typeA n–1 ,B n ,C n , andD n . These quantum vector fields are functionals over the complex quantum groupA. In the special caseA 1 it is shown that this Hopf algebra coincides withU q sl(2, ).  相似文献   

2.
We construct complex quantum groups associated with the Lie algebras of typeA n–1 ,B n ,C n andD n which are considered as real algebras. Following the ideas of Faddeev, Reshetikhin and Takhtayan, we obtain the Hopf algebras of regular functionalsU R , on these real complexified quantum groups. Theq-analogues of the left invariant vector fields of the quantum enveloping algebras are defined. These quantum vector fields are functionals over the corresponding real formA of the complex quantum groupA. The equivalence of the Hopf algebra of regular functionals and the algebra of complex quantum vector fields is shown by factorizing the vector fields uniquely into a triangular and a unitary part and identifying them with the corresponding elements of the algebra of regular functionals. In the special exampleA 1 , we derive theq-deformed real complexified enveloping algebraU q sl(2, ) with six generators.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June, 1992.Based on the papers: [i]Drabant B., Schlieker M., Weich W., and Zumino B.: PreprintLMU-TPW 1991-5 (to appear in Commun. Math. Phys.) [ii]Chryssomalakos C., Drabant B., Schlieker M., Weich W., and Zumino B.: Preprint UCB 92/03 (to appear in Commun. Math. Phys.) [iii]Drabant B., Juro B., Schlieker M., Weich W., and Zumino B.: Preprint MPI-Ph/92-39 (submitted to Lett. Math. Phys.)  相似文献   

3.
We study the canonical quantization of the SU(n) WZNW model. Decoupling the chiral dynamics requires an extended state space including left and right monodromies as independent variables. In the simplest (n = 2) case we explicitly show that the zero modes of the monodromy extended SU(2) WZNW model give rise to a quantum group gauge theory in a finite-dimensional Fock space. We define the subspace of Uq(sl(2)) ⊗ Uq(sl(2))-invariant vectors on which the monodromy invariance is also restored and construct the physical space applying a generalized cohomology condition.  相似文献   

4.
We discuss the parametrization of quantum groups in terms of independent operators. We find that this consideration leads to the parametrization ofSU q(2) in terms of aq-oscillator plus a commuting phase. The commuting phase is naturally identified with the subgroupU(1) and the remaining cosetSU q(2)/U(1)=CP q(1) consists of aq-oscillator. For unitary quantum groupsSU q (n), the analogous construction results in the quantum projective spaceSU q(n+1)/U q (n)=CP q (n) being identified with then-dimensionalq-oscillator. This yields a nonlinear action of the quantum groupSU q(n+1) on then-dimensionalq-oscillator.  相似文献   

5.
The classical Frobenius-Schur duality gives a correspondence between finite dimensional representations of the symmetric and the linear groups. The goal of the present paper is to extend this construction to the quantum toroidal setup with only elementary (algebraic) methods. This work can be seen as a continuation of [J, D1 and C2] (see also [C-P and G-R-V]) where the cases of the quantum groups U q (sl(n)), Y(sl(n)) (the Yangian) and U q (sl(n)) are given. In the toroidal setting the two algebras involved are deformations of Cherednik's double affine Hecke algebra introduced in [C1] and of the quantum toroidal group as given in [G-K-V]. Indeed, one should keep in mind the geometrical construction in [G-R-V] and [G-K-V] in terms of equivariant K-theory of some flag manifolds. A similar K-theoretic construction of Cherednik's algebra has motivated the present work. At last, we would like to lay emphasis on the fact that, contrary to [J, D1 and C2], the representations involved in our duality are infinite dimensional. Of course, in the classical case, i.e.,q=1, a similar duality holds between the toroidal Lie algebra and the toroidal version of the symmetric group. The authors would like to thank V. Ginzburg for a useful remark on a preceding version of this paper. Communicated by M. Jimbo  相似文献   

6.
7.
We establish the connection between certain quantum algebras and generalizedClifford algebras (GCA). To be precise, we embed the quantum tori Lie algebraand U q (sl(2)) in GCA.  相似文献   

8.

We construct representations of the quantum algebras Uq,q(gl(n)) and Uq,q(sl(n)) which are in duality with the multiparameter quantum groups GLqq(n), SLqq(n), respectively. These objects depend on n(n − 1)/2+ 1 deformation parameters q, qij (1 ≤ i< jn) which is the maximal possible number in the case of GL(n). The representations are labelled by n − 1 complex numbers ri and are acting in the space of formal power series of n(n − 1)/2 non-commuting variables. These variables generate quantum flag manifolds of GLqq(n), SLqq(n). The case n = 3 is treated in more detail.

  相似文献   

9.
The quantum group structure of 2D gravity recently put forward by one of us (J.-L. G.) is used to study quantum gravity on the strip. The boundary conditions, previously studied by A. Neveu and this author become easy to implement when one introduces the universal family of chiral operators associated withU q (sl(2)). A general formula for inverse powers of the metric-tensor operator is thereby derived. It contains a new universal matrixA, acting in representation-space, which obeys identities involving theR matrix, the Clebsch-Gordon coefficients, and the co-products ofU q (sl(2)). The physical meaning of these identities is to ensure that these powers of the metric are local and closed by fusion.  相似文献   

10.
We give the Heisenberg realization for the quantum algebra U q (sl n ), which is written by theq-difference operator on the flag manifold. We construct it from the action of U q (sl n ) on theq-symmetric algebraA q (Mat n ) by the Borel-Weil-like approach. Our realization is applicable to the construction of the free field realization for U q [2].  相似文献   

11.
We give explicit realization for the quantum enveloping algebras U q(B n). In these formulae the generators of the algebra are expressed by means of 2n–1 canonical q-boson pairs and one auxiliary representation of U q(B n–1)  相似文献   

12.
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.  相似文献   

13.
It is shown that the quantum supergroup U q (osp(1/2n)) 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.  相似文献   

14.
Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groupsU q (g). They have the same FRT generatorsl ± but a matrix braided-coproductL=LL, whereL=l + Sl , and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matricesBM q(2); it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum doubleD(U q (sl 2)) (also known as the quantum Lorentz group) is the semidirect product as an algebra of two copies ofU q (sl 2), and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups.  相似文献   

15.
An analog of the minimal unitary series representations for the deformed Virasoro algebra is constructed using vertex operators of the quantum affine algebra Uq(sl2). A similar construction is proposed for the elliptic algebra Aq,p(sl2).  相似文献   

16.
We characterize the finite-dimensional representations of the quantum affine algebra U q ( n+1) (whereq × is not a root of unity) which are irreducible as representations of U q (sl n+1). We call such representations small. In 1986, Jimbo defined a family of homomorphismsev a from U q (sl n+1) to (an enlargement of) U q (sl,n+1), depending on a parametera ·. A second family,ev a can be obtained by a small modification of Jimbo's formulas. We show that every small representation of U q ( n+1) is obtained by pulling back an irreducible representation of U q (sl n+1) byev a orev a for somea ·.  相似文献   

17.
The structure of all discrete series of unitary irreducible representations of the U q (u(3, 1)) and U q (u(n, 1)) noncompact quantum algebras are investigated with the aid of extremal projection operators and the q-analog of the Mickelsson-Zhelobenko algebra Z(g, g′) q . The orthonormal basis constructed in the infinite-dimensional space of irreducible representations of the U q (u(n, 1)) ⊇ U q (u(n)) algebra is the q-analog of the Gelfand-Graev basis in the space of the corresponding irreducible representations of the u(n, 1) ⊇ u(n) classical algebra.  相似文献   

18.
The nonstandardU z sl(2, IR) quantum algebra is considered together with other nonstandard algebras sharing the same universalR-matrix as well as a fixed Hopf subalgebra. Some boson realizations for these nonstandard algebras are obtained which are later used in order to compute in a simplified way their (finite and infinite dimensional) representations. In the limit when the deformation parameterz vanishes these realizations turn into the well known (one or two-boson) Gelfand-Dyson realizations for the corresponding classical Lie algebras.  相似文献   

19.
When the deformation parameter is a root of unity, the centre of a quantum group can be described by a set of generators and non trivial relations. In the case ofU q (sl(N)), these relations simply derive from the expressions of the deformed Casimir operators. In the case ofU q (osp(1|2)), the relation is simple if we use an operator which anticommutes with the fermionic generators and whose square is the quadratic Casimir. This operator also simplifies the classification of finite dimensional irreducible representations. In the case ofU q (sl(1|2)), the relations derive from the (infinite set of) standard Casimir operators.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.  相似文献   

20.
In defining quantum superalgebras, extra relations need to be added to the Serre-like relations. They are obtained for sl q (m, n) and osp q (m, 2n) usingq-oscillator representations.Supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada.  相似文献   

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