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1.
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   

2.
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) ).  相似文献   

3.
4.
The dually conjugate Hopf algebrasFun p,q (R) andU p,q (R) associated with the two-parametric (p,q)-Alexander-Conway solution (R) of the Yang-Baxter equation are studied. Using the Hopf duality construction, the full Hopf structure of the quasitriangular enveloping algebraU p,q (R) is extracted. The universal ?-matrix forsFun p,q (R) is derived. While expressing an arbitrary group element of the quantum group characterized by the noncommuting parameters in a representation independent way, the ?-matrix generalizes the familiar exponential relation between a Lie group and its Lie algebra. The universal ?-matrix and the FRT matrix generators,L (±), forU p,q (R) are derived from the ?-matrix.  相似文献   

5.
A quantum algebraU p, q (,H,X ±) associated with a nonstandardR-matrix with two deformation parameters (p, q) is studied and, in particular, its universal -matrix is derived using Reshetikhin's method. Explicit construction of the (p, q)-dependent nonstandardR-matrix is obtained through a coloured generalized boson realization of the universal -matrix of the standardU p, q(gl(2)) corresponding to a nongeneric case. General finite dimensional coloured representation of the universal -matrix ofU p, q(gl(2)) is also derived. This representation, in nongeneric cases, becomes a source for various (p, q)-dependent nonstandardR-matrices. Superization ofU p, q(,H,X ±) leads to the super-Hopf algebraU p, q(gl(1/1)). A contraction procedure then yields a (p, q)-deformed super-Heisenberg algebraU p, q(sh(1)) and its universal -matrix.  相似文献   

6.
Invertible universal ?-matrices of quantum Lie algebras do not exist at roots of unity. However, quotients exist for which intertwiners of tensor products of representations always exist, i.e. ?-matrices exist in the representations. One of these quotients, which is finite-dimensional, has a universal ?-matrix. In this Letter we answer the following question: under which condition are the different quotients of U q (sl(2)) (Hopf)-equivalent? In the case when they are equivalent, the universal ?-matrix of the one can be transformed into a universal ?-matrix of the other. We prove that this happens only whenq 4 = 1, and we explicitly give the expressions for the automorphisms and for the transformed universal ?-matrices in this case.  相似文献   

7.
We define theq-version of the Weyl group for quantized universal enveloping algebras of simple Lie group and we find explicit multiplicative formulas for the universalR-matrix.Supported in part by the Department of Energy under Grant DE-FG02-88ER25065  相似文献   

8.
We establish an explicit algebra isomorphism between the quantum reflection algebra for the Uq([^(sl2)]) R{U_q(\widehat{sl_2}) R}-matrix and a new type of current algebra. These two algebras are shown to be two realizations of a special case of tridiagonal algebras (q-Onsager).  相似文献   

9.
A general method is developed for constructing quantum group invariants and determining their eigenvalues. Applied to the universalR-matrix this method leads to the construction of a closed formula for link polynomials. To illustrate the application of this formula, the quantum groupsU q (E 8),U q (so(2m+1) andU q (gl(m)) are considered as examples, and corresponding link polynomials are obtained.  相似文献   

10.
Crystal algebra     
We define the crystal algebra, an algebra which has a base of elements of crystal bases of a quantum group. The multiplication is defined by the tensor product rule of crystal bases. A universal n-colored crystal algebra is defined. We study the relation between those algebras and the tensor algebras of the crystal algebra of U q (sl(2)) and give a presentation by generators and relations for the case of U q (sl(n)).  相似文献   

11.
The spectrum of the transfer matrices corresponding to trigonometrical Bazhanov-Jimbo R matrices is found. The Bethe equations characterizing the eigenvalues of the transfer matrices are written down in terms of root systems. Using the generalization of the Bethe equations for Kac-Moody algebras D inf4 sup(3) , G inf2 sup(1) , E inf6 sup(1) and E inf6 sup(2) , we give conjectures for the eigenvalues of the corresponding transfer matrices.  相似文献   

12.
A compact form for the universalR-matrix of U q (sl n ) is derived and illustrated by simple applications.  相似文献   

13.
For the quantum groupGL p,q (2) and the corresponding quantum algebraU p,q (gl(2)) Fronsdal and Galindo [Lett. Math. Phys.27 (1993) 59] explicitly constructed the so-called universalT-matrix. In a previous paper [J. Phys. A28 (1995) 2819] we showed how this universalT-matrix can be used to exponentiate representations from the quantum algebra to get representations (left comodules) for the quantum group. Here, further properties of the universalT-matrix are illustrated. In particular, it is shown how to obtain comodules of the quantum algebra by exponentiating modules of the quantum group. Also the relation with the universalR-matrix is discussed.Presented at the 4th International Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.  相似文献   

14.
Intertwining relations for the quantumR-matrix of theSU p,q (2) invariant spin chain are obtained and the corresponding face model is deduced. An important difference is seen to arise due to the asymmetry generated by the parametersp andq, which leads to a asymmetric face model. An algebraic Bethe ansatz is set up and solved with the help of these intertwining vectors.  相似文献   

15.
Nonstandard q-deformed algebras U q(so3) and U q(so4), which can be embedded into U q(sl3) and U q(sl4) and are coideals in them, are considered. It is shown how to multiply finite dimensional representations of U q(so3) when q is positive. Homomorphisms from U q(so3) and U q(so4) to the q-oscillator algebras are given. By making use of these homomorphisms, irreducible representations of U q(so3) and U q(so4) for q equal to a root of unity are obtained.  相似文献   

16.
We present an integral formula for the universal R-matrix of quantum affine algebra U q with Drinfeld comultiplication. We show that the properties of the universal R-matrix follow from the factorization properties of the cycles in proper configuration spaces. For general g we conjecture that such cycles exist and unique. For U q we describe precisely the cycles and present a new simple expression for the universal R-matrix as a result of calculation of corresponding integrals.  相似文献   

17.
The Borel-Weil (BW) construction for unitary irreps of a compact Lie group is extended to a construction of all unitary irreps of the quantum groupU q(n). Thisq-BW construction uses a recursion procedure forU q(n) in which the fiber of the bundle carries an irrep ofU q(n–1)×U q(1) with sections that are holomorphic functions in the homogeneous spaceU q(n)/U q(n–1)×U q(1). Explicit results are obtained for theU q(n) irreps and for the related isomorphism of quantum group algebras.Supported in part by the National Science Foundation, No. PHY-9008007  相似文献   

18.
We address the problem of duality between the colored extension of the quantized algebra of functions on a group and that of its quantized universal enveloping algebra, i.e., its dual. In particular, we derive explicitly the algebra dual to the colored extension of GL q(2) using the colored RLL relations and exhibit its Hopf structure. This leads to a colored generalization of the R-matrix procedure to construct a bicovariant differential calculus on the colored version of GL q(2). In addition, we also propose a colored generalization of the geometric approach to quantum group duality given by Sudbery and Dobrev.  相似文献   

19.
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   

20.
We provide a braid group action on theq-deformed Weyl algebraW q (n). The restriction of this action to the representations ofU q (A n–1 ) andU q (C n ) inW q (n) is seen to agree with the braid group action introduced by Lusztig on these quantum algebras.Supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada.  相似文献   

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