Representations of Multiparameter Quantum Groups
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ELEMENTARY PARTICLES AND FIELDS Theory
Representations of Multiparameter Quantum Groups∗ V. K. Dobrev** Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, 1784 Bulgaria Received April 17, 2018
Abstract—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 < j ≤ n) 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. DOI: 10.1134/S1063778818060121
1. INTRODUCTION About 30 years passed since the advent of quantum groups at center-stage of modern mathematical physics, cf., e.g., [1–10]. Yet the field is growing stronger every day, cf. a recent review in [11]. With the present paper we contribute to the continuing in-depth studies of quantum groups by constructing representations of multiparameter quantum groups on the example of such multiparameter deformations of the group GL(n). We follow the approach of [12] as adopted to quantum groups in [13, 14] (see also [11]). We start with the multiparameter quantum groups GLqq (n), introduced by Sudbery [15] depending on n(n − 1)/2 + 1 deformation parameters q, qij (1 ≤ i < j ≤ n) which is the maximal possible number in the case of GL(n). We construct representations of the quantum algebras Uq,q (gl(n)) and Uq,q (sl(n)) (found in [16]) which are in duality with the multiparameter quantum groups GLqq (n), SLqq (n), respectively. 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). We treat the case n = 3 in greater detail. 2. MULTIPARAMETRIC DEFORMATION OF GL(n) Here we use the quantum group deformation of GL(n) introduced by Sudbery [15]. That deforma∗ **
The text was submitted by the author in English. E-mail: [email protected]
tion depends on the maximal possible number of parameters: N = n(n − 1)/2 + 1. We denote these N parameters by q and qij , 1 ≤ i < j ≤ n, and also for shortness by the pair q, q. The standard oneparameter deformation is obtained by setting qij = q, ∀i, j. Explicitly, we consider an n × n quantum matrix M with non-commuting matrix elements aij , 1 ≤ i, j ≤ n. The matrix quantum group A ≡ GLqq (n) is generated by the matrix elements aij with the following commutation relations [15]: aij ai = pai aij , for j < , aij akj = rakj aij , for i < k, pai akj = rakj ai , for i < k, j < ,
(1a) (1b) (1c)
rqak aij − (qp)−1 aij ak = λai akj , for i < k, j < ,
(1d)
λ = q − 1/q.
(1e)
2
p = qj /q ,
r = 1/qik ,
Considered as a bialgebra, it has the standard comultiplication δA and couni
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