q-Deformed Orthogonal and Pseudo-Orthogonal Algebras and Their Representations

Deformed orthogonal and pseudo-orthogonal Lie algebras are constructed which differ from deformations of Lie algebras in terms of Cartan subalgebras and root vectors and which make it possible to construct representations by operators acting according to Gel’fand–Tsetlin-type formulas. Unitary representations of the q-deformed algebras Uq(son,1) are found. AMS subject classifications (1980). 16A58, 16A64, 17B10, 81D99. 1. In his Letter [1], M. Jimbo defined a q-deformation Uq(g) of any simple Lie algebra g by means of its Cartan subalgebra and root elements. M. Rosso has shown in [2] that to every integral highest weight there corresponds an irreducible finite-dimensional representation of Uq(g). For the q-deformed algebra Uq(sl(n,C)), finite-dimensional irreducible representations were explicitly constructed by M. Jimbo [3] through the q-analogue of the Gelfand–Tsetlin formulas. If one constructs the algebra Uq(so(n,C)) according to Jimbo’s formulas, then (as well as for the nondeformed case) it is impossible to derive irreducible finite-dimensional representations in terms of the formulas of the Gel’fand–Tsetlin type. In explicitly constructing representations of the Lie algebras of the orthogonal groups, one uses the generators Ik,k−1 = Ek,k−1 − Ek−1,k , where Eis is the matrix with the elements (Eis)jr = δijδsr. The purpose of this Letter is to propose another deformation Uq(so(n,C)) of the orthogonal algebras which allows one to construct the q-analogue of the Gel’fand–Tsetlin formulas for them. With the help of ∗-operations in Uq(so(n,C)) , it is possible to introduce a compact deformed algebra Uq(son) and pseudo-orthogonal deformed algebras Uq(sor,s), r + s = n. We derive ‘unitary’ representations (that is, ∗-representations) of the deformed Lorentz algebras Uq(son,1) . It turns out that, unlike the classical Lie algebra so(n, 1), for the algebras Uq(son,1) there also appears a continuous ‘unitary’ series (strange series) of representations. Under q → 1, this series disappears (goes to infinity). 2. By the quantum algebra Uq(so(n,C)), n ≥ 3, we shall mean the complex associative algebra generated by the elements Ii,i−1, i = 2, . . . , n , which obey the relations [Ii,i−1, Ij,j−1] = 0 if | i− j |> 1, (1)