Induced Representations of U_{q}so(5)

Preeti Parashar

Max-Planck Institute fur Mathematik Vivatsgasse 7, 53111 Bonn, Germany

Abstract

We study the reduction of the left regular representation of the quantum algebra U_{q}so(5). This yields induced representations of U_{q}so(5) on some quotient space of the matrix quantum group SO_{q}(5). To facilitate the construction of a suitable basis the Gauss decomposition of SO_{q}(5) is worked out explicitly. We then investigate the behaviour of these representations for a generic q as well as when q is an N-th root of unity leading to some finite dimensional irreducible representations of U_{q}so(5). Two basic intertwining operators are also constructed via the right action.

I. INTRODUCTION

A canonical method for the construction of differential operators intertwining representations of semisimple Lie groups was proposed in [1]. This rather algebraic procedure was generalized to the case of quantum groups also. The left regular representation , its reduction to infinite family of reducible and irreducible representaions and the q- difference intertwining operators were studied on many quantum algebras , namely, the Lorentz quantum algebra [2], the Euclidean algebra U_{q}e(2) [3] (as a contaction of U_{q}sl(2)) and generalizations to U_{q}sl(3) [4] and U_{q}sl(n) [5], [6]. Recently induced representations of the three dimensional quantum orthogonal algebra U_{q}so(3) have also been obtained [7].

In this paper we investigate the case of U_{q}so(5, C I) , a quantization of the 5 dimensional complex special orthogonal Lie algbra of rank two. For simplicity we shall prefer to call it just U_{q}so(5) (and similarly for the dual group). Besides being computationally horrendous, it presents some interesting features which would facilitate generalization to the n-dimensional case and also help us to understand better the important relation between representation theory and the geometry of quantum groups. On the other hand, one of the real forms of this algebra obtained by conjugation is the non-compact quantum Anti-de Sitter (AdS) algebra U_{q}so(3, 2) which happens to be the symmetry algebra of the non-commutative AdS space. This aroused a lot of interest specially after the AdS - Conformal field theory duality conjecture was formulated, see for example [8] and references therein for physical applications. This connection motivates us further to look at U_{q}so(5) in somewhat more detail but mainly from the point of view of representation theory.

We start out by defining the quantum matrix group SO_{q}(5) and the quantum algebra U_{q}so(5) in Sec. II. The duality pairings between these two Hopf algebras are explicitly given. In Sections III and IV we present the left and right regular representations of U_{q}so(5) on the matrix elements (generators) of the dual group SO_{q}(5). Recall that the classical SO(5) group is defined by 10 independent generators, whereas here our quantum group SO_{q}(5) has 25 generators. We must therefore, redefine these generators to get only 10 independent ones. The monomials in these whould give us a suitable basis to extend the left and right action. This is achieved by Gauss decomposition of the quantum matrix belonging to SO_{q}(5) and is explicitly worked out in Sec. V.

Next, in Sec. VI we carry out the procedure of reduction of the left regular representation. As in the classical case the left and right actions commute ( due to associativity).

Therefore, we use the right covariance to reduce the left regular representation i.e., we select the eigenspaces of the right regular representation of some particular generators of the quantum algebra U_{q}so(5) . This basically implies imposing the condition that the Cartan generators act as a multiplication by a constant and that some of the aforementioned generators act as a multiplication by zero. In this way we obtain an infinite family of induced representations of U_{q}so(5) realized on the non-commutative coset space of SO_{q}(5). In Sec. VII we investigate the reducibility of these induced representaions when q is in general a complex number and also when q is an N-th root of unity. This leads to some finite dimensional irreducible representations under some restricted values of the labels which parametrize them. The right action of the remaining generators on the basis is presented in Sec. VIII. These remaining U_{q}so(5) generators yield ( via the right action ) interesting intertwining operators which we give in Sec. IX . The explicit commutation relations of the quantum group SO_{q}(5) are contained in the Appendix. We conclude the paper with some remarks.

II. SO_{q}(5) AND THE DUAL U_{q}so(5)

We begin by defining the matrix quantum group SO_{q}(5) and its corresponding quantum universal enveloping algebra U_{q}so(5) which can be recovered from the general formulae of [9].

SO_{q}(5) is generated by I and elements T = (t_{ij}) , i, j = 1, ..5 obeying the following commutaion relations :

RT_{1}T_{2} = T_{2}T_{1}R  (2.1)
T CT^{t}C^{−1} = CT^{t}C^{−1}T = I  (2.2)

where R is a quantum 5^{2} × 5^{2} invertible matrix, T_{1} = T ⊗I , T_{2} = I⊗T, I is the identity 5 × 5 matrix, and C is an (antidiagonal) 5 × 5 matrix. The coalgebra structure is given by the following co-product Δ, co-unit ε, and antipode S, :

Δ(t_{ij}) = ∑_{k=1}^{5} t_{ik} ⊗ t_{kj} ,  (2.3a)
ε(t_{ij}) = δ_{ij} ,  (2.3b)
S(T) = C T^{t} C^{−1} ,  (2.3c)

Using this compact matrix form relations (2.2) can be rewritten in the general form:

T S(T) = S(T) T = I  (2.4)

The quantum R- matrix is given by

\begin{aligned}R = & q ∑_{i≠i′}^{5} e_{ii} ⊗ e_{ii} + e_{3,3} ⊗ e_{3,3} + ∑_{i≠j,j′}^{5} e_{ii} ⊗ e_{jj} \\ & + q^{−1} ∑_{i≠i′}^{5} e_{i′i′} ⊗ e_{ii} + λ ∑_{i>j}^{5} e_{ij} ⊗ e_{ji} − λ ∑_{i>j}^{5} q^{ρ_{i}−ρ_{j}}e_{ij} ⊗ e_{i′j′} , \end{aligned}  (2.5)

with λ = q − q^{−1}, (ρ_{1}, ....ρ_{5}) = (3/2, 1/2, 0, −1/2, −3/2) and i′ = 5 + 1 − i. For SO_{q}(5) the quantum R matrix is a lower triangular 25 × 25 matrix whose non-zero elements are explicitly given below (R_{ij}^{kl} = e_{ik} ⊗ e_{jl}) :

\begin{matrix}R_{ii}^{ii} = q , i ≠ 3 , R_{ii′}^{ii′} = q^{−1} , i ≠ 3 , R_{33}^{33} = 1 , \\ R_{ij}^{ij} = 1 , i ≠ j, j′ , R_{ij}^{ji} = λ , i > j , i ≠ j′ , \\ R_{33}^{24} = R_{42}^{33} = −λq^{−1/2} = α , R_{42}^{24} = −λ(q^{−1} − 1) = β , \\ R_{42}^{15} = R_{51}^{24} = −λq^{−2} = γ , R_{51}^{15} = −λ(q^{−3} − 1) = δ , \\ R_{33}^{15} = R_{51}^{33} = −λq^{−3/2} = η , R_{24}^{15} = R_{51}^{42} = −λq^{−1} = ξ. \end{matrix}  (2.6)

The corresponding antidiagonal C-matrix is:

C = ( \begin{matrix}0 & 0 & 0 & 0 & q^{−3/2} \\ 0 & 0 & 0 & q^{−1/2} & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & q^{1/2} & 0 & 0 & 0 \\ q^{3/2} & 0 & 0 & 0 & 0 \end{matrix} ) ,  C^{2} = I  (2.7)

Inorder to derive the explicit commutation relations among the 25 generators t_{ij} of SO_{q}(5) we need to substitute the above R and C matrices in the defining relations (2.1) and (2.2) . By doing so we get an overwhelming number of relations (≈ 400) . These have been condensed to less then 40 in such a way that transparency is not lost. They are given in the Appendix as they would be essential for the Gauss decomposition in section V.

The quantum Universal enveloping algebra U_{q}so(5) dual to SO_{q}(5) is generated by the unit 1 and the functionals l_{ij}^{+} with i < j; l_{ij}^{−} with i > j; l_{ii}^{+} and l_{ii}^{−}, where i, j = 1, ..., 5.

When arranged in upper- and lower-triangular matrices L^{±}, respectively, they are defined by the duality conditions [9]

(L^{±}, T_{1} … T_{m}) = R_{1}^{±} … R_{m}^{±} , for m = 1, 2, …;  (2.8)

where for 1 ≤ ℓ ≤ m, T_{ℓ}^{±} act in the ℓ^{th} factor; R_{ℓ}^{±} act in the (0, ℓ)^{th} factor of (C I ^{5})^{⊗(m+1)}; R^{+} = P RP , R^{−} = R^{−1} and P ∈ Mat(C I ^{5} ⊗ C I ^{5}) is the permutation matrix.

The commutation relations are written in a compact form

R^{+}L_{1}^{±}L_{2}^{±} = L_{2}^{±}L_{1}^{±}R^{+} , R^{+}L_{1}^{+}L_{2}^{−} = L_{2}^{−}L_{1}^{+}R^{+}  (2.9)

Additional constraints are:

l_{ii}^{+}l_{ii}^{−} = l_{ii}^{−}l_{ii}^{+} = 1, i = 1, .., 5 ;  (2.10)
l_{11}^{+}....l_{55}^{+} = l_{11}^{−}....l_{55}^{−} = 1 ;  (2.11)
L^{±}C^{t}(L^{±})^{t}(C^{−1})^{t} = C^{t}(L^{±})^{t}(C^{−1})^{t}L^{±} = I.  (2.12)

The complete Hopf algebra structure is given by

Δ(l_{ij}^{±}) = ∑_{k=1}^{5} l_{ik}^{±} ⊗ l_{kj}^{±}  (2.13)
ε(l_{ij}^{±}) = δ_{ij}  (2.14)
S(L^{±}) = C^{t}(L^{±})^{t}(C^{−1})^{t}  (2.15)

Since we shall be needing the explicit form of the antipode in defining the left action in the next section, we give it below.

S(L^{+}) = ( \begin{matrix}l_{55}^{+} & ql_{45}^{+} & q^{3/2}l_{35}^{+} & q^{2}l_{25}^{+} & q^{3}l_{15}^{+} \\ 0 & l_{44}^{+} & q^{1/2}l_{34}^{+} & ql_{24}^{+} & q^{2}l_{14}^{+} \\ 0 & 0 & l_{33}^{+} & q^{1/2}l_{23}^{+} & q^{3/2}l_{13}^{+} \\ 0 & 0 & 0 & l_{22}^{+} & ql_{12}^{+} \\ 0 & 0 & 0 & 0 & l_{11}^{+} \end{matrix} )  (2.16)

S(L^{−}) = ( \begin{matrix}l_{55}^{−} & 0 & 0 & 0 & 0 \\ q^{−1}l_{54}^{−} & l_{44}^{−} & 0 & 0 & 0 \\ q^{−3/2}l_{53}^{−} & q^{−1/2}l_{43}^{−} & l_{33}^{−} & 0 & 0 \\ q^{−2}l_{52}^{−} & q^{−1}l_{42}^{−} & q^{−1/2}l_{32}^{−} & l_{22}^{−} & 0 \\ q^{−3}l_{51}^{−} & q^{−2}l_{41}^{−} & q^{−3/2}l_{31}^{−} & q^{−1}l_{21}^{−} & l_{11}^{−} \end{matrix} )  (2.17)

The l_{ij}^{±} can be expressed in terms of more popular generators H_{i}, X_{i}^{±}, i = 1, 2 via:

\begin{matrix}q^{H_{2}} = l_{11}^{+}(l_{22}^{+})^{−1}, & q^{H_{1}} = (l_{22}^{+})^{2}, \\ X_{2}^{−} = λ^{−1}q^{1/2}l_{12}^{+}(l_{11}^{+})^{−1/2}(l_{22}^{+})^{−1/2}, & X_{1}^{−} = λ^{−1}\sqrt{[}2]_{q}q^{1/2}l_{23}^{+}(l_{22}^{+})^{−1/2}, \\ X_{2}^{+} = −λ^{−1}q^{−1/2}l_{21}^{−}(l_{11}^{+})^{1/2}(l_{22}^{+})^{1/2}, & X_{1}^{+} = −λ^{−1}\sqrt{[}2]_{q}l_{32}^{−}(l_{22}^{+})^{1/2}. \end{matrix}  (2.18)

They satisfy the following commutation relations:

\begin{matrix}[H_{i}, H_{j}] = 0, [H_{i}, X_{j}^{±}] = ±(a_{ij})X_{j}^{±}, i, j = 1, 2, \\ [X_{i}^{+}, X_{j}^{−}] = δ_{ij}[H_{i}]_{q_{i}} , q_{1} = q, q_{2} = q^{2}, \\ (X_{1}^{±})^{3}X_{2}^{±} − [3]_{q_{1}}(X_{1}^{±})^{2}X_{2}^{±}X_{1}^{±} + [3]_{q_{1}}X_{1}^{±}X_{2}^{±}(X_{1}^{±})^{2} − X_{2}^{±}(X_{1}^{±})^{3} = 0, \\ (X_{2}^{±})^{2}X_{1}^{±} − [2]_{q_{2}}X_{2}^{±}X_{1}^{±}X_{2}^{±} + X_{1}^{±}(X_{2}^{±})^{2} = 0, \end{matrix}  (2.19)

where [x]_{q_{i}} = \frac{(q_{i}^{x/2}−q_{i}^{−x/2})}{(q_{i}^{1/2}−q_{i}^{−1/2})} and a_{ij} = ( \begin{matrix}2 & −2 \\ −1 & 2 \end{matrix} ) is the Cartan matrix for so(5). The

duality between U_{q}so(5) and SO_{q}(5) is given by the pairing (2.8) between the generators H_{i}, X_{i}^{±}and the T matrix:

\begin{matrix}⟨H_{1}, T ⟩ =( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −2 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ), ⟨H_{2}, T ⟩ =( \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & −1 \end{matrix} ), \\ ⟨X_{1}^{+}, T ⟩ = \sqrt{[}2]_{q_{1}} ( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ), ⟨X_{1}^{−}, T ⟩ = \sqrt{[}2]_{q_{1}} ( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ), \\ ⟨X_{2}^{+}, T ⟩ =( \begin{matrix}0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & −1 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ), ⟨X_{2}^{−}, T ⟩ =( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 & 0 \end{matrix} ). \end{matrix} (2.20)

To complete the Hopf algebra structure, the other maps are given as (i = 1, 2):

Δ(H_{i}) = 1 ⊗ H_{i} + H_{i} ⊗ 1 ,  (2.21a)
Δ(X_{i}^{±}) = q_{i}^{H_{i}/4} ⊗ X_{i}^{±} + X_{i}^{±} ⊗ q_{i}^{−H_{i}/4} ,  (2.21b)
ε(H_{i}) = 0 ,  ε(X_{i}^{±}) = 0 ,  (2.21c)
S(H_{i}) = −H_{i} ,  S(X_{i}^{±}) = −q_{i}^{−H_{i}/4}X_{i}^{±}q_{i}^{H_{i}/4} ,  (2.21d)

III. LEFT REGULAR REPRESENTATION The left regular representation of U_{q}so(5) is defined in a standard way as

π_{L}(X) t_{ij} = ∑_{k=1}^{5} ⟨S(X), t_{ik}⟩ t_{kj}, X ∈ U_{q}so(5)  (3.1)

which when q = 1 can be seen as the infinitesimal version of

π_{L}(T′) T = T′^{−1} T, T′ , T ∈ SO(5)  (3.2)

In the present case it is explicitly given as:

\begin{matrix}π_{L}(H_{1})T = 2 ( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ −t_{21} & −t_{22} & −t_{23} & −t_{24} & −t_{25} \\ 0 & 0 & 0 & 0 & 0 \\ t_{41} & t_{42} & t_{43} & t_{44} & t_{45} \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ) \\ π_{L}(H_{2})T = ( \begin{matrix}−t_{11} & −t_{12} & −t_{13} & −t_{14} & −t_{15} \\ t_{21} & t_{22} & t_{23} & t_{24} & t_{25} \\ 0 & 0 & 0 & 0 & 0 \\ −t_{41} & −t_{42} & −t_{43} & −t_{44} & −t_{45} \\ t_{51} & t_{52} & t_{53} & t_{54} & t_{55} \end{matrix} ) \\ π_{L}(X_{1}^{+})T = q_{1}^{−1/2}\sqrt{[}2]_{q_{1}} ( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ −t_{31} & −t_{32} & −t_{33} & −t_{34} & −t_{35} \\ t_{41} & t_{42} & t_{43} & t_{44} & t_{45} \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ) \\ π_{L}(X_{1}^{−})T = q_{1}^{1/2}\sqrt{[}2]_{q_{1}} ( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ −t_{21} & −t_{22} & −t_{23} & −t_{24} & −t_{25} \\ t_{31} & t_{32} & t_{33} & t_{34} & t_{35} \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ) \\ π_{L}(X_{2}^{+})T = q_{2}^{−1/2} ( \begin{matrix}−t_{21} & −t_{22} & −t_{23} & −t_{24} & −t_{25} \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ t_{51} & t_{52} & t_{53} & t_{54} & t_{55} \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ) \\ π_{L}(X_{2}^{−})T = q_{2}^{1/2} ( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ −t_{11} & −t_{12} & −t_{13} & −t_{14} & −t_{15} \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ t_{41} & t_{42} & t_{43} & t_{44} & t_{45} \end{matrix} ) \end{matrix}  (3.3)

supplemented by the following action on the unit element of SO_{q}(5):

π_{L}(H_{i}) I = 0, π_{L}(X_{i}^{±}) I = 0, i = 1, 2  (3.4)

The action π_{L} can be extended on arbitrary monomials by making use of the following twisted derivation rule consistent with the coproduct and the representation structure:

π_{L}(x)a b = m (π_{L}(σ ∘ Δ(x))(a ⊗ b))  (3.5)

where m is the multiplication map and σ is the permutation operator. Explicitly we obtain

π_{L}(H_{i})a b = a · π_{L}(H_{i}) b + π_{L}(H_{i}) a · b ,  (3.6a)
π_{L}(X_{i}^{±})a b = π_{L}(X_{i}^{±}) a · π_{L}(q_{i}^{H_{i}/4}) b + π_{L}(q_{i}^{−H_{i}/4}) a · π_{L}(X_{i}^{±}) b  (3.6b)

Further we have :

\begin{matrix}π_{L}(xy) = & π_{L}(x) · π_{L}(y) , \\ π_{L}(αx + βy) = & απ_{L}(x) + βπ_{L}(y) ,  α, β ∈ C I , x, y ∈ U_{q}so(5). \end{matrix}  (3.7)

IV. RIGHT REGULAR REPRESENTATION

Analogous to the previous section we now introduce the right regular representation of U_{q}so(5) on SO_{q}(5) as

π_{R}(X) t_{ij} = ∑_{k=1}^{5} t_{ik} ⟨X , t_{kj}⟩, X ∈ U_{q}so(5)  (4.1)

which in the classical case is the infinitesimal counterpart of

π_{R}(T′) T = T T′ , T , T′ ∈ SO(5)  (4.2)

The explicit expressions are given below in matrix form:

\begin{matrix}π_{R}(H_{1})T = 2 ( \begin{matrix}0 & t_{12} & 0 & −t_{14} & 0 \\ 0 & t_{22} & 0 & −t_{24} & 0 \\ 0 & t_{32} & 0 & −t_{34} & 0 \\ 0 & t_{42} & 0 & −t_{44} & 0 \\ 0 & t_{52} & 0 & −t_{54} & 0 \end{matrix} ) \\ π_{R}(H_{2})T = ( \begin{matrix}t_{11} & −t_{12} & 0 & t_{14} & −t_{15} \\ t_{21} & −t_{22} & 0 & t_{24} & −t_{25} \\ t_{31} & −t_{32} & 0 & t_{34} & −t_{35} \\ t_{41} & −t_{42} & 0 & t_{44} & −t_{45} \\ t_{51} & −t_{52} & 0 & t_{54} & −t_{55} \end{matrix} ) \\ π_{R}(X_{1}^{+})T = \sqrt{[}2]_{q_{1}} ( \begin{matrix}0 & 0 & t_{12} & −t_{13} & 0 \\ 0 & 0 & t_{22} & −t_{23} & 0 \\ 0 & 0 & t_{32} & −t_{33} & 0 \\ 0 & 0 & t_{42} & −t_{43} & 0 \\ 0 & 0 & t_{52} & −t_{53} & 0 \end{matrix} ) \\ π_{R}(X_{1}^{−})T = \sqrt{[}2]_{q_{1}} ( \begin{matrix}0 & t_{13} & −t_{14} & 0 & 0 \\ 0 & t_{23} & −t_{24} & 0 & 0 \\ 0 & t_{33} & −t_{34} & 0 & 0 \\ 0 & t_{43} & −t_{44} & 0 & 0 \\ 0 & t_{53} & −t_{54} & 0 & 0 \end{matrix} ) \\ π_{R}(X_{2}^{+})T = ( \begin{matrix}0 & t_{11} & 0 & 0 & −t_{14} \\ 0 & t_{21} & 0 & 0 & −t_{24} \\ 0 & t_{31} & 0 & 0 & −t_{34} \\ 0 & t_{41} & 0 & 0 & −t_{44} \\ 0 & t_{51} & 0 & 0 & −t_{54} \end{matrix} ) \\ π_{R}(X_{2}^{−})T = ( \begin{matrix}t_{12} & 0 & 0 − t_{15} & 0 \\ t_{22} & 0 & 0 − t_{25} & 0 \\ t_{32} & 0 & 0 − t_{35} & 0 \\ t_{42} & 0 & 0 − t_{45} & 0 \\ t_{52} & 0 & 0 − t_{55} & 0 \end{matrix} ) \end{matrix}  (4.3)

Similar to the left action, the right action of the generators of U_{q}so(5) on the unit element of SO_{q}(5) is zero.

The twisted derivation rule is given by:

π_{R}(x)a b = m (π_{R}(Δ(x))(a ⊗ b))  (4.4)

which in the present situation translates into:

π_{R}(H_{i})a b = a · π_{R}(H_{i}) b + π_{R}(H_{i}) a · b ,  (4.5a)
π_{R}(X_{i}^{±})a b = π_{R}(X_{i}^{±}) a · π_{R}(q_{i}^{−H_{i}/4}) b + π_{R}(q_{i}^{H_{i}/4}) a · π_{R}(X_{i}^{±}) b  (4.5b)

Note that since π_{R} is a representation, relations similar to (3.7) hold for the right action also.

V. GAUSS DECOMPOSITION OF SO_{q}(5)

We know that the classical SO(5) group is defined in terms of 10 generators and the Poincare-Birkoff-Witt (PBW) basis is defined as monomials in these generators. We have seen in Sec. II that the quantum analogue SO_{q}(5) has 25 generators t_{ij} obeying hundred’s of commutation relations among them, which gives us lot of freedom to introduce the PBW basis. Here we shall adopt the method of Gauss decomposition to achieve it.

The Gauss decomposition for the quantum T matrix of the SO_{q}(5) group is

T = T_{L} T_{D} T_{U}  (5.1)

where T_{L} = (l_{ij}) is a strictly lower-triangular matrix with units on the main diagonal (l_{ii} = 1), T_{D} = diag (A_{ii}) and T_{U} = (u_{ij}) is a strictly upper-triangular matrix (u_{ii} = 1). All these are 5 × 5 matrices i.e., i, j = 1, ..., 5.

After very lengthy and cumbersome computations we find that the matrix elements can be described in terms of generators of which only ten are independent. The diagonal generators A_{ii} are assumed to be invertible.

\begin{matrix}A_{11} = t_{11}, A_{22} = t_{22} − t_{21}t_{11}^{−1}t_{12} = t_{22} − l_{21}A_{11}u_{12}, \\ A_{33} = t_{33} − l_{31}A_{11}u_{13} − l_{32}A_{22}u_{23} = 1, \\ A_{44} = A_{22}^{−1}, A_{55} = A_{11}^{−1}, \end{matrix}  (5.2)

\begin{matrix}u_{12} = t_{11}^{−1}t_{12}, u_{13} = t_{11}^{−1}t_{13}, u_{14} = t_{11}^{−1}t_{14}, \\ u_{15} = t_{11}^{−1}t_{15} = −u_{12}u_{14} − q[2]^{−1}u_{13}^{2}, \\ u_{23} = A_{22}^{−1}(t_{23} − t_{21}t_{11}^{−1}t_{13}), u_{24} = −[2]^{−1}u_{23}^{2} \\ u_{25} = −u_{24}u_{12} − q^{−1/2}u_{23}u_{13} − q^{−1}u_{14}, u_{34} = −q^{1/2}u_{23} \\ u_{35} = q^{1/2}u_{23}u_{12} − q^{−1/2}u_{13}, u_{45} = −u_{12}, \end{matrix}  (5.3)

\begin{matrix}l_{21} = t_{21}t_{11}^{−1}, l_{31} = t_{31}t_{11}^{−1}, l_{41} = t_{41}t_{11}^{−1}, \\ l_{51} = t_{51}t_{11}^{−1} = −l_{21}l_{41} − q[2]^{−1}l_{31}^{2}, \\ l_{32} = (t_{32} − t_{31}t_{11}^{−1}t_{12})A_{22}^{−1}, l_{42} = −[2]^{−1}l_{32}^{2}, \\ l_{52} = −l_{21}l_{42} − q^{1/2}l_{31}l_{32} − ql_{41}, l_{43} = −q^{−1/2}l_{32}, \\ l_{53} = q^{−1/2}l_{21}l_{32} − q^{1/2}l_{31}, l_{54} = −l_{21}. \end{matrix}  (5.4)

where [2] = \frac{(q−q^{−1})}{(q^{1/2}−q^{−1/2})}.

It turns out that A_{33} commutes with all the other generators, hence we have set it to 1. So the ten independent generators are A_{11}, A_{22}, u_{12}, u_{13}, u_{14}, u_{23} , l_{21}, l_{31}, l_{41} and l_{32}. Below we give the commutation relations among them (i, j = 2, 3, 4) by making use of the relations between the t_{ij} from the Appendix:

\begin{matrix}A_{11}u_{1i} = qu_{1i}A_{11}, A_{11}l_{i1} = ql_{i1}A_{11}, \\ A_{11}u_{23} = u_{23}A_{11}, A_{11}l_{32} = l_{32}A_{11}, \\ [u_{ki}, l_{jl}] = 0, k, l = 1, 2, k < i, j > l, A_{11}A_{22} = A_{22}A_{11}, \\ A_{22}u_{12} = q^{−1}u_{12}A_{22}, A_{22}l_{21} = q^{−1}l_{21}A_{22}, \\ A_{22}u_{13} = u_{13}A_{22}, A_{22}l_{31} = l_{31}A_{22}, \\ A_{22}u_{14} = qu_{14}A_{22}, A_{22}l_{41} = ql_{41}A_{22}, \\ A_{22}u_{23} = qu_{23}A_{22}, A_{22}l_{32} = ql_{32}A_{22}, \\ u_{12}u_{13} = qu_{13}u_{12}, l_{21}l_{31} = ql_{31}l_{21}, \\ u_{12}u_{14} = u_{14}u_{12} − λ[2]^{−1}u_{13}u_{13}, l_{21}l_{41} = l_{41}l_{21} − λ[2]^{−1}l_{31}l_{31}, \\ u_{12}u_{23} = q^{−1}u_{23}u_{12} + q^{−1}λu_{13}, l_{21}l_{32} = q^{−1}l_{32}l_{21} + λl_{31}, \\ u_{13}u_{14} = qu_{14}u_{13}, l_{31}l_{41} = ql_{41}l_{31}, \\ u_{13}u_{23} = u_{23}u_{13} − q^{−1/2}λu_{14}, l_{31}l_{32} = l_{32}l_{31} − q^{1/2}λl_{41}, \\ u_{14}u_{23} = qu_{23}u_{14}, l_{41}l_{32} = ql_{32}l_{41}. \end{matrix}  (5.5)

There is an inherent symmetry in the relations for u′s and l′s. Note that the diagonal generators A_{ii} form an abelian subgroup while A_{22}, u_{23} and l_{32} generate the subgroup SO_{q}(3). The above commutation relations obtained by Gauss decomposition are much simpler in form and easier to handle and would thus result in a convenient basis for SO_{q}(5).

VI. INDUCED REPRESENTATIONS

In this section we shall use the right action to reduce the left regular representation (which is highly reducible) . But before we do so we need to construct a basis from the above 10 independent generators . Let us choose the following ordering. Then the representation space will have elements which are formal power series in the basis :

ϕ = ∑ l_{21}^{n_{1}}l_{31}^{n_{2}}l_{41}^{n_{3}}l_{32}^{n_{4}}A_{11}^{n_{5}}A_{22}^{n_{6}}u_{12}^{n_{7}}u_{13}^{n_{8}}u_{14}^{n_{9}}u_{23}^{n_{10}}  (6.1)

Here the summation is over n_{i}, i = 1, 2, ..., 10. They take values in the set of nonnegative integers ZZ_{+} , except n_{5} and n_{6} which belong to the set of integers ZZ.

We shall now impose the condition of (infinitesimal) right covariance on this independent basis of SO_{q}(5). This essentially means annihilation by the raising operators X_{i}^{+} and scalar multiplication by exponentials of the Cartan generators H_{i} (similar to a highest weight module). Therefore, the right covariance conditions for the holomorphic representations with respect to X_{1}^{+}, X_{2}^{+}, K_{1} and K_{2} where K_{1} = q_{1}^{H_{1}/2}, K_{2} = q_{2}^{H_{2}/2} are given by :

π_{R}(X_{i}^{+}) ϕ = 0, i = 1, 2  (6.2a)
π_{R}(K_{1}) l_{21}^{n_{1}}....u_{23}^{n_{10}} = q^{n_{6}+n_{7}−n_{9}−n_{10}} l_{21}^{n_{1}}....u_{23}^{n_{10}}  (6.2b)
π_{R}(K_{2}) l_{21}^{n_{1}}....u_{23}^{n_{10}} = q^{n_{5}−n_{6}−2n_{7}−n_{8}+n_{10}} l_{21}^{n_{1}}....u_{23}^{n_{10}}  (6.2c)

All the l′s and A′s are annihilated by the right action of X_{1}^{+} and X_{2}^{+} whereas we get nontrivial action on the u′s . Thus (6.2 )implies that the function ϕ does not depend on u′s and hence it is reduced to :

ϕ = ∑_{{n_{1},..n_{4}∈Z Z_{+} \atop n_{5},n_{6}∈Z Z}} l_{21}^{n_{1}}l_{31}^{n_{2}}l_{41}^{n_{3}}l_{32}^{n_{4}}A_{11}^{n_{5}}A_{22}^{n_{6}}  (6.3)

For convenience let us denote by Φ the above ordered monomial i.e.,

Φ =: l_{21}^{n_{1}}l_{31}^{n_{2}}l_{41}^{n_{3}}l_{32}^{n_{4}}A_{11}^{n_{5}}A_{22}^{n_{6}}  (6.4)

Then the right covariance conditions on the reduced basis become

π_{R}(X_{i}^{+}) Φ = 0  (6.5a)
π_{R}(K_{1}) Φ = q^{n_{6}} Φ  (6.5b)
π_{R}(K_{2}) Φ = q^{n_{5}−n_{6}} Φ  (6.5c)

So the monomial Φ forms a basis in the space of common solutions of the equations (6.5.) The above algebra with PBW basis Φ may be viewed as the q-deformation of the quotient manifold SO(5)/U , where U is the subgroup of upper diagonal matrices with 1 on the diagonal.

Next, we shall reduce the left regular representation on the eigenspaces of the right regular representation obtained above. For this we need the left action of K_{1} and K_{2} on the reduced basis Φ. This turns out to be :

π_{L}(K_{1}) Φ = q^{−n_{1}+n_{3}+n_{4}−n_{6}} Φ  (6.6a)
π_{L}(K_{2}) Φ = q^{2n_{1}+n_{2}−n_{4}−n_{5}+n_{6}} Φ  (6.6b)

In order to proceed for the calculation of the left action of X_{i}^{+} and X_{i}^{−} on Φ , it would be convenient to first find the action on the powers of the individual generators. This is obtained by iterating the twisted derivation rule (3.6 )which yields

π_{L}(X_{1}^{+}) l_{21}^{n_{1}} = − \sqrt{[}2] q^{−n_{1}/2} [n_{1}]_{q^{2}} l_{21}^{n_{1}−1} l_{31}  (6.7a)
π_{L}(X_{1}^{+}) l_{31}^{n_{2}} = \sqrt{[}2] q^{−n_{2}/2} [n_{2}]_{q} l_{31}^{n_{2}−1} l_{41}  (6.7b)
π_{L}(X_{1}^{+}) l_{41}^{n_{3}} = 0  (6.7c)
π_{L}(X_{1}^{+}) l_{32}^{n_{4}} = \sqrt{[}2]^{−1} q^{−1} [n_{4}]_{q} l_{32}^{n_{4}+1}  (6.7d)
π_{L}(X_{1}^{+}) A_{11}^{n_{5}} = 0  (6.7e)
π_{L}(X_{1}^{+}) A_{22}^{n_{6}} = − \sqrt{[}2] q^{(n_{6}−2)/2} [n_{6}]_{q^{2}} l_{32} A_{22}^{n_{6}}  (6.7f)

π_{L}(X_{2}^{+}) l_{21}^{n_{1}} = q^{−3/2} [n_{1}]_{q^{2}} l_{21}^{n_{1}+1}  (6.8a)
π_{L}(X_{2}^{+}) l_{31}^{n_{2}} = q^{−(n_{2}+3)/2} [n_{2}]_{q^{2}} l_{21} l_{31}^{n_{2}}  (6.8b)
π_{L}(X_{2}^{+}) l_{41}^{n_{3}} = − [2]^{−1} q^{−(2n_{3}+1)/2} [n_{3}]_{q^{2}} l_{31}^{2} l_{41}^{n_{3}−1}  (6.8c)
π_{L}(X_{2}^{+}) l_{32}^{n_{4}} = − q^{(n_{4}−3)/2} [n_{4}]_{q^{2}} l_{21} l_{32}^{n_{4}} +
+ q^{n_{4}−1} [n_{4}]_{q} l_{31} l_{32}^{n_{4}−1} + (q^{n_{4}} − 1)[n_{4} − 1]_{q} l_{41}l_{32}^{n_{4}−2} (6.8d)
π_{L}(X_{2}^{+}) A_{11}^{n_{5}} = − q^{(n_{5}−3)/2} [n_{5}]_{q^{2}} l_{21} A_{11}^{n_{5}}  (6.8e)
π_{L}(X_{2}^{+}) A_{22}^{n_{6}} = q^{−(n_{6}+3)/2} [n_{6}]_{q^{2}} l_{21} A_{22}^{n_{6}}  (6.8f)

π_{L}(X_{1}^{−}) l_{21}^{n_{1}} = 0  (6.9a)
π_{L}(X_{1}^{−}) l_{31}^{n_{2}} = − \sqrt{[}2] q^{(2−n_{2})/2} [n_{2}]_{q} l_{21} l_{31}^{n_{2}−1}  (6.9b)
π_{L}(X_{1}^{−}) l_{41}^{n_{3}} = \sqrt{[}2] q^{(2−n_{3})/2} [n_{3}]_{q^{2}} l_{31} l_{41}^{n_{3}−1}  (6.9c)
π_{L}(X_{1}^{−}) l_{32}^{n_{4}} = − \sqrt{[}2] q [n_{4}]_{q} l_{32}^{n_{4}−1}  (6.9d)
π_{L}(X_{1}^{−}) A_{11}^{n_{5}} = 0  (6.9e)
π_{L}(X_{1}^{−}) A_{22}^{n_{6}} = 0  (6.9f)

π_{L}(X_{2}^{−}) l_{21}^{n_{1}} = −q^{3/2} [n_{1}]_{q^{2}} l_{21}^{n_{1}−1}  (6.10a)
π_{L}(X_{2}^{−}) l_{31}^{n_{2}} = 0  (6.10b)
π_{L}(X_{2}^{−}) l_{41}^{n_{3}} = 0  (6.10c)
π_{L}(X_{2}^{−}) l_{32}^{n_{4}} = 0  (6.10d)
π_{L}(X_{2}^{−}) A_{11}^{n_{5}} = 0  (6.10e)
π_{L}(X_{2}^{−}) A_{22}^{n_{6}} = 0  (6.10f)

Using the above expressions and again using the twisted derivation rules (3.6 )we obtain the left action of the generators on the entire monomial Φ:

π_{L}(X_{1}^{+}) Φ = − q^{(−n_{1}+n_{3}+n_{4}−n_{6})/2}\sqrt{[}2][n_{1}]_{q^{2}} Φ(n_{1} − 1, n_{2} + 2)  (6.11)

+ q^{(n_{1}−n_{2}+n_{3}+n_{4}−n_{6})/2}\sqrt{[}2][n_{2}]_{q} Φ(n_{2} − 1, n_{3} + 1) + q^{(n_{1}−n_{3}+n_{6}−2)/2}\sqrt{[}2][n_{4}/2 − n_{6}]_{q^{2}} Φ(n_{4} + 1)

π_{L}(X_{2}^{+}) Φ =q^{(−n_{2}+n_{4}+n_{5}−n_{6}−3)/2}[n_{1} + n_{2} − n_{4} − n_{5} + n_{6}]_{q^{2}} Φ(n_{1} + 1)

− q^{(−2n_{1}−n_{2}−2n_{3}+3n_{4}+3n_{5}−3n_{6}−1)/2}[2]^{−1}[n_{3}]_{q^{2}} Φ(n_{2} + 2, n_{3} − 1) + q^{(−2n_{1}−n_{2}−2n_{3}+2n_{4}+3n_{5}−3n_{6}−2)/2}[n_{4}]_{q} Φ(n_{2} + 1, n_{4} − 1) + (q^{(−2n_{1}−n_{2}−n_{5}+n_{6})/2} + q^{(−2n_{1}−n_{2}+n_{4}+n_{5}−n_{6}−2)/2}[n_{5} − n_{6}]_{q^{2}}λ)

× (q^{n_{4}} − 1)[n_{4} − 1]_{q} Φ(n_{3} + 1, n_{4} − 2)  (6.12)

π_{L}(X_{1}^{−}) Φ =q^{(n_{1}−n_{2}+n_{3}+n_{4}−n_{6}+2)/2}\sqrt{[}2][n_{2}]_{q} Φ(n_{1} + 1, n_{2} − 1)
+ q^{(n_{1}−n_{3}+n_{4}−n_{6}+2)/2}\sqrt{[}2][n_{3}]_{q^{2}} Φ(n_{2} + 1, n_{3} − 1)
− q^{(n_{1}−n_{3}−n_{6}+2)/2}\sqrt{[}2][n_{4}]_{q} Φ(n_{4} − 1)  (6.13)
π_{L}(X_{2}^{−}) Φ = −q^{(n_{2}−n_{4}−n_{5}+n_{6}+3)/2}[n_{1}]_{q^{2}} Φ(n_{1} − 1)  (6.14)

Therefore, we have obtain induced representations of U_{q}so(5) on the coset space of SO_{q}(5) which are parametrized by the integers n_{5} and n_{6} . The non-commutative representation space is spanned by functions Φ obeying the covariance conditions (6.5.) Notice that the dependence of Φ on n_{5} and n_{6} remain constant under the above transformations. Hence it is appropriate to label our representaions by these two integers. For the remaining n′s the left action induces an increase or decrease in some of these powers. For e.g., Φ(n_{1}−1, n_{2}+1) appearing on the r.h.s. means that the power of the generator l_{21} is lowered by one while that of l_{31} is raised by one, keeping the other powers constant. More explicitly

Φ(n_{1} − 1, n_{2} + 1) = l_{21}^{n_{1}−1}l_{31}^{n_{2}+1}l_{41}^{n_{3}}l_{32}^{n_{4}}A_{11}^{n_{5}}A_{22}^{n_{6}}.

VII. REDUCIBILITY FOR q GENERIC AND ROOT OF UNITY

The induced representations obtained above are in general irreducible for complex q and n_{5}, n_{6}. However, they become reducible for q ∈ C I and some specific values of n_{5} and n_{6}. The representation space decomposes into two invariant subspaces as we see below.

Setting π_{L}(X_{1}^{−}) Φ = 0 implies that n_{2} = n_{3} = n_{4} = 0 which gives us the corresponding lowest weight vector, while setting π_{L}(X_{1}^{+}) Φ = 0 yields the highest weight vector under the condition n_{1} = n_{2} = 0 and n_{4} = 2n_{6}. We call this invariant subspace V_{1}. The parameter n_{6} which upto now belonged to the set of integers is restricted to only non-negative integers, i.e. n_{6} ∈ ZZ_{+}.

Similarly for the second subspace V_{2} , by imposing π_{L}(X_{2}^{±}) Φ = 0 we get the lowest weight vector when n_{1} = 0 and the highest weight vector when n_{3} = n_{4} = 0 and n_{1} + n_{2} = n_{5} − n_{6}. Since n_{1} and n_{2} are both non-negative integers, this restricts the values of n_{5} − n_{6} ≥ 0. Thus we get finite dimensional irreducible representations of the quantum algebra U_{q}so(5) obtained as subrepresentations of the elementary representations realized on the quotient group of SO_{q}(5) and parametrized by n_{5} and n_{6}.

Next, we wish to observe the behaviour of our representations when q is a root of unity, ie q^{N} = 1 for N = 3, 4, 5, .... For this we substitute q = e^{2πi/N} and find conditions on the labels n_{1}, ...n_{6} for which the representations become reducible ( i.e. r.h.s. becomes zero as was done above for the general q ). One has to be careful here since we would be dealing with two different q-numbers [n_{j}]_{q} and [n_{j}]_{q^{2}}. Hence we obtain:

π_{L}(X_{1}^{+}) Φ = 0, n_{1} = k_{1}N′, n_{2} = k_{2}N , n_{4}/2 − n_{6} = k′N′  (7.1)

where k_{j} (j = 1, 2, 3, 4) = 0, 1, 2, 3, ......, k′ = 0, ±1, ±2, ...., N′ = N for odd N and N/2 for even N. Also [2] = \frac{sin(2π/N)}{sin(π/N)} ≠ 0 since N = 3, 4, .....

π_{L}(X_{1}^{−}) Φ = 0, n_{2} = k_{2}N, n_{3} = k_{3}N′, n_{4} = k_{4}N  (7.2)
π_{L}(X_{2}^{+}) Φ = 0 , n_{1} + n_{2} − n_{4} − n_{5} + n_{6} = kN′ , n_{3} = k_{3}N′, n_{4} = k_{4}N  (7.3)

where k = 0, ±1, ±2, ....... .

π_{L}(X_{2}^{−}) Φ = 0, n_{1} = k_{1}N′  (7.4)

It is clear that in this case we have an enlarged centre as elements of the form (l_{ij})^{N} would belong to the centre of the group. Analogously elements of the form (X_{i}^{±})^{N} would lie in the centre of the quantum algebra U_{q}so(5).

Let us now analyse the non-trivial conditions given above. The third condition in (7.1) can be written as n_{4} = 2(k′N′ + n_{6}). We know that n_{4} is a non negative integer while n_{6} is an integer. Hence two possibilities arise:

(a) n_{6} ≥ 0, k′ = 0, 1, 2, ....  (7.5a)
(b) n_{6} < 0, k′N′ ≥ −n_{6}  (7.5b)

The first condition in (7.3) can be written as n_{1} + n_{2} = N + n_{5} − n_{6} + kN′. Since l.h.s. is always a non-negative number, there are three main possibilities:

1. If n_{5} − n_{6} > 0 then k = 0, 1, 2....( N and N′ are positive valued). k can also have negative values provided N + n_{5} − n_{6} ≥ −KN′.

2. If n_{5} − n_{6} = 0 then k = 0, 1, 2, .... No negative values are allowed here.

3. If n_{5} − n_{6} < 0 then kN′ + N ≥ −(n_{5} − n_{6}).

There could be further many sub-cases of each of the three possibilities mentioned above:

1. n_{5} − n_{6} > 0 is possible when:

n_{5} > 0, n_{6} > 0, n_{5} > n_{6}  (7.6a)
n_{5} < 0, n_{6} < 0, n_{5} > n_{6}  (7.6b)
n_{5} > 0, n_{6} < 0  (7.6c)
n_{5} > 0, n_{6} = 0  (7.6d)
n_{5} = 0, n_{6} < 0  (7.6e)

2. n_{5} − n_{6} = 0 is possible when:

n_{5} = 0, n_{6} = 0  (7.7a)
n_{5} = n_{6} ≠ 0  (7.7b)

3. n_{5} − n_{6} < 0 is possible when:

n_{5} > 0, n_{6} > 0, n_{5} < n_{6}  (7.8a)
n_{5} < 0, n_{6} < 0, n_{5} < n_{6}  (7.8b)
n_{5} < 0, n_{6} > 0  (7.8c)
n_{5} < 0, n_{6} = 0  (7.8d)
n_{5} = 0, n_{6} > 0  (7.8e)

In this way we obtain finite dimensional irreps under certain restrictions on the values of the parameters labelling them. As is clear from the above analysis, the representation theory which is quite similar to the classical case for a generic q changes drastically when q is taken to be a root of unity.

VIII. RIGHT ’ACTION’

For the sake of completeness we evaluate the right action on the monomial Φ. The covariance condition (6.5a) implies that the right action of the raising generators X_{i}^{+} on the monomial Φ is zero. Thus we need to compute it only for the remaining lowering generators.

Right action of X_{1}^{−}:
π_{R}(X_{1}^{−}) l_{21}^{n_{1}} = 0  (8.1a)
π_{R}(X_{1}^{−}) l_{31}^{n_{2}} = 0  (8.1b)
π_{R}(X_{1}^{−}) l_{41}^{n_{3}} = 0  (8.1c)
π_{R}(X_{1}^{−}) l_{32}^{n_{4}} = \sqrt{[}2] q^{(2−n_{4})/2} [n_{4}]_{q} l_{32}^{n_{4}−1} A_{22}^{−1}  (8.1d)
π_{R}(X_{1}^{−}) A_{11}^{n_{5}} = 0  (8.1e)
π_{R}(X_{1}^{−}) A_{22}^{n_{6}} = \sqrt{[}2] q^{(1−n_{6})/2} [n_{6}]_{q^{2}}A_{22}^{n_{6}} u_{23}  (8.1f)
Right action of X_{2}^{−}:
π_{R}(X_{2}^{−}) l_{21}^{n_{1}} = q^{(3−2n_{1})/2} [n_{1}]_{q^{2}}l_{21}^{n_{1}−1} A_{11}^{−1} A_{22}  (8.2a)
π_{R}(X_{2}^{−}) l_{31}^{n_{2}} = q^{(2−n_{2})/2} ([n_{2}]_{q} l_{31}^{n_{2}−1} l_{32} + (q − q^{1−n}) [n_{2} − 1]_{q} l_{31}^{n_{2}−2} l_{41})
× A_{11}^{−1} A_{22}  (8.2b)
π_{R}(X_{2}^{−}) l_{41}^{n_{3}} = − [2]^{−1} q^{(3−2n_{3})/2} [n_{3}]_{q^{2}} l_{41}^{n_{3}−1} l_{32}^{2} A_{11}^{−1} A_{22}  (8.2c)
π_{R}(X_{2}^{−}) l_{32}^{n_{4}} = 0  (8.2d)
π_{R}(X_{2}^{−}) A_{11}^{n_{5}} = q^{(1−n_{5})/2} [n_{5}]_{q^{2}} A_{11}^{n_{5}} u_{12}  (8.2e)
π_{R}(X_{2}^{−}) A_{22}^{n_{6}} = − q^{(1+n_{6})/2} [n_{6}]_{q^{2}} A_{22}^{n_{6}} u_{12}  (8.2f)

So the action on the entire monomial becomes:

π_{R}(X_{1}^{−}) Φ = \sqrt{[}2] q^{(2−n_{4}−n_{6})/2} [n_{4}]_{q} Φ(n_{4} − 1, n_{6} − 1)
+ \sqrt{[}2] q^{(1−n_{6})/2} [n_{6}]_{q^{2}} Φ u_{23}  (8.3)

π_{R}(X_{2}^{−}) Φ = q^{(−2n_{1}−2n_{2}+2n_{4}−n_{5}+n_{6}+3)/2} [n_{1}]_{q^{2}} Φ(n_{1} − 1, n_{5} − 1, n_{6} + 1)
+ q^{(−n_{2}−2n_{3}+2n_{4}−n_{5}+n_{6}+2)/2} [n_{2}]_{q}Φ(n_{2} − 1, n_{4} + 1, n_{5} − 1, n_{6} + 1)

+ q^{(2n_{4}−n_{5}+n_{6})/2}(q − q^{1−n_{2}}) [n_{2} − 1]_{q} Φ(n_{2} − 2, n_{3} + 1, n_{5} − 1, n_{6} + 1) − q^{(−2n_{3}+2n_{4}−n_{5}+n_{6}+3)/2} [2]^{−1} [n_{3}]_{q^{2}} Φ(n_{3} − 1, n_{4} + 2, n_{5} − 1, n_{6} + 1)

+ q^{(−n_{5}+n_{6}+1)/2} [n_{5} − n_{6}]_{q^{2}} Φ u_{12}  (8.4)

We notice the appearance of terms involving the variables u_{ij}, which is natural since there is no intrinsic right action of X_{i}^{−} on the induced representations. Such actions are needed only as input in singular vectors in which all such terms involving the variables u_{ij} will vanish. Thus, it would be fine to neglect them already now provided they do not produce any wanted terms (i.e. terms involving l′s and A′s) when we act repeatedly with π_{R}(X_{i}^{−}). For this we should check the action of X_{i}^{−} on u_{ij}.

Right action of X_{1}^{−} on the u′s:
π_{R}(X_{1}^{−}) u_{12} = \sqrt{[}2] u_{13}  (8.5a)
π_{R}(X_{1}^{−}) u_{13} = − \sqrt{[}2] u_{14}  (8.5b)
π_{R}(X_{1}^{−}) u_{14} = 0  (8.5c)
π_{R}(X_{1}^{−}) u_{23} = − \sqrt{[}2](1 − q^{−1/2} [2]^{−1}) u_{23}^{2}  (8.5d)

Right action of X_{2}^{−} on the u′s:

π_{R}(X_{2}^{−}) u_{12} = − q^{1/2} u_{12}^{2}  (8.6a)
π_{R}(X_{2}^{−}) u_{13} = − u_{12} u_{13}  (8.6b)
π_{R}(X_{2}^{−}) u_{14} = q^{1/2} [2]^{−1} u_{13}^{2}  (8.6c)
π_{R}(X_{2}^{−}) u_{23} = q (u_{12} u_{23} − u_{13})  (8.6d)

We see that the u_{23} term in (8.3) would not disappear under the action of X_{1}^{−}. Similarly, the u_{12} term in (8.4) would still remain under the action of X_{2}^{−}.

IX. INTERTWINERS

There are various partial equivalences among the reducible representaions. These partial equivalences are realized by intertwining operators for which a canonical derivation was given in [1]. We shall follow this approach without going into details as here we would be interested in obtaining only the simplest intertwiners.

Let us denote our representation space obtained in Sec. VI by Vn̄ where n̄ = (n_{5}, n_{6}) since it is characterized by the integers n_{5}, n_{6}. The corresponding representaions are similarly denoted by π_{L} = πn̄ . Then the intertwiner is a map from the space Vn̄ to another space V_{}_{n̄′} labelled by n′_{5} and n′_{6} such that

I_{i} πn̄(X_{i}^{−}) = π_{}_{n̄′}(X_{i}^{−}) I_{i}  (9.1)

where n̄′ is expected to be −n̄ − 1. According to the general prescription [1], these operators should be given by (π_{R}(X_{i}^{−}))^{s} and the parameter s is expected to be equal to the dimension of the subspace. It is easy to check this directly. Let s ∈ IN and suppose that Φ′ = (π_{R}(X_{i}^{−}))^{s} Φ where Φ′ belongs to V_{}_{n̄′} . Extending the right covariance condition (6.5a) leads to π_{R}(X_{i}^{+}) Φ′ = 0. Below we calculate this explicitly for X_{1}^{−} :

π_{R}(X_{1}^{+}) Φ′ = π_{R}(X_{1}^{+}) (π_{R}(X_{1}^{−}))^{s} Φ
= [π_{R}(X_{1}^{+}) , (π_{R}(X_{1}^{−}))^{s}] Φ
= π_{R} ([X_{1}^{+} , (X_{1}^{−})^{s}]) Φ
= π_{R} ([s]_{q} (X_{1}^{−})^{s−1} (q^{−(s−1)/2} K_{1} − q^{(s−1)/2} K_{1}^{−1})/(q^{1/2} − q^{−1/2})) Φ
= [s]_{q} π_{R}(X_{1}^{−})^{s−1} ((q^{−(s−1)/2+n_{6}} − q^{(s−1)/2−n_{6}})/(q^{1/2} − q^{−1/2})) Φ
= − [s]_{q} [s − 1 − 2n_{6}]_{q} π_{R}(X_{1}^{−})^{s−1} Φ  (9.2)

If q is a generic complex number then, as expected, this quantity becomes zero only for s = 2n_{6} + 1. Next, employing the other conditions of right covariance we have π_{R}(K_{i}) Φ′ = q^{n̄′} Φ′, i.e.,

\begin{aligned}π_{R}(K_{1}) Φ′ & = π_{R}(K_{1}) (π_{R}(X_{1}^{−}))^{s} Φ \\ & = π_{R} (q^{−s} (X_{1}^{−})^{s} K_{1}) Φ \\ & = q^{−s} π_{R}(X_{1}^{−})^{s} q^{n_{6}} Φ \\ & = q^{n_{6}−s} Φ′ \end{aligned}  (9.3)

So, n′_{6} = n_{6} − s = −n_{6} − 1. Then the intertwining operator I_{1} (up to a multiplicative nonzero constant) is :

I_{1} = (π_{R}(X_{1}^{−}))^{2n_{6}+1}  (9.4)

Similar calculations for the other generator X_{2}^{−} yield

π_{R}(X_{2}^{+}) Φ′ = −[s]_{q^{2}} [s − 1 − n_{5} + n_{6}]_{q^{2}} π_{R}(X_{2}^{−})^{s−1} Φ  (9.5)

which vanishes when s = n_{5} − n_{6} + 1. On the other hand,

π_{R}(K_{2}) Φ′ = q^{n_{5}−n_{6}−2s} Φ′  (9.6)

So, n′_{5} −n′_{6} = n_{5} −n_{6} −2s = (−n_{5} −1)+(n_{6} −1) and thus the second intertwining operator I_{2} is indeed given by

I_{2} = (π_{R}(X_{2}^{−}))^{n_{5}−n_{6}+1}.  (9.7)

The finite-dimensional invariant subspaces V_{1} and V_{2} discussed earlier are the kernels of these intertwining operators.

X. CONCLUDING REMARKS

In this paper we have obtained induced representations of the quantum algebra U_{q}so(5) realized on some (non-commutative) coset space of the dual quantum group SO_{q}(5) . We remark that although we have assumed integer values for the parameters n_{5} and n_{6} which characterize them, the formulae (of left action) define a representaion of U_{q}so(5) also for arbitrary complex values. However, only for integer values , they comprise a deformation of representations which are integrable to a representaion of the group SO_{q}(5) . We have also investigated the reducibility of these representations under restricted values of n_{5}, n_{6} which lead to finite dimensional irreps for q generic and for root of unity.

Two basic intertwiners corresponding to the two simple roots were shown to be expressible in terms of powers of the right action. It would be interesting to find out other mixed or composite intertwiners corresponding to the non-simple roots. For that we would need to compute the powers of the right action explicitly , i.e. by iteration of the results presented in Sec. VIII. In this way one can obtain whole new families of intertwiners. By a suitable redefinition of the generators it should be possible to realize them as q- difference operators. This yields a natural q-deformation of several equations in mathematical physics, which can be viewed as intertwiner between some representations of the underlying symmetry algebra. Also it would be interesting to explore the connection with the non-compact quantum Anti de Sitter algebra U_{q}so(3, 2) in the context of induced representations and q-difference intertwiners. We hope to address all these issues in a sequel.

ACKNOWLEGDEMENTS I thank Prof. V. K. Dobrev for suggesting this problem and for useful discussions at various stages of this work. Financial support by CSIR (New Delhi) and infrastructure provided by Institute of Physics, Bhubaneswar (India), where part of the work was done, is gratefully acknowledged. I also wish to express my gratitude to the Max-Planck Institute for Mathematik in Bonn for a visiting faculty position.

Appendix A. COMMUTATION RELATIONS OF SO_{q}(5)

Here we give the relations which the 25 elements t_{ij} obey and which follow from (2.1) and (2.2) using (2.6) and (2.7). This explicit form is also necessary for the verification of the Gauss decomposition of Section III. The relations are:

t_{ij} t_{iℓ} = q t_{iℓ} t_{ij} ,  i ≠ 3 ,  j < ℓ , j′ ≠ ℓ  (A.1)
t_{ij} t_{kj} = q t_{kj} t_{ij} ,  j ≠ 3 ,  i < k , i′ ≠ k  (A.2)
t_{ij} t_{ij′} = q^{2} (t_{ij′} t_{ij} + λ t_{i,j′+1} t_{i,j−1} ) ,  i ≠ 3 ,  j = 1, 2  (A.3)
t_{ij} t_{i′j} = q^{2} (t_{i′j} t_{ij} + λ t_{i′+1,j} t_{i−1,j} ) ,  j ≠ 3 ,  i = 1, 2  (A.4)
t_{3j} t_{3ℓ} = t_{3ℓ} t_{3j} + α t_{2ℓ} t_{4j} + η t_{1ℓ} t_{5j} ,  j < ℓ , j′ ≠ ℓ  (A.5)
t_{i3} t_{k3} = t_{k3} t_{i3} + α t_{k2} t_{i4} + η t_{k1} t_{i5} ,  i < k, i′ ≠ k  (A.6)
t_{3j}^{2} = −q^{−1} [2] t_{2j} t_{4j} − q^{−2} [2] t_{1j} t_{5j} ,  j ≠ 3  (A.7)
t_{i3}^{2} = −q^{−1} [2] t_{i2} t_{i4} − q^{−2} [2] t_{i1} t_{i5} ,  i ≠ 3  (A.8)
t_{ij} t_{kℓ} = t_{kℓ} t_{ij} + λ t_{kj} t_{iℓ} ,  i < k , i′ ≠ k , j < ℓ , j′ ≠ ℓ  (A.9)

t_{ij} t_{kℓ} = t_{kℓ} t_{ij} ,  i < k , i′ ≠ k , j > ℓ , j′ ≠ ℓ  (A.10)
t_{ij} t_{i′ℓ} = q (t_{i′ℓ} t_{ij} + λ t_{i′j} t_{iℓ}) + λ t_{i−1,j} t_{i′+1,ℓ}  i = 1, 2 , j < ℓ , j′ ≠ ℓ (A.11)
t_{ij} t_{i′ℓ} = q (t_{i′ℓ} t_{ij} + λ t_{i′+1,ℓ} t_{i−1,j} ),  i = 1, 2 , j > ℓ , j′ ≠ ℓ  (A.12)
t_{ij} t_{kj′} = q (t_{kj′} t_{ij} + λ t_{ij′} t_{kj}) ,  i < k , i′ ≠ k , j = 1, 2  (A.13)
t_{ij′} t_{kj} = q^{−1} (t_{kj} t_{ij′} − λ t_{k,j−1} t_{i,j′+1}) ,  i < k , i′ ≠ k , j = 1, 2  (A.14)
t_{i2} t_{k4} = q^{−1}t_{k4} t_{i2} + α t_{k3} t_{i3} + β t_{k2} t_{i4} + γ t_{k1} t_{i5} ,  i < k , i′ ≠ k  (A.15)
t_{11} t_{55} = t_{55} t_{11} + q (ξt_{54} t_{12} + ηt_{53} t_{13} + γt_{52} t_{14} + δt_{51} t_{15} )  (A.16)
t_{12} t_{54} = t_{54} t_{12} + q (αt_{53} t_{13} + βt_{52} t_{14} + γt_{51} t_{15} )  (A.17)
t_{13} t_{53} = q (t_{53} t_{13} + αt_{52} t_{14} + ηt_{51} t_{15} )  (A.18)
t_{14} t_{52} = t_{52} t_{14} − λt_{51} t_{15}  (A.19)
t_{15} t_{51} = t_{51} t_{15}  (A.20)
t_{21} t_{45} = t_{45} t_{21} + q (ξt_{44} t_{22} + ηt_{43} t_{23} + γt_{42} t_{24} + δt_{41} t_{25} ) + λt_{11} t_{55}  (A.21)
t_{22} t_{44} = t_{44} t_{22} + q (αt_{43} t_{23} + βt_{42} t_{24} + γt_{41} t_{25} ) + λt_{12} t_{54}  (A.22)
t_{23} t_{43} = q (t_{43} t_{23} + αt_{42} t_{24} + ηt_{41} t_{25} ) + λt_{13} t_{53}  (A.23)
t_{24} t_{42} = t_{42} t_{24} + λ (t_{52} t_{14} − t_{51} t_{15} − t_{41} t_{25} )  (A.24)
t_{25} t_{41} = t_{41} t_{25} + λt_{51} t_{15}  (A.25)
t_{31} t_{35} = q (t_{35} t_{31} + αt_{25} t_{41} + ηt_{15} t_{51} )  (A.26)
t_{32} t_{34} = q (t_{34} t_{32} + αt_{24} t_{42} + ηt_{14} t_{52} ) + λt_{31} t_{35}  (A.27)

The additional constraints are (i = 1, 2, 3, 4, 5)

αt_{23} t_{43} + ηt_{13} t_{53} = αt_{32} t_{34} + ηt_{31} t_{35}  (A.28)
q^{3} t_{i5} t_{51} + q^{2} t_{i4} t_{52} + q^{3/2} t_{i3} t_{53} + q t_{i2} t_{54} + t_{i1} t_{55} = δ_{i1}  (A.29)
q^{2} t_{i5} t_{41} + q t_{i4} t_{42} + q^{1/2} t_{i3} t_{43} + t_{i2} t_{44} + q^{−1} t_{i1} t_{45} = δ_{i2}  (A.30)
q^{3/2} t_{i5} t_{31} + q^{1/2} t_{i4} t_{32} + t_{i3} t_{33} + q^{−1/2} t_{i2} t_{34} + q^{−3/2} t_{i1} t_{35} = δ_{i3}  (A.31)
q t_{i5} t_{21} + t_{i4} t_{22} + q^{−1/2} t_{i3} t_{23} + q^{−1} t_{i2} t_{24} + q^{−2} t_{i1} t_{25} = δ_{i4}  (A.32)

t_{i5} t_{11} + q^{−1} t_{i4} t_{12} + q^{−3/2} t_{i3} t_{13} + q^{−2} t_{i2} t_{14} + q^{−3} t_{i1} t_{15} = δ_{i5}  (A.33)
t_{55} t_{1i} + q^{−1} t_{45} t_{2i} + q^{−3/2} t_{35} t_{3i} + q^{−2} t_{25} t_{4i} + q^{−3} t_{15} t_{5i} = δ_{1i}  (A.34)
q t_{54} t_{1i} + t_{44} t_{2i} + q^{−1/2} t_{34} t_{3i} + q^{−1} t_{24} t_{4i} + q^{−2} t_{14} t_{5i} = δ_{2i}  (A.35)
q^{3/2} t_{53} t_{1i} + q^{1/2} t_{43} t_{2i} + t_{33} t_{3i} + q^{−1/2} t_{23} t_{4i} + q^{−3/2} t_{13} t_{5i} = δ_{3i}  (A.36)
q^{2} t_{52} t_{1i} + q t_{42} t_{2i} + q^{1/2} t_{32} t_{3i} + t_{22} t_{4i} + q^{−1} t_{12} t_{5i} = δ_{4i}  (A.37)
q^{3} t_{51} t_{1i} + q^{2} t_{41} t_{2i} + q^{3/2} t_{31} t_{3i} + q t_{21} t_{4i} + t_{11} t_{5i} = δ_{5i}  (A.38)

In fact we can obtain the set of relations (A.34) - (A.38) from (A.29) - (A.33) by q → q^{−1} and t_{ij} t_{kl} → t_{lk} t_{ji}.

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