Classification theorem on irreducible representations of the q-deformed algebra

The aim of this paper is to give a complete classification of irreducible finite dimensional representations of the nonstandard q-deformation U'_q(so(n)) (which does not coincide with the Drinfeld-Jimbo quantum algebra U_q(so(n)) of the universal enveloping algebra U(so(n,C)) of the Lie algebra so(n,C) when q is not a root of unity. These representations are exhausted by irreducible representations of the classical type and of the nonclassical type. Theorem on complete reducibility of finite dimensional representations of U'_q(so(n)) is proved.


Classification theorem on irreducible representations
This q-deformation was first constructed in [8]. It permits one to construct the reductions of U q (so n,1 ) and U q (so n+1 ) onto U q (so n ). The q-deformed algebra U q (so n ) leads for n = 3 to the q-deformed algebra U q (so 3 ) defined by Fairlie [4]. The cyclically symmetric algebra, similar to Fairlie's one, was also considered somewhat earlier by Odesskiȋ [31].
In the classical case, the imbedding SO(n) ⊂ SU(n) (and its infinitesimal analogue) is of great importance for nuclear physics and in the theory of Riemannian symmetric spaces. It is well known that in the framework of quantum groups and Drinfel'd-Jimbo quantum algebras one cannot construct the corresponding embedding. The algebra U q (so n ) allows to define such an embedding [29], that is, it is possible to define the embedding U q (so n ) ⊂ U q (sl n ), where U q (sl n ) is the Drinfel'd-Jimbo quantum algebra.
As a disadvantage of the algebra U q (so n ), we have to mention the difficulties with Hopf algebra structure. Nevertheless, U q (so n ) turns out to be a coideal in U q (sl n ) (see [29]) and this fact allows us to consider tensor products of finite-dimensional irreducible representations of U q (so n ) for many interesting cases (see [13]).
The algebra U q (so n ) and its representations are interesting in many cases. Main directions of interest are the following: (1) the theory of orthogonal polynomials and special functions (especially, the theory of q-orthogonal polynomials and basic hypergeometric functions); this direction is not well worked out; some ideas of such applications can be found in [23]; (2) the algebra U q (so n ) (especially its particular case U q (so 3 )) is related to the algebra of observables in 2 + 1 quantum gravity on the Riemmanian surfaces (see papers [2,5,28]); (3) a quantum analogue of the Riemannian symmetric space SU(n)/ SO(n) is constructed by means of the algebra U q (so n ); this construction is fulfilled in the paper [29] (see also [24]); (4) a q-analogue of the theory of harmonic polynomials (q-harmonic polynomials on quantum vector space R n q ) is constructed by using the algebra U q (so n ); in particular, a q-analogue of different separations of variables for the q-Laplace operator on R n q is given by means of this algebra and its subalgebras; this theory is contained in the papers [17,30]; (5) the algebra U q (so n ) also appears in the theory of links in the algebraic topology (see [1]); (6) the algebra U q (so n ) is connected with Yangians (see [27] and references therein); (7) a new quantum analogue of the Brauer algebra is connected with the algebra U q (so n ) (see [26]).
A large class of finite-dimensional irreducible representations of the algebra U q (so n ) was constructed in [8]. The formulas of action of the generators of U q (so n ) upon the basis (which is a q-analogue of the Gel'fand-Tsetlin basis) are given there. A proof of these formulas and some of their corrections were given in [6]. However, the finite-dimensional irreducible representations described in [6,8] are representations of the classical type. They are q-deformations of the corresponding irreducible representations of the Lie algebra so n , that is, at q → 1 they turn into representations of so n . N. Z. Iorgov and A. U. Klimyk 227 The algebra U q (so n ) has other classes of finite-dimensional irreducible representations, which have no classical analogue. These representations are singular at the limit q → 1. They are described in [15]. The description of these representations for the algebra U q (so 3 ) is given in [9]. A classification of irreducible * -representations of real forms of the algebra U q (so 3 ) is given in [33]. The representation theory of U q (so n ) when q is a root of unity is studied in [16].
In this paper, we deal with classification of finite-dimensional irreducible representations of the algebra U q (so n ) when q is not a root of unity. As mentioned above, there were constructed irreducible representations of the algebra U q (so n ) belonging to the classical and to the nonclassical types. However, it was not known that these representations exhaust all irreducible finite-dimensional representations. We started to study this problem in [22]. We show there that these representations are determined by the so-called highest weights (which were defined in [22] and differ from highest weights in the theory of quantized universal enveloping algebras). However, we do not know a correspondence between known representations of the classical and nonclassical types and highest weights. In the present paper, we develop an approach to the problem of classification from other points of view. Namely, we prove that each irreducible finite-dimensional representation of U q (so n ) belongs to the set of representations of the classical type or to the set of representations of the nonclassical type constructed before. For proving this, we use our previous results on structure of the algebra U q (so n ) (tensor operators, Wigner-Eckart theorem, etc.). We also need the theorem on complete reducibility of finite-dimensional representations of U q (so n ). This theorem is proved in this paper. Some ideas from the theory of representations of the Lie algebra so n (C) and its real forms are also used.
Note that the problem of classification of irreducible finite-dimensional representations of U q (so n ) is much more complicated than in the case of Drinfel'd-Jimbo quantum algebras since in U q (so n ) we do not have an analogue of a Cartan subalgebra and root elements. The set of all irreducible finite-dimensional representations of U q (so n ) is wider than in the case of U q (so n ).
Everywhere below we assume that q is not a root of unity.
2. The q-deformed algebra U q (so n ) The universal enveloping algebra U(so n (C)) is generated by the elements But in order to generate the algebra U(so n (C)), it is enough to take only the elements I 21 ,I 32 ,...,I n,n−1 . It is a minimal set of elements necessary for generating U(so n (C)). These elements satisfy the relations The following theorem is true for U(so n (C)) (see [21]): the enveloping algebra U(so n (C)) is isomorphic to the complex associative algebra (with a unit element) generated by the elements I 21 ,I 32 ,...,I n,n−1 satisfying the above relations.

Classification theorem on irreducible representations
We make a q-deformation of these relations by fulfilling the deformation of the integer 2 as 2 → [2] q := (q 2 − q −2 )/(q − q −1 ) = q + q −1 . As a result, we obtain the relations The q-deformed algebra U q (so n ) is defined as the complex unital (i.e., with a unit element) associative algebra generated by elements I 21 ,I 32 ,...,I n,n−1 satisfying relations (2.2), (2.3), and (2.4). It is a q-deformation of the universal enveloping algebra U(so n (C)), different from the Drinfel'd-Jimbo quantized universal enveloping algebra U q (so n ). For this algebra, the inclusions U q (so n ) ⊃ U q (so n−1 ) and U q (sl n ) ⊃ U q (so n ) are constructed, where U q (sl n ) is the well-known Drinfel'd-Jimbo quantum algebra (see the introduction). An analogue of the skew-symmetric matrices I i j = E i j − E ji , i > j, constituting a basis of the Lie algebra so n (C), can be introduced into U q (so n ) (see [7,30]). For k > l + 1, they are defined recursively by the formulas (2.5) The elements I kl , k > l, satisfy the commutation relations I lr ,I kl q = I kr , I kl ,I kr q = I lr , I kr ,I lr q = I kl for k > l > r, (2.6) I kl ,I sr = 0 for k > l > s > r, k > s > r > l, (2.7) I kl ,I sr q = q − q −1 I lr I ks − I kr I sl for k > s > l > r. (2.8) For q = 1, they coincide with the corresponding commutation relations for the Lie algebra so n (C). The algebra U q (so n ) can be also defined as a unital associative algebra generated by I kl , 1 ≤ l < k ≤ n, satisfying the relations (2.6), (2.7), and (2.8). In fact, the relations (2.6), (2.7), and (2.8) can be reduced to the relations (2.2), (2.3), and (2.4) for I 21 ,I 32 ,...,I n,n−1 .
The Poincaré-Birkhoff-Witt theorem for the algebra U q (so n ) can be formulated as follows (a proof of this theorem is given in [16] form a basis of the algebra U q (so n ). In U q (so n ), the commutative subalgebra Ꮽ generated by the elements I 21 ,I 43 ,I 65 ,..., I n−1,n−2 (or I n,n−1 ) can be separated. So, this subalgebra is generated by n/2 elements, where n/2 is an integral part of the number n/2. However, there exist no root elements in the algebra U q (so n ) with respect to this commutative subalgebra. This leads to the fact that properties of U q (so n ) are not similar to those of the Drinfel'd-Jimbo algebra U q (so n ). N. Z. Iorgov and A. U. Klimyk 229

Irreducible representations of the classical and nonclassical types
In this section, we give known facts on irreducible representations of U q (so n ), which will be used below. The corresponding references are given in the introduction.
Two types of irreducible finite-dimensional representations are known for U q (so n ): (a) representations of the classical type; (b) representations of the nonclassical type.
Known irreducible representations of the classical type are q-deformations of the irreducible finite-dimensional representations of the Lie algebra so n . There is a one-to-one correspondence between these irreducible representations of the algebra U q (so n ) and irreducible finite-dimensional representations of the Lie algebra so n . Moreover, formulas for representations of the classical type of U q (so n ) turn into the corresponding formulas for the representations of Lie algebra so n at q → 1.
There exists no classical analogue for representations of the nonclassical type: representation operators T(a), a ∈ U q (so n ), have singularities at q = 1.
We describe known irreducible finite-dimensional representations of the algebras U q (so n ), n ≥ 3, which belong to the classical type. As in the classical case, they are given by sets m n of n/2 numbers m 1,n ,m 2,n ,...,m n/2 ,n (here n/2 denotes the integral part of n/2), which are all integral or all half-integral and satisfy the dominance conditions for n = 2k + 1 and n = 2k, respectively. These representations are denoted by T mn . We take a q-analogue of the Gel'fand-Tsetlin basis in the representation space, which is obtained by successive reduction of the representation T mn to the subalgebras U q (so n−1 ),U q (so n−2 ), ...,U q (so 3 ), U q (so 2 ) := U(so 2 ). As in the classical case, its elements are labelled by the Gel'fand-Tsetlin tableaux where, as in the nondeformed case, the components of m s and m s−1 satisfy the "betweenness" conditions Sometimes, the basis elements, defined by a tableau {α n }, are denoted as |α n−1 or as |m n−1 ,α n−2 , that is, we will omit the first row m n in a tableau. It is convenient to introduce the so-called l-coordinates l j,2p+1 = m j,2p+1 + p − j + 1, l j,2p = m j,2p + p − j, (3.4) 230 Classification theorem on irreducible representations for the numbers m i,k . The operator T mn (I 2p+1,2p ) of the representation T mn of U q (so n ) acts upon Gel'fand-Tsetlin basis elements, labelled by (3.2), as (3.5) and the operator T mn (I 2p,2p−1 ) acts as The numbers in square brackets in formulas (3.6), (3.7), (3.8), and (3.9) mean q-numbers defined by It is seen from formula (3.9) that the coefficient C 2p−1 vanishes if m p,2p ≡ l p,2p = 0.
N. Z. Iorgov and A. U. Klimyk 231 The following assertion is well known [8]: the representations T mn are irreducible. The representations T mn and T m n are pairwise nonequivalent for m n = m n .
Irreducible finite-dimensional representations of the nonclassical type are given by sets := ( 2 , 3 ,..., n ), i = ±1, and by sets m n consisting of n/2 half-integral (but not integral) numbers m 1,n ,m 2,n ,...,m n/2 ,n that satisfy the dominance conditions m 1,n ≥ m 2,n ≥ ··· ≥ m n/2 ,n ≥ 1 2 . (3.12) These representations are denoted by T ,mn . For a basis in the representation space, we use an analogue of the basis of the previous case. Its elements are labelled by tableaux (3.2), where the components of m s and m s−1 satisfy the "betweenness" conditions (3.13) The corresponding basis elements are denoted by the same symbols as in the previous case. The l-coordinates for m j,s are introduced by the same formulas as before.
The operator T ,mn (I 2p+1,2p ) of the representation T ,mn of U q (so n ) acts upon the basis elements |α n by the formulas T ,mn I 2p+1,2p α n = δ mp,2p,1/2 (3.14) where the summation in the last sum must be from 1 to p − 1 if m p,2p = 1/2, and the operator T mn (I 2p,2p−1 ) acts as The following assertion is true (see [15]): the representations T ,mn are irreducible. The representations T ,mn and T ,m n are pairwise nonequivalent for ( ,m n ) = ( ,m n ). For any admissible ( ,m n ) and m n , the representations T ,mn and T m n are pairwise nonequivalent. Remark 3.1. As in the case of irreducible representations of the Lie algebra so n , it follows from the explicit description of irreducible representations T mn and T ,mn of U q (so n ) that the restriction of T mn onto the subalgebra U q (so n−1 ) decomposes into a direct sum of irreducible representations of this subalgebra belonging to the classical type, and the restriction of T ,mn onto U q (so n−1 ) decomposes into a direct sum of irreducible representations belonging to the nonclassical type. Formulas for the representations determine explicitly these decompositions.

Vector operators and Wigner-Eckart theorem
In this section, we define vector operators for irreducible representations of U q (so n ) and give the Wigner-Eckart theorem for them. This information will be used under proving our main results.
The algebra U q (so n ) is not a Hopf algebra. For this reason, we cannot define a tensor product of its representations. However, U q (so n ) can be embedded into the Hopf algebra U q (sl n ) (see [29,30]). Using this embedding, a tensor product of the irreducible representations T 1 and T of U q (so n ) is determined, where T 1 is a vector representation (i.e., a representation of the classical type characterized by the numbers (1,0, ...,0)) and T is an arbitrary irreducible finite-dimensional representation [13]. The decomposition of this tensor product into irreducible constituents is given by the formulas as in the classical case if the representation T belongs to the classical type (i.e., the decomposition of T 1 ⊗ T mn contains the irreducible representations of the classical type characterized by m + j n , m − j n , j = 1,2,..., n/2 , and also the representation T mn if n = 2k + 1 and m k,2k+1 = 0). For the representations T = T ,mn of the nonclassical type, we have As before, m ± j n is the set of numbers m n with m jn replaced by m jn ± 1, respectively. Note that each representation T m n and each representation T ,m n for which m n does not satisfy the dominance conditions must be omitted. Proofs of these decompositions can be found in [14]. As in the case of quantized universal enveloping algebras (see [20,Chapter 7]), decompositions of the above tensor products are fulfilled by means of matrices whose entries are called Clebsch-Gordan coefficients.
We define a vector operator (it is a set of n operators), which transforms under the vector representation of the algebra U q (so n ). This operator acts on a linear space Ᏼ on which some representation T of U q (so n ) acts. We will consider only the case when Ᏼ is a finite-dimensional space. We also suppose that Ᏼ decomposes into a direct sum of irreducible invariant (with respect to U q (so n )) subspaces, where only irreducible representations of the classical type or only irreducible representations of the nonclassical type are realized. This assumption is explained by the fact that a vector operator cannot map a subspace on which an irreducible representation of the classical type is realized into a subspace on which a representation of the nonclassical type is realized, or vise versa.
The set A r , r = 1,2,...,n, of operators on Ᏼ is called a vector operator for the algebra We represent the space Ᏼ as a direct sum of irreducible invariant (with respect to U q (so n )) subspaces where ᐂ ,mn,i is a subspace, on which an irreducible representation of U q (so n ) characterized by and m n is realized, and i separates multiple irreducible representations of U q (so n ) in the decomposition. If irreducible representations belong to the classical type, then must be omitted.
We take a Gel'fand-Tsetlin basis in each subspace ᐂ ,mn,i and denote these basis vectors by | ,m n ,i,α , where α ≡ α n−1 are the corresponding Gel'fand-Tsetlin tableaux. Then the subspaces can be defined. The Wigner-Eckart theorem for vector operators {A j } (proved in [14]) states that the matrix elements of A j are of the form where C ,m n−1 ,α j; ,mn−1,α are Clebsch-Gordan coefficients of the tensor product T 1 ⊗ T ,mn (these coefficients are given in an explicit form in [14]), and ,m n−1 ,i |A| ,m n−1 ,i are called reduced matrix elements of the vector operator {A j }. These reduced matrix elements depend only on numbers characterizing the representations and on the indices separating multiple representations, and are independent of basis elements of irreducible invariant subspaces. They are also independent of the number j of the operator A j . In the above formulas, must be omitted if we deal only with representations of the classical type.
Due to the formulas for decompositions of the tensor products T 1 ⊗ T mn and T 1 ⊗ T ,mn , we find that matrix elements ,m n ,i ,α |A j | ,m n ,i,α can be nonvanishing only if = and also m n = m ±s n or m n = m n (since only for these cases, the corresponding Clebsch-Gordan coefficients can be nonvanishing). Due to the above formulas for decompositions of tensor products of representations, a vector operator cannot map a subspace of an irreducible representation of the classical type (of the nonclassical type) into subspaces on which irreducible representations of the nonclassical type (of the classical type) are realized. Therefore, in matrix elements (4.6), both indices and exist or both are absent.
We can define the operators which have matrix elements coinciding with reduced matrix elements of the tensor oper- representation T, these operators satisfy the following relations: Proposition 4.1. Let ξ ∈ Ᏼ belong to a subspace Ᏼ mn of the irreducible representation T mn of U q (so n ). Then A m + j n mn ξ and A m − j n mn ξ belong to some subspaces Ᏼ m + j n and Ᏼ m − j n of Ᏼ, on which the irreducible representations T m + j n and T m − j n of U q (so n ) are realized, respectively. All the vectors A m ± j n mn (T mn (a)ξ), a ∈ U q (so n ), also belong to these subspaces Ᏼ m ± j n , respectively. Proof. The assertion follows from the definition of vector operators and from formula (4.6).

Auxiliary propositions
As stated above, the algebra U q (so n ) has a commutative subalgebra Ꮽ generated by the elements I 2s,2s−1 , s = 1,2,...,r, where r = n/2 is the integral part of n/2.
Proof. This proposition is true for the algebra U q (so 3 ). It follows from complete reducibility of finite-dimensional representations of U q (so 3 ) (see [12]) and from the fact that representations of the classical and of the nonclassical types exhaust all irreducible representations of U q (so 3 ) (see [11]). Each of the elements I 21 ,I 43 ,...,I 2k,2k−1 can be included into some subalgebra U q (so 3 ) as one of its generating elements. Therefore, each of the operators T(I 2 j,2 j−1 ), j = 1,2,...,k, can be diagonalized and has eigenvalues indicated in assertion (b). This means that these operators are semisimple. Semisimple operators on a finite-dimensional space can be simultaneously diagonalized if they commute with each other. The proposition is proved.
In Propositions 5.3, 5.4, and 5.5 below, we suppose that the following assumption is fulfilled: each finite-dimensional representation of U q (so n−1 ) is completely reducible and irreducible finite-dimensional representations of U q (so n−1 ) are exhausted by the irreducible representations of the classical and nonclassical types described in Section 3. Note that for U q (so 3 ) and U q (so 4 ), this assumption is true (see [10,11,12]).
Proposition 5.3. The restriction of any irreducible finite-dimensional representation T of the algebra U q (so n ) onto the subalgebra U q (so n−1 ) is completely reducible representation of U q (so n−1 ) and decomposes into irreducible representations of this subalgebra which belong only to the classical type or only to the nonclassical type.
Proof. The restriction of T to the subalgebra U q (so n−1 ) is completely reducible due to the assumption. Let T↓ U q (son−1) = i R i , where R i are irreducible representations of U q (so n−1 ), and let Ᏼ = i ᐂ i be the corresponding decomposition of the space Ᏼ of the representation T. The subspaces ᐂ i are invariant with respect to the operators T(I j, j−1 ), j = 2,3,...,n − 1, corresponding to the elements of U q (so n−1 ). Only the operator T(I n,n−1 ) maps vectors of any of the subspaces ᐂ i to linear combinations of vectors from other subspaces ᐂ i . Since the representation T is irreducible, then acting repeatedly by T(I n,n−1 ) upon any vector of any subspace ᐂ i we obtain linear combinations of vectors from all other subspaces ᐂ i . Let some irreducible representation R i0 of U q (so n−1 ) in the decomposition of T belong to the classical type. We state then that all other representations R i in the decomposition belong to the classical type. This follows from the following reasoning. We take the operators T(I n,s ), s = 1,2,...,n − 1. It follows from the commutation relations (2.6), (2.7), and (2.8) for the elements I r,s , r > s, given in Section 2, that these operators constitute a vector operator for the subalgebra U q (so n−1 ) (generated by I 21 ,I 32 ,...,I n−1,n−2 ) acting on the space Ᏼ. Then due to the Wigner-Eckart theorem, the action of operators T(I n,s ), s = 1,2,...,n − 1, on vectors of ᐂ i0 gives linear combinations of vectors of subspaces ᐂ i on which only irreducible representations of the classical type are realized. Repeated application of T(I n,s ) again gives representations of the same type. Therefore, in this case, all representations R i belong to the classical type. If R i0 belongs to the nonclassical type, then (by the same reasoning) all representations R i belong to the nonclassical type. The proposition is proved.
We write down the decomposition T↓ U q (son−1) = i R i from the above proof in the form T↓ U q (son−1) = mn−1 d mn−1 T mn−1 if the decomposition contains representations of the classical type, where T mn−1 are irreducible representations of U q (so n−1 ) from Section 3 and d mn−1 are multiplicities of these representations. If the decomposition contains irreducible representations of the nonclassical type, then T↓ U q (son−1) = ,mn−1 d ,mn−1 T ,mn−1 , where T ,mn−1 are irreducible representations of the nonclassical type. Proof. The operators T(I n,s ), s = 1,2,...,n − 1, constitute a vector operator for the subalgebra U q (so n−1 ). Now the proposition follows from the Wigner-Eckart theorem. Proof. The proposition is true for the algebra U q (so 4 ). Namely, eigenvalues of T(I 21 ) and T(I 43 ) of an irreducible representation T of U q (so 4 ) are of the classical type if T is a representation of the classical type and of the nonclassical type if T is a representation of the nonclassical type (see [10]). We restrict the representation T of U q (so n ) successively to U q (so n−1 ),U q (so n−2 ),...,U q (so 4 ) and decompose it into irreducible constituents. (Moreover, the chain of these subalgebras can be taken in such a way that the last subalgebra U q (so 4 ) contains any two fixed neighbouring operators from Proposition 5.1(a).) Applying Proposition 5.3 at the first step, we obtain in the decomposition of T irreducible representations of U q (so n−1 ) all belonging to the classical type or all belonging to the nonclassical type. Due to the assumption before Proposition 5.3 and Remark 3.1 at the end of Section 3, on each next step, we obtain only irreducible representations of the classical type or only irreducible representations of the nonclassical type, described in Section 3. Thus, restriction of T onto any subalgebra U q (so 4 ) decomposes into irreducible representations of U q (so 4 ) all belonging to the classical type or all belonging to the nonclassical type. Our proposition follows from this assertion. The proposition is proved.
An irreducible representation T of U q (so n ) for which all the operators T(I 2i,2i−1 ), i = 1,2,..., n/2 , have eigenvalues of the classical type (of the nonclassical type) is called a representation of the classical type (of the nonclassical type). The algebra U q (so n ) does not have irreducible finite-dimensional representations of other types. In Section 3, irreducible representations of the classical and of the nonclassical type are given. But we do not know yet that they exhaust all irreducible representations of these types. Our aim is to prove that the irreducible representations of Section 3 exhaust all irreducible finitedimensional representations of U q (so n ).

Reduced matrix elements for the classical type representations
The theorem on classification of irreducible finite-dimensional representations of the algebra U q (so n ) will be proved by means of mathematical induction. Namely, we make an assumption on irreducible finite-dimensional representations of the subalgebra U q (so n−1 ) (which is true for the subalgebra U q (so 4 )) and then prove that this assumption is true for the algebra U q (so n ). Assumption 6.1. Each finite-dimensional representation of U q (so n−1 ) is completely reducible and irreducible finite-dimensional representations of U q (so n−1 ) are exhausted by irreducible representations of the classical and nonclassical types described in Section 3.
This assumption is true for the algebras U q (so 3 ) and U q (so 4 ) (see [10,11]). As we know from the previous section, irreducible finite-dimensional representations T of U q (so n ) are divided into two classes-irreducible representations of the classical type and irreducible representations of the nonclassical type. For deriving the theorem on classification of irreducible representations belonging to the classical type, we need the results on reduced matrix elements of the tensor operator T(I n,r ), k = 1,2,...,n − 1, for the subalgebra U q (so n−1 ).
Let T be an irreducible finite-dimensional representation of U q (so n ) belonging to the classical type. According to our assumption and Proposition 5.3, this representation decomposes under the restriction onto the subalgebra U q (so n−1 ) as a direct sum of irreducible representations of the classical type from Section 3. For the space Ᏼ of the representation T, we have where ᐂ mn−1,i is a linear subspace, on which the irreducible representation T mn−1 of U q (so n−1 ) from Section 3 is realized, and i separates multiple irreducible representations of U q (so n−1 ) in the decomposition. Let We take a Gel'fand-Tsetlin basis in each subspace ᐂ mn−1,i and denote these basis vectors by |m n−1 ,i,α , where α ≡ α n−2 are the corresponding Gel'fand-Tsetlin tableaux. Then the subspaces can be defined. We know from Proposition 5.4 that the operator T(I n,n−1 ) maps the vector |m n−1 ,i,α into a linear combination of vectors of the subspaces ᐂ mn−1 and ᐂ m ±s n−1 , s = 1,2,...,k, where n − 1 = 2k or n − 1 = 2k + 1. Since the operator T(I n,n−1 ) commutes with all the operators T(I s,s−1 ), s = 2,3,...,n − 2 (i.e., with operators corresponding to elements of the subalgebra U q (so n−2 )), it maps the subspace ᐂ α mn−1 into a sum of subspaces ᐂ α m n−1 with the same α.
Theorem 6.4. If the above assumption is true, then the restriction of an irreducible representation T of U q (so n ) to the subalgebra U q (so n−1 ) contains each irreducible representation of this subalgebra not more than once.
Proof. We prove the theorem for the algebra U q (so 2p+2 ). For the algebra U q (so 2p+1 ), the proof is the same. We consider the decomposition (6.25) where d m2p+1 denotes a multiplicity of the representation T m2p+1 in the decomposition. The decomposition Ᏼ = m2p+1,α ᐂ α m2p+1 corresponds to the decomposition (6.25), where, as in Section 4, α numerates elements of the Gel'fand-Tsetlin basis for the representation T m2p+1 . Let T m 2p+1 ≡ T m max 2p+1 be a maximal irreducible representation of U q (so 2p+1 ) in the decomposition (6.25), that is, such that ρ j (m 2p+1 ) = 0, j = 1,2,..., p. Due to the relations (6.8), (6.9), and (6.10), the operators ρ i and ρ j , as well as the operators ρ i and τ j , i = j, and the operators τ i and τ j , commute (up to a constant) with each other. For this reason, each of the parameters l i,2p+1 , i = 1,2,..., p, in the set of the representations T m2p+1 from the decomposition (6.25) runs over some set of numbers independent of values of other parameters l j,2p+1 , j = i.
Let ξ j,i = τ j (m −i 2p+1 )ξ i , i = 1,2,..., p. As above, it is shown that the subspace ᐂ ir We continue this reasoning further applying successively the operators τ j and ρ j with appropriate values of the numbers m 2p+1 . Due to the relations (6.8), (6.9), and (6.10), the operators ρ i and ρ j , as well as the operators ρ i and τ j , i = j, and the operators τ i and τ j , commute (up to a constant) with each other. Therefore, as a result of such continuation, we obtain the set of subspaces ᐂ ir m2p+1 of the representation space Ᏼ, on which nonequivalent irreducible representations of the subalgebra U q (so 2p+1 ) are realized and which consist of eigenvectors of the operators σ(m 2p+1 ). These subspaces are mapped by the operators ρ i and τ i into subspaces of this set. We consider the subspace Ᏼ of the space Ᏼ, which is a direct sum of these subspaces ᐂ ir m2p+1 . It follows from the expression (6.4) for T(I 2p+2,2p+1 ) that this operator leaves Ᏼ invariant. Due to irreducibility of the representation T, we have Ᏼ = Ᏼ. This completes the proof for the algebra U q (so 2p+2 ). As is noted above, for U q (so 2p+1 ), the proof is the same. The only difference is that instead of relations (6.8), (6.9), (6.10), (6.11), (6.12), (6.13), (6.14), (6.15), and (6.16), we have to use relations (6.17), (6.18), (6.19), (6.20), (6.21), and (6.22). The theorem is proved.
Remark 6.5. We have seen under proving Theorem 6.4 that in the set of the representations T m2p+1 from the decomposition (6.25) each of the parameters m i,2p+1 , i = 1,2,..., p, runs over some set of numbers independent of values of other parameters m j,2p+1 , j = i. It is easy to show by means of formula (6.27) that in an irreducible representation T of U q (so 2p+2 ) each m i,2p+1 , i = 1,2,..., p, takes all values from the set m min i,2p+1 ,m min i,2p+1 + 1,...,m max i,2p+1 without any omitting. A similar assertion is true for irreducible finitedimensional representations of U q (so 2p+1 ).
We derive from (6.17), (6. is independent of l i,2p . Therefore, the expression β j l j,2p = ρ j m 2p τ j m + j 2p q lj,2p + q −lj,2p q lj,2p+1 + q −lj,2p−1 × r = j l r,2p l r,2p − 1 − l j,2p l j,2p + 1 l r,2p + 1 l r,2p − l j,2p l j,2p + 1 (6.57) depends only on l j,2p . Then we rewrite the relations (6.21) and (6.22) for β j (l j,2p ) and in the same way as in Proposition 6.6, using the equalities (6.41) and (6.43), derive the following proposition. Separating ρ j (m 2p ) and τ j (m + j 2p ) from β j (l j,2p ) as in the previous case, for the operator T(I 2p+1,2p ) of an irreducible representation T of U q (so 2p+1 ), we obtain  Thus, we derived an explicit form of the operator T(I n,n−1 ) of an irreducible representation of U q (so n ). In order to obtain a classification of irreducible representations of the classical type, we have (by using (6.53) and (6.59)) to derive a domain of the parameters l 1n ,l 2n ,...,l pn , p = n/2 .

Reduced matrix elements for the nonclassical type representations
We assume that Assumption 6.1 of Section 6 is acting.
Proposition 7.1. Let T be an irreducible finite-dimensional representation of U q (so n ) belonging to the nonclassical type. Then the decomposition of T↓ U q (son−1) into irreducible constituents contains irreducible representations T ,mn−1 with the same .
Proof. The proposition follows from Proposition 5.4 and from the fact that the decomposition of the tensor products T 1 ⊗ T ,mn−1 (where T 1 is a vector representation) into irreducible constituents contains irreducible representations of the nonclassical type with coinciding with in T ,mn−1 . The proposition is proved.
Let T be such as in Proposition 7.1 and let Ᏼ be a space on which T acts. Let where ᐂ ,mn−1,i is a linear subspace, on which an irreducible representation T ,mn−1 of U q (so n−1 ) is realized, and i separates multiple irreducible representations in the decomposition. We also introduce the subspaces We take a Gel'fand-Tsetlin basis in each subspace ᐂ ,mn−1,i and denote the basis vectors by | ,m n−1 ,i,α , where α ≡ α n−2 are the corresponding Gel'fand-Tsetlin tableaux. Let 2 + · σ 2 ,m 2p+1 = −E, (7.13) where i = j, E is the unit operator on ᐂ α ,m2p+1 and k is a fixed number from the set {1, 2,..., p}.
The irreducible representations T ,m2p+1 of U q (so 2p+1 ) under restriction to U q (so 2p ) decompose into irreducible representations T ,m2p of this subalgebra such that the numbers m 2p satisfy the inequalities determined by the Gel'fand-Tsetlin tableaux. Under this, each of the numbers l r,2p runs over a certain set of values. Assuming that none of l r,2p , r = p, is a constant for the representation T ,m2p+1 , we equate in (7.13) terms with the same dependence on [l r,2p ] 2 + and obtain the relations If k parameters l r,2p , r = p, are constant for the representation T ,m2p+1 , then the number of the relations (7.14), (7.15), and (7.16) is decreased by k.
In a similar way it is proved that ρ i ( ,m 2p ) and τ i ( ,m 2p ) from formula (7.5) satisfy the relations If the values of l s,2p are fixed in the considered representations of U q (so 2p+1 ), then the number of relations, which follow from (8.5) and the number of G i are decreased by the number of fixed l s,2p . Thus, as before, we get G i = 0, i = 2,3,..., p and, therefore,σ = 0.
We have proved thatσ(m) = 0 for all irreducible representations T m of U q (so 2p+1 ), contained in the representation T↓ U q (son) . This means that all operators σ (T) (m) are diagonal and the further proofs of complete reducibility are conducted in the same way as in the previous subcase.
Case 3 is proved in the same way as Case 2 and we omit this proof. The theorem is proved.
Corollary 8.2. If irreducible finite-dimensional representations of U q (so n−1 ) are exhausted by irreducible representations of Section 3, then each finite-dimensional representation of U q (so n ) is completely reducible.

Classification theorems
Suppose that Assumption 6.1 of Section 6 is acting.
Proposition 9.1. If Assumption 6.1 of Section 6 is true, then irreducible finite-dimensional representations T of U q (so n ) such that the restriction T↓ U q (son−1) contains in the decomposition into irreducible components only representations of the classical type of U q (so n−1 ) are exhausted by the representations of the classical type from Section 3.
Proof. We prove the proposition when n = 2p + 2. For n = 2p + 1, the proof is similar.
In order to prove the proposition for representations of the algebra U q (so 2p+1 ), we use the formula of Proposition 6.7 and formula (6.59) instead of formulas (6.40) and (6.53). The proposition is proved.
Proposition 9.2. If Assumption 6.1 of Section 6 is true, then irreducible finite-dimensional representations T of U q (so n ) such that the restriction T↓ U q (son−1) contains in the decomposition into irreducible components only representations of the nonclassical type of U q (so n−1 ) are exhausted by the representations of the nonclassical type of Section 3.

N. Z. Iorgov and A. U. Klimyk 261
The proof of this proposition is the same as that of Proposition 9.1. Theorem 9.3. Irreducible finite-dimensional representations of the algebra U q (so n ) are exhausted by representations of the classical type and of the nonclassical type from Section 3.
Proof. For the algebra U q (so n−1 ) ≡ U q (so 4 ), Assumption 6.1 of Section 6 is true (see [10]). Now the theorem is easily proved by induction taking into account Theorem 8.1 and Propositions 9.1 and 9.2. The theorem is proved.
Corollary 9.4. Each finite-dimensional representation of U q (so n ) is completely reducible.
Proof. This assertion follows from Corollary 8.2 of Section 8 and from Theorem 9.3.