aa r X i v : . [ m a t h . QA ] M a y DUALITIES FOR SPIN REPRESENTATIONS
HANS WENZL
Abstract.
Let S be the spinor representation of U q so N , for N odd and q not a rooot ofunity. We show that the commutant of its action on S ⊗ n is given by a representation of thenonstandard quantum group U ′− q so n . For N even, an analogous statement also holds for S = S + ⊕ S − the direct sum of the irreducible spinor representations of U ′ q so N , with thecommutant given by U ′− q o n , a Z / U ′− q so n . Similar statements also hold forfusion tensor categories with q a root of unity. The decomposition of tensor powers of the vector representations of classical Lie groupswas successfully studied in papers by Schur, Weyl and Brauer. These classical duality resultswere extended to Drinfeld-Jimbo quantum groups ([14], [4], [24]) which have had applicationsin a number of fields such as tensor categories, low-dimensional topology and von Neumannalgebras. But it seems that an intrinsic description of the commutant of the action of spingroups on tensor powers of spinor representations has been found only fairly recently in [27].The current paper deals with the missing cases in that paper. Moreover, we give simpler andmore general proofs also for the cases already covered.The inspiration for the approach in the paper [27] came from a paper by Hasegawa [10], seeSection 6.1 for more details. He showed (as a special case of a far more general construction)that the obviously commuting actions of O ( N ) and SO ( n ) on R N ⊗ R n can be extendedto commuting actions of the corresponding spin groups on the Clifford algebra Cl ( N n ) ∼ = Cl ( N ) ⊗ n . For N even, this vector space isomorphism can actually be made into an algebrahomomorphism, with Cl ( N ) ∼ = End( S ) for the spinor representation S of O ( N ). Hence thecommutant of the P in ( N ) action on S ⊗ n is given by a representation of Spin ( n ). This resultcould be extended to prove a duality on S ⊗ n between a semidirect product U of the quantumgroup U q so N with Z / q -deformation U ′ q so n of the universal envelopingalgebra U so n . The latter has been studied before in particular by Klimyk and his coauthors,see e.g. [8], and it has also appeared in work of Noumi and Sugitani [20] and Letzter [17].But for N odd, it was not possible to prove such a simple fact, due to the fact that theClifford algebra is not simple in that case. Our main new result in this paper is a directdescription of the commutant of the action of U q so N on S ⊗ n via a representation of thenonstandard orthogonal quantum group U ′− q so n . This can be quite easily seen for q = 1using the element C = P i e i ⊗ e i ⊂ Cl ( N ) ⊗ , where { e i } is an orthonormal basis for R N .We then extend this result to the quantum group case by using Hayashi’s q -Clifford algebra.Here is the paper in more detail: We first review some basics about Clifford algebras andspin representations S and produce a canonical element C ∈ End( S ⊗ ) which commutes with the spin group action. We review basic facts about the algebras U ′ q so n and their representa-tions in the second section. In particular, we show that we obtain a representation of U ′− so n on S ⊗ n which commutes with the pin group action. In the third section we review Hayashi’sspin representations of U q so N into his q -Clifford algebra and we construct the q -analogs of thecommuting elements C . It is shown in the following section that these can be used to define arepresentation of U ′− q so n on S ⊗ n . We then obtain first and second fundamental theorems forEnd U ( S ⊗ n ), where U can be U q so N or, for N even, it can also be U q so N ⋊ Z /
2. We concludewith some remarks about related work and applications.
Acknowledgements : Part of the work on this paper was done while the author enjoyed thehospitality and support of the Max Planck Institute of Mathematics in Bonn. He would liketo thank Catharina Stroppel and Daniel Tubbenhauer for stimulating discussions. He wouldalso like to thank Doron Gepner for his interest and encouragement.1.
Clifford Algebras and spinor representations
Basic Definitions.
Let V be a finite dimensional real inner product space. Then itsClifford algebra is the associative complex algebra generated by the elements of V subject tothe relation v · w + w · v = 2( v, w )1 . Let { e , ... e N } be an orthonormal basis of the inner product space V . Then the Cliffordalgebra Cl = Cl ( N ) corresponding to V can also be defined via generators, denoted by e i aswell, and relations e i e j + e j e i = 2 δ ij , for 1 ≤ i, j ≤ N. If N = 2 k is even, it will be convenient to use a second presentation in terms of generators ψ j and ψ † j , 1 ≤ j ≤ k defined by(1.1) ψ j = 12 ( e j − + ie j ) , ψ † j = 12 ( e j − − ie j ) . Then a generator ψ j anticommutes with all other generators except ψ † j where we have ψ j ψ † j + ψ † j ψ j = 1 , ≤ j ≤ k. Spinor representations.
Let us assume N = 2 k to be even. It is well-known that Cl ( N ) has dimension 2 N and it is isomorphic to M N/ , where M d denotes the d × d matrices.Let S be a simple Cl ( N )-module, of dimension 2 N/ . The action of an element g in theorthogonal group O ( N ) on V induces an automorphism α g on Cl ( N ), for each g ∈ O ( N ). Asany automorphism of the d × d matrices is inner, we obtain a projective representation g u g of O ( N ) on the Cl ( N )-module S . This projective representation can be made into an honestrepresentation of the universal covering groups P in ( N ) of O ( N ). By restriction, the module S becomes a P in ( N − S . It decomposes into the directsum of two simple projective O ( N − S + ⊕ ˜ S − . These two modules are isomorphic UALITIES FOR SPIN REPRESENTATIONS 3 as projective SO ( N − Lemma 1.1.
Let N = 2 k be even. Define f r = ( − i ) r ( r − / e e ... e r ∈ Cl ( N ) , for ≤ r ≤ N .Then we have(a) f r e i = ( − r e i f r for i > r and f r e i = ( − r − e i f r for i ≤ r ,(b) f r = 1 and f r f s = ( − r ( s − r ) f s f r for r < s .(c) f = e e = ( ψ ψ † − ψ † ψ ) and f N = ( − k − Q kj =1 ( ψ j ψ † j − ψ † j ψ j ) .(d) α g ( f r ) = det ( g ) f r and u g f r = det ( g ) f r u g for g ∈ O ( r ) , ≤ r ≤ N . P roof.
Parts (a) - (c) are straightforward. Part (d) is checked by an explicit calculationfor N = 2. The same calculation also works for g ∈ O ( N ) which is the identity matrix exceptfor a 2 × O ( N ), the claim follows.1.3. Explicit description.
We will need a more explicit description of the spin module S .Observe that Cl = Cl + Cl − , where Cl + and Cl − are the subalgebras generated by 1 and theelements ψ † i for Cl + , and by 1 and the elements ψ i for Cl − . We define S = Cl/I for N = 2 k even, where I is the left ideal generated by the ψ i ’s. S has a basis x ( m ), where m ∈ R k with m i ∈ { , } for 1 ≤ i ≤ k , and where x ( m ) = ( ψ † ) m ( ψ † ) m ... ( ψ † k ) m k mod I. If we assign to the highest weight vector m (0) the weight ǫ = ( , , ... ), the vector x ( m )would have the weight µ with µ i = − m i . We also define the quantity m { r } = r X j =1 m j . Then it follows that ψ † j x ( m ) = 0 if m j = 1, and ψ † j x ( m ) = ( − m { j − } x ( ¯ m j ) , if m j = 0 , where ¯ m j coincides with m except in the j -th coordinate, which is replaced by 1 − m j . Observethat the weight of x ( ¯ m j ) differs from the one for x ( m ) only in the j -th coordinate, by a sign.The action of ψ j is given by the adjoint of ψ † j .If N = 2 k + 1 is odd, we can make S as before into a Cl (2 k + 1)-module as follows. Theactions of e i with i ≤ k resp ψ j and ψ † j with j ≤ k is as before. The action of e k +1 is givenby e k +1 x ( m ) = ± f k x ( m ) = ± ( − m { k } x ( m ) , where we obtain a Cl (2 k + 1) action for each choice of the sign. HANS WENZL
Tensor products.
Let S be the Cl ( N )-module as described in the previous subsection,for both N odd and N even. In the following, we will identify elements of Cl ( N ) with itsimage in End( S ). We define elements C ∈ End( S ⊗ ) by C = 12 N X i =1 e i ⊗ e i . Using the definitions in 1.1, we obtain12 ( e j − ⊗ e j − + e j ⊗ e j ) = ψ j ⊗ ψ † j + ψ † j ⊗ ψ j . Using this and the definitions in the last section, we can also write for N = 2 k + 1(1.2) C = 12 f k ⊗ f k + k X j =1 ψ j ⊗ ψ † j + ψ † j ⊗ ψ j , while for N = 2 k even C is as above without the first summand. We will also need thewell-known isomorphisms of P in ( N ) modules given by(1.3) N = 2 k : S ⊗ ∼ = k M j =0 ∧ j V, N = 2 k + 1 : S ⊗ ∼ = k M j =0 ∧ j V, where V = C N is the vector representation of O ( N ). Lemma 1.2. (a) If N = 2 k , the element C has exactly the integer eigenvalues j satisfying − k ≤ j ≤ k .(b) If N = 2 k + 1 , the element C has the eigenvalues ε ( − j ( k + − j ) , ≤ j ≤ k with thesign ε depending on the choice of the Cl ( N ) -module S . P roof.
Observe that( ψ j ⊗ ψ † j + ψ † j ⊗ ψ j ) x ( m ) ⊗ x ( n ) = ( − m { j − }− n { j − } x ( ¯ m j ) ⊗ x (¯ n j ) , if m j + n j = 1, and it is equal to 0 otherwise. Let now ρ = ( k − i ) i ∈ R k , and let ¯ m be definedby ¯ m j = 1 − m j , 1 ≤ j ≤ k . Let(1.4) v = X m ( − ( m,ρ ) x ( m ) ⊗ x ( ¯ m ) . Then the x ( m ) ⊗ x ( ¯ m ) coordinate of Cv is given by( − ( m,ρ ) X j ( − m { j − }− ( j − m j − m,ρ ) = ( − ( m,ρ ) ( − k − k. This shows that ( − k − k is an eigenvalue for C . One obtains an eigenvector with eigenvalue( − k k by multiplying the x ( m ) ⊗ x ( ¯ m )-coordinate of the vector v by the scalar ( − m { k } .One can similarly define eigenvectors v with eigenvalues ± j by the sum of vectors x ( m ) ⊗ x ( n ) UALITIES FOR SPIN REPRESENTATIONS 5 where m i + n i = 1 for 1 ≤ i ≤ j and where m ( i ) = n ( i ) = 0 for i > j . These are all possibleeigenvalues in view of 1.3 and Lemma 1.3.It follows from Lemma 1.1,(c) that f N x ( m ) ⊗ f N x ( n ) = ( − m { k } + n { n } x ( m ) ⊗ x ( n ). Itfollows from this and statement (a) that the eigenvalues of C are given by ( − j ( k + − j ),1 ≤ j ≤ k . Lemma 1.3.
The element C commutes with the action of P in ( N ) on S ⊗ . Moreover, if C = C ⊗ ∈ End( S ⊗ ) and C = 1 ⊗ C ∈ End( S ⊗ ) then we have C C + 2 C C C + C C = C , C C + 2 C C C + C C = C .P roof. As g. ( P Ni =1 e i ⊗ e i ) = P Ni =1 e i ⊗ e i ∈ V ⊗ for any g ∈ O ( N ), the correspondingelement P Ni =1 e i ⊗ e i ∈ Cl ( n ) ⊗ commutes with the action of P in ( N ) on S ⊗ . The firststatement follows from this and Lemma 1.1,(d).Let [ A, B ] + = AB + BA for any two elements A, B of a ring. Then it follows from therelations that(1.5) N X i,j =1 [ e i , e j ] + = N X i,j =1 δ i,j . Using this, we obtain[ C , C ] + = 14 N X i,j =1 e i ⊗ [ e i , e j ] + ⊗ e j = 12 N X i =1 e i ⊗ ⊗ e i . We obtain in the same fashion[ C , [ C , C ] + ] + = 14 N X i,j =1 [ e i , e j ] + ⊗ e i ⊗ e j = 14 r X i,j =1 δ ij ⊗ e i ⊗ e j = C . This proves the first identity in the statement. The proof of the second identity goes the sameway. 2.
Representations of U ′ q so n We assume throughout this paper all the algebras to be defined over the field of complexnumbers, with q not being a root of unity. See Section 5.5 for more general rings.2.1. The algebras U ′ q so n and U ′ q o n . We shall need a q -deformation U ′ q so n of the universalenveloping algebra of the Lie algebra so n . It is defined via generators B i , 1 ≤ i < n andrelations B i B j = B j B i for | i − j | > B i B i ± − ( q + q − ) B i B i ± B i + B i ± B i = B i ± , with the choice of sign in the indices the same for all terms. This algebra was defined, indepen-dently from each other, by Gavrilik and Klimyk [8], by Letzter [17] and by Noumi and Sugitani[20]. Its finite-dimensional representations were classified by Klimyk and collaborators (see[12] and references there). HANS WENZL
Theorem 2.1.
Let q not be a root of unity. Then there are two series of finite-dimensionalirreducible representations of U ′ q so n , where k = ⌊ n/ ⌋ :(a) The classical representations are q -deformations of the representations of so n . Theyare labeled by the dominant integral weights of so n . They are given by all vectors λ =( λ , λ , ..., λ k ) , where all coefficients are either integers or they all are congruent to mod Z , and such that λ ≥ λ ≥ ... ≥ λ k ≥ (for n odd), or λ ≥ λ ≥ ... ≥ | λ k | (for n even).They have the same dimensions as the corresponding so n representations.(b) The nonclassical representations are labelled by all dominant integral weights of so n whose coefficients are not integers. Their dimensions are − k times the dimension of thecorresponding classical representations for n odd, and − k times the dimension of the cor-responding classical representations for n even. For each such weight, we have n − non-equivalent representations of U ′ q so n . They can be obtained from each other by multiplying thematrices for various generators by − . For n even, it suffices to consider only representationswith highest weights λ for which λ k > , together with the just mentioned operation of signchanges.Remark . The construction of the representations of U ′ q so n by Klimyk et al essentially is a q -version of the construction of representations of so n via Gelfand-Zetlin bases, see [19]. Thisapproach takes advantage of the fact that a simple so n -module, viewed as an so n − moduledecomposes into a direct sum of mutually nonisomorphic simple so n − modules, see e.g. [19],Section 4.1 and 4.2 for details. This also determines the decomposition of a simple classicalU ′ q so n module into a direct sum of mutually nonisomorphic U ′ q so n − modules.The decomposition of a simple non-classical U ′ q so n module V λ into a direct sum L V µ ofsimple U ′ q so n − modules can be described similarly: If n = 2 k is even, µ runs through allweights µ satisfying(2.1) λ ≥ µ ≥ λ ≥ ... ≥ µ k − ≥ λ k > , where for n = 2 k + 1 odd we have(2.2) λ ≥ µ ≥ λ ≥ ... ≥ µ k − ≥ λ k > µ k > . All quantities in these inequalities are half-integers, i.e. congruent to 1/2 mod Z . For classicalrepresentations, the restriction rules are almost the same. We only need to replace λ k by | λ k | for n = 2 k , and µ k by | µ k | for n = 2 k + 1 in the inequalities above, see [19].We will also need an analog of the full orthogonal group in this setting. We define the algebra U ′ q o n by adding an additional generator F to the generators of U ′ q so n with the relations F = 1 , F B = − B F and F B i = B i F for i > . Remark .
1. It is well-known that the Lie algebra so n can be defined via generators L i = E i,i +1 − E i +1 ,i , 1 ≤ i < n , with E i,j matrix units. It is then easy to check that the maps B j
7→ √− L j , 1 ≤ j < n and F diag ( − , , , ...
1) define representations of U ′ so n = U so n and U ′ o n respectively. UALITIES FOR SPIN REPRESENTATIONS 7
2. It is also clear from the representation in the first remark that any irreducible repre-sentation of the group O ( N ) defines a representation of U ′ o n by viewing the image of F as agroup element, and identifying U ′ so n with the universal enveloping algebra of so n . We willsee in this paper that these representations also exist and remain irreducible for generic q .2.2. Homomorphism onto Temperley-Lieb algebras.
The Temperley-Lieb algebra
T L n is given by generators e i , 1 ≤ i < n and relations e i = e i , e i e j = e j e i for | i − j | > e i e i ± e i = q + q − e i . It is well-known that End U q sl ( V ⊗ n ) is isomorphic to T L n in ourparametrization for V = C . The proof of the following proposition is a straightforward, ifmoderately tedious calculation. It is a special case of our main results Theorem 5.2 and 5.3. Proposition 2.4.
The map B i q + q − − ( q + q − ) e i , ≤ i < n defines an algebra homo-morphism from U ′− q so n onto T L n , which induces non-classical representations of U ′− q so n . Representations of U ′ q so . Similarly as for the case of the Lie algebra sl , it is easy towrite down explicit irreducible representations for U ′ q so . The ( N + 1)-dimensional irreducible classical representation V N of U ′ q so can be described as follows. We fix a basis { v j , ≤ j ≤ N } of weight vectors. Then the actions of B and B are given by(2.3) B v j = [ N/ − j ] v j , B v j = v j +1 + α j − ,j v j − , where α j − ,j = [ N + 1 − j ][ j ]( q N/ − j + q j − N/ )( q N/ − j +1 + q j − N/ − ) . Similarly, for N odd, we can describe an ( N + 1) / non-classical repre-sentation of U ′ q so with respect to a basis { v j , ≤ j < ( N − / } by(2.4) B v j = [ N/ − j ] + v j , B v j = v j +1 + α + j − ,j v j − , j < ( N − / , where(2.5) α + j − ,j = [ N + 1 − j ][ j ]( q N/ − j − q j − N/ )( q N/ − j +1 − q j − N/ − ) . If j = ( N − /
2, the action on v j by B is as in 2.4, while we have B v ( N − / = ± [( N + 1) / i ( q / − q − / ) v ( N − / + α +( N − / , ( N − / v ( N − / , where the representation with the minus sign in the formula above is equivalent to the repre-sentation with the plus sign, after replacing B with − B . We will consider representations ofthe algebra U ′− q so n . If we choose ( − q ) / = iq , the eigenvalues of B in the ( N +1) / k +1-dimensional nonclassical representation will be ( − k − j ( q N − j − q j − N ) / ( q − q − ), 0 ≤ j ≤ k . Lemma 2.5. (a) We can make V N into a U ′− q so -module by leaving the action of B thesame, and replacing B by ˜ B whose action on V N is given by ˜ B v j = ( − j B v j .(b) If N is even, the representation of U ′− q so in (a) is isomorphic to its classical repre-sentation with highest weight [ N/ . If N is odd, it decomposes into the direct sum of two HANS WENZL non-classical irreducible representations of U ′− q so with highest weight [ N/ + = i ( q N/ + q − N/ ) / ( q − q − ) . P roof.
Obviously the actions of B and ˜ B on V N are the same, while one checks easilythat ˜ B B ˜ B v j = − B B B v j for all basis vectors of V N . This implies (a), using Theorem2.1.If N is even, it is easy to check that v N is a highest weight vector with weight [ N/ − q forthe U ′− q so action on V N given via ˜ B and B which remains irreducible. For N odd, ˜ B actsvia the same scalar on v j as on v N − j . Hence we obtain two highest weight vectors (in the senseof [28]) with highest weight [ N/ + , where the q -number here is defined for − q . It followsthat V N decomposes into the direct sum of two irreducible non-equivalent U ′− q so -modules(see e.g. [28], Theorem 3.8 for details).2.4. Homomorphisms.
Let S be the spinor module as described in Section 1.3 and let C and ˜ C be as in Section 1.4. We define elements C i = 1 ⊗ ⊗ ... ⊗ C ⊗ ... ⊗ ∈ End( S ⊗ n ) , ≤ i < n, ˜ C i = 1 ⊗ ⊗ ... ⊗ ˜ C ⊗ ... ⊗ ∈ End( S ⊗ n ) , ≤ i < n, i odd , where the element C on the right hand side acts on the i -th and ( i + 1)-st factor of S ⊗ n .Moreover, we define ˜ C i = C i for 1 < i < n . Proposition 2.6.
For both N odd and even, the map B i C i defines a homomorphisms of U ′− so n into End
P in ( N ) ( S ⊗ n ) . For N even, the map B i ˜ C i for i odd and B i C i for i even defines a homomorphism of U so n into End
P in ( N ) ( S ⊗ n ) . P roof.
The first statement follows from Lemma 1.3 and the discussion before this proposi-tion. The second statement was already shown in [27]. It can also be deduced from the firststatement using Lemma 2.5. 3.
Quantum groups q -Clifford algebra. We follow the paper [11] by Hayashi, with some minor modifica-tions. For a somewhat more conceptual approach to q -Clifford algebras, see [7] and Section6.3. The q -Clifford algebra Cl q (2 k ) coincides with the ordinary Clifford algebra in the sensethat it is again generated by elements ψ i and ψ † i , 1 ≤ i ≤ k satisfying the relations for theusual Clifford algebra. In particular, ψ i ψ † i and ψ † i ψ i are idempotents which annihilate eachother and add up to 1. The dependency on q will be reflected by additional elements ω i defined by(3.1) ω i = ψ i ψ † i + q − ψ † i ψ i . Then it is clear that ω ± i ω ± j = ω ± j ω ± i and that(3.2) ω i ψ i = ψ i = qψ i ω i , ψ † i ω i = ψ † i = qω i ψ † i . UALITIES FOR SPIN REPRESENTATIONS 9
Homomorphisms into q -Clifford algebras. We are now defining maps from the quan-tum groups U q so k and U q so k +1 , i.e. of Lie type D k and B k into Cl q ( N ). Here we use thedefinition of the quantum groups as in [18] or in [13], Section 4.3. In particular, the innerproduct on the weight lattice is normalized such that ( α i , α i ) = 2 for every short root. Themaps appeared before in [11]. However, our normalizations are not always the same as in thatpaper, so we give the explicit maps below as follows: For 1 ≤ i ≤ k − K i ω i ω − i +1 , E i ψ i ψ † i +1 , F i ψ i +1 ψ † i . For type B k , we also define(3.4) K k qω k , E k ψ k f k , F k f k ψ † k , while for type D k we define(3.5) K k q ω k − ω k , E k ψ k − ψ k , F k ψ † k − ψ † k , Proposition 3.1. (see [11] ) (a) The assignments in 3.3 and 3.5 define a representation of U q so N , N = 2 k even into Cl q ( N ) ∼ = End( S ) .(b) The assignments in 3.3 and 3.4 define a representation of U q so k +1 into Cl q (2 k ) ∼ =End( S ) . P roof.
Statement (a) was proved in [11]. The assignments in (b) differ from the ones in[11] only by multiplying the images of E k and F k by f k from the right and from the left. Itis not hard to check that this still satisfies the quantum group relationis, as f k = 1 and f k commutes with the images of the lower indexed generators. Remark .
1. Observe that we have defined a representation of U q so N for N = 2 k . Thiswill make it easier to deal with the odd- and even-dimensional cases at the same time.2. One can check that for N = 2 k + 1 odd the vector x (0) is a highest weight vector withweight ε = (1 , , ..., x ( m ) has weight µ with µ i = − m i , 1 ≤ i ≤ k .If N = 2 k even, we also have the highest weight ε − = x ( m ) with m i = δ i,k . The weight of x ( m ) can be determined as in the odd-dimensional case.3.3. Commuting objects, Lie type D . We now define the q -deformations of the operators C of the previous section for quantum groups. As before, we define them as elements of Cl q ( N ) ⊗ acting on End( S ⊗ ). For Lie type D k , we define(3.6) C = k X i =1 Ω − i − ψ i ⊗ Ω i − ψ † i + Ω − i − ψ † i ⊗ Ω i − ψ i , where Ω r = Q rj =1 ω j , 1 ≤ r ≤ k . We leave it to the reader to check that(3.7) (Ω − j − ⊗ Ω j − )( ψ j ⊗ ψ † j + ψ † j ⊗ ψ j ) x ( m ) ⊗ x ( n ) = ( − q ) m { j − }− n { j − } x ( ¯ m j ) ⊗ x (¯ n j ) , if m j + n j = 1, and it is equal to 0 otherwise. Lemma 3.3.
The operator C defined in Eq 3.6 commutes with the action of U q so k on S ⊗ . P roof.
Let us first do this for type D . We shall use the coproduct defined by∆( E i ) = K / i ⊗ E i + E i ⊗ K − / i ;it is well-known that this is equivalent to the coproduct defined in [18] and [13], using theautomorphism defined by E i E i K / i , F i K − / i F i , K i K i , 1 ≤ i ≤ k . Using Def. 3.3,we obtain ∆( E ) = ω ω − ⊗ ψ ψ † + ψ ψ † ⊗ ω − ω . Let C = ψ ⊗ ψ † + ω − ψ † ⊗ ω ψ . We then obtain[∆( E ) , C ] = ω ω − ψ ⊗ ψ ψ † ψ † − ψ ω ω − ⊗ ψ † ψ ψ † + ψ ψ † ω − ψ ⊗ ω − ω ω ψ † − ω − ψ ψ ψ † ⊗ ω ψ † ω − ω = − ω − ψ ⊗ ( ψ ψ † + q − ψ † ψ ) ψ † + ψ ( qψ † ψ + ψ ψ † ) ⊗ ω ψ † = − ω − ψ ⊗ ω ψ † + ω − ψ ⊗ ω ψ † = 0 , (3.8)where we used the relations 3.1 and 3.2 after the definition of the q -Clifford algebra. Onesimilarly also shows that the commutant of ∆( E ) with ψ † ⊗ ψ + ω − ψ ⊗ ω ψ † is equal to 0.This shows that [∆( E ) , C ] = 0. The statement for F can be shown by a similar calculation,or it can be deduced by the following argument: The transpose map T induced by ψ i ψ Ti = ψ † i , ψ † i ( ψ † i ) T = ψ i induces an algebra antiautomorphism of Cl q which induces on the image of U q so N the algebraanti-automorphism defined by E i E Ti = F i , F i F Ti = E i , K i K Ti = K i , which is compatible with the Hopf algebra structure. As C T = C , it follows[∆( F i ) , C ] = [∆( E Ti ) , C T ] = − [∆( E i ) , C ] T = 0 . The commutation relation of C with ∆( E ) is shown by a similar calculation, from whichfollows the claim for F by the previous argument.For the general case, one observes that ∆( E i ) trivially commutes with all summands of C except the ones indexed by i and i + 1. The proof that ∆( E i ) commutes with these remainingsummands is essentially the same as the one for the case D .3.4. Commuting objects, Lie type B . For Lie type B k , we define the operator C ∈ End( S ⊗ ) by (compare with Section 1.4 for q = 1)(3.9) C = 1[2] (Ω − k f k ⊗ Ω k f k ) + k X i =1 Ω − i − ψ i ⊗ Ω i − ψ † i + Ω − i − ψ † i ⊗ Ω i − ψ i . Lemma 3.4.
The operator C defined in Eq 3.9 commutes with the action of U q so k +1 on S ⊗ . UALITIES FOR SPIN REPRESENTATIONS 11
P roof.
The fact that the images of generators labeled by i < k commute with C exceptfor the first summand follows from the proof of Lemma 3.3. One then checks that ψ i ψ † i +1 commutes with ω i ω i +1 by a direct calculation; this implies that it also commutes with Ω k . Itis now easy to check that ∆( E i ) also commutes with the first summand in the definition of C .To finish the proof, recall that we have (with N = 2 k )∆( E k ) = q / ω k ⊗ ψ k f N + ψ k f N ⊗ q − / ω − k . We will use relations 3.1 and 3.2, which also imply ψ k − Ω k − = q − Ω k − ψ k − . We obtain[∆( E k ) , Ω − k f N ⊗ Ω k − f N ] == q / ω k Ω − k f N ⊗ [ ψ k f N Ω k f N − Ω k f N ψ k f N ]+ [ ψ k f N Ω − k f N − Ω − k f N ψ k f N ] ⊗ q − / ω − k Ω k f N = ω k Ω − k f N ⊗ ( q − / + q / )Ω k ψ k + ( q / + q − / )Ω − k ψ k ⊗ Ω k f N ω − k = ( q + q − ) ( q − / ω k Ω − k f N ⊗ Ω k ψ k + q / Ω − k ψ k ⊗ Ω k − f N ω − k − ) . (3.10)We also have [∆( E k ) , (Ω − k − ⊗ Ω k − )( ψ k ⊗ ψ † k + ψ † k ⊗ ψ k )] == q / ω k Ω − k − ψ k ⊗ ψ k f N Ω k − ψ † k − Ω − k − ψ k q / ω k ⊗ Ω k − ψ † k ψ k f N + ψ k f N Ω − k − ψ † k ⊗ q − / ω − k Ω k − ψ k − Ω − k − ψ † k ψ k f N ⊗ q − / Ω k − ψ k ω − k = − Ω − k − ψ k ⊗ Ω k − f N ( q / ψ k ψ † k + q − / ψ † k ψ k ) − ( q − / ψ k ψ † k + q / ψ † k ψ k )Ω − k − f N ⊗ Ω k − ψ k = − q / Ω − k ψ k ⊗ Ω k − f N ω k − q − / ω − k Ω − k − f N ⊗ Ω k − ψ k . (3.11)Obviously, ∆( E k ) commutes with the first k − C . It follows from the identity ω k Ω − k = ω − k Ω − k − and the last two calculationsthat ∆( E k ) also commutes with the remaining summands of C . The commutation with ∆( F k )can be deduced from this using the transposition map T as in the proof of Lemma 3.3.4. Relations
The main result of this section will be to give an algebraic description of the centralizer ofthe action of U q so N on S ⊗ n . It is possible, and fairly straightforward, to extend the proof in[27] to the additional cases treated here. However, that proof was somewhat indirect. So wedecided to give another proof here which, basically, is a direct calculation. While not quite asstraightforward as the proof for q = 1 in Lemma 1.3, it would still seem to be an improvementover the one in [27]. Basic relations.
Let a, b be any elements in an associative algebra, and let v be anyinvertible element in its ground ring. Then we define(4.1) lhs v ( a ; b ) = a b + ( v + v − ) aba + ba . It will be convenient to introduce the notation c i, + = Ω − i − ψ i , c i, − = Ω − i − ψ † i , d i, + = Ω i − ψ i , d i, − = Ω i − ψ † i . Using this, we can write the commuting operators C from Lemma 3.3 and Lemma 3.4 as(4.2) C = k X i =1 C ( i ) and C = ˜ C ( k + 1) + k X i =1 C ( i )where C ( i ) = c i, + ⊗ d i, − + c i, − ⊗ d i, + , ˜ C ( k + 1) = 1[2] Ω − k f N ⊗ Ω k f N . It is straightforward to check the following relations, where ε, κ ∈ {±} and q ε = q ± :(4.3) d i,ε c j,κ = ( − q ε c j,κ d i,ε i < j, − q κ c j,κ d i,ε i > j. We obtain from the relations so far the following equation which will be useful later: d i,ε d i, − ε c j,κ + ( q + q − ) d i,ε c j,κ d i, − ε + c j,κ d i,ε d i, − ε =(4.4) = ( i > j, (1 − q ε ) d i,ε d i, − ε c j,κ i < j. Technical lemma.Lemma 4.1.
Using notation defined in 4.1 we have(a) lhs q ( C ⊗
1; 1 ⊗ C ) = P i,k lhs q ( C ( i ) ⊗
1; 1 ⊗ C ( k )) ,(b) lhs q ( C ( i ) ⊗
1; 1 ⊗ C ( k )) = i > k (Ω − i − ⊗ Ω i − ⊗ ⊗ C ( k )) i = k, ([Ω − i − ⊗ Ω i − − Ω − i ⊗ Ω i ] ⊗ ⊗ C ( k )) i < k.P roof. It will be convenient to write C = P u c u ⊗ d u for (a). (This is not quite consistentwith our previous notation, but should not lead to confusion). It then follows that(4.5) lhs q ( C ⊗
1; 1 ⊗ C ) = X u,v,w c u c v ⊗ [ d u d v c w + ( q + q − ) d u c w d v + c w d u d v ] ⊗ d w . Let c u = c i,ε and let c v = c j, ˜ ε with i = j . Observe that the claim is proved if for given index w the summand for our given indices u and v cancels with the one with c u = c j, ˜ ε and c v = c i,ε . UALITIES FOR SPIN REPRESENTATIONS 13
Let d u = d i, − ε and d v = d j, − ˜ ε , and c u and c v as at the beginning of this paragraph. It followsfrom 4.3 that c u c v = − q ± ε c v c u , d u d v = − q ± ε d v d u , with matching signs in the exponents. Let us choose the labeling such that c u c v = − q c v c u ,and hence also d u d v = − q d v d u . Using the commutation relations above these two summandsadd up to c u c v ⊗ ( q + q − )[ q − d u d v c w + d u c w d v − q − d v c w d u + q − c w d u d v ] ⊗ d w . Then our claim will follow if we can show that the middle factor M in this tensor product isequal to 0.First observe that d u d v = − q d v d u implies that d u = d i, + and d v = d j, ± or d v = d i, − and d u = d j, ± with i < j . Let us consider the case d u = d i, + = ω i − ψ i . Then one checks that c w d u = − q − d u c w is only possible for c w = c a, − for a < i . As i < j , we then also have c w d v = − q d v c w , which forces M = 0. One similarly shows in the second case with d v = d i, − that c w d v = − q d v c w would imply c w d u = − q d u c w . This completes the proof for claim (a).For part (b), we only need to consider the cases with c u = c i,ε and c v = c i, − ε , as c i,ε = 0.Using 4.3, one checks that c i,ε c i, − ε ⊗ [ d i, − ε d i,ε c j,κ + ( q + q − ) d i, − ε c j,κ d i,ε + c j,κ d i, − ε d i,ε ] ⊗ d j, − κ == ( i > j, (1 − q ε )(Ω − i − ψ εi ψ − εi ⊗ Ω i − ψ − εi ψ εi ⊗ ⊗ c j,κ ⊗ d j, − κ ) i < j. Adding up these quantities for all possible choices of ε and κ , we obtain 0 for i > j . Using(1 − q ) ψ i ψ † i ⊗ ψ † i ψ i + (1 − q − ) ψ † i ψ i ⊗ ψ i ψ † i = 1 ⊗ − ω i ⊗ ω − i , we similarly obtain the claim for i < j . The claim for i = j follows from a direct calculation. Proposition 4.2.
Let C = C ⊗ and C = 1 ⊗ C . Then we have C C + ( q + q − ) C C C + C C = C .P roof. It follows from Lemma 4.1(b) that for fixed j k X i =1 lhs q ( C ( i ) ⊗
1; 1 ⊗ C ( j )) = 1 ⊗ C ( j ) . The claim follows from this and Lemma 4.1(a) for type D . For Lie type B k , we have to add˜ C ( k + 1) to the expression for type D k , see 4.2. Setting c k +1 ,ε = Ω − k f N and d k +1 ,ε = Ω k f N ,one checks that Eq 4.3 and 4.4 also hold if one of the indices is k + 1. One deduces the resultsof Lemma 4.1(a) and, except for i = j = k + 1, also of part (b) from this. Observe that c k +1 ,ε commutes with d k +1 ,ε . One calculates that lhs q ( ˜ C ( k + 1) ⊗
1; 1 ⊗ ˜ C ( k + 1)) = (Ω − k ⊗ Ω k ⊗ ⊗ ˜ C ( k + 1)) . The claim can be deduced from this, using Lemma 4.1, as it was done for type D . First and Second Fundamental Theorem
Preliminaries.
We consider the spinor module S as in Section 1.2 for U = U q so k +1 ,for U = U q so k and for U = U q so k ⋊ Z/
2, where the Z / k in all these cases, and its weights are given by allpossible vectors ω = ( ± , ± , ..., ± ) ∈ R k . As all weights have multiplicity 1, tensoring anirreducible highest weight module V λ by S is given by(5.1) V λ ⊗ S ∼ = ⊕ µ V µ , where the summation goes over all dominant weights µ of the form µ = λ + ω . If U = U q so k ⋊ Z/
2, this has to be slightly modified, see [27]. A first fundamental theorem has beenproved for End U ( S ⊗ n ) for this case as well as for U = U q so k +1 in [27], Theorem 3.3. We willreview the method used there, a modification of which will be used also for the missing caseto be proved here. In all of these cases, we have(5.2) S ⊗ n = S ⊗ nold ⊕ S ⊗ nnew , where S ⊗ nold is a direct sum of irreducible modules V λ which have already appeared in smallertensor powers of S (for which λ < n/
2) and where S ⊗ nnew is a direct sum of irreducible modules V λ which have not appeared before (for which λ = n/ C to be a C -linear rigid braided tensor category,see e.g. [23] for precise definitions. The only property we will need is the fact that for everyobject X in C there exists an object ¯ X and morphisms ι X : → X ⊗ ¯ X , ˜ d X : X ⊗ ¯ X → such that T r ( a ) = ˜ d X ( a ⊗ ι X for any a ∈ End( X ). It is well-known that this holds for C =Rep U q so N . Proposition 5.1.
Let V be a self-dual rigid object in the braided spherical tensor category C and let p = ι V ˜ d V . Let E n = End C ( V ⊗ n ) . Then End( V ⊗ n ) old = ( E n − ⊗ p n − ( E n − ⊗ . P roof.
The p in the statement can be normalized to be a projection. It then satisfiesexactly the same properties as the p in the proof of Proposition 4.10 in [26] for k = 2, even if V is not necessarily simple. The claim follows from this.5.2. First fundamental theorem.
We study S ⊗ nnew by induction on the rank of U . It followsfrom the relations that the subalgebra of U q so N , generated by E i , F i and K ± i , 2 ≤ i ≤ k isisomorphic to U q so N − , for N = 2 k or N = 2 k + 1. It will be convenient to denote U q so N or U q so N ⋊ Z / U ( N ), the spinor module S of U ( N ) by S ( N ), and the centralizer algebraEnd U ( S ⊗ n ) by E ( N ) n . We have the following well-known facts which are easy to check:(a) We have the isomorphism of U ( N −
2) modules S ( N ) ∼ = S ( N − ⊕ S ( N − , where S ( N − is spanned by the weight vectors with weights ( , ω ′ ), with ω ′ a weight of S ( N − v ∈ S ( N − ⊗ n ⊂ S ( N ) ⊗ n . Then v is a highest weight vector for U ( N −
2) ofweight λ ′ if and only if it is a highest weight vector for U ( N ) of weight ( n , λ ′ ). UALITIES FOR SPIN REPRESENTATIONS 15
Theorem 5.2. (First Fundamental Theorem) The algebra
End U ( S ⊗ n ) is generated by theelements a i , i = 1 , , ... n − , with a ∈ End U ( S ⊗ ) . P roof.
We only need to consider the case U = U q so k . The statement was proved in [27],Theorem 3.3 for the other cases.For getting the induction on the rank k going, we define Spin (2) to be the Z / SO (2). Its irreducible representations are labeled by half integers. In this case S is the directsum of two 1-dimensional representations with weights ± , which is obviously self-dual. Let f ∈ End( S ) act via ± ± . It follows from the tensor productrules that f ⊗ n − again acts via ± ± n .The tensor product rules also show that this is S ⊗ nnew . The claim now follows for SO (2) fromthis and Proposition 5.1 by induction on n .We similarly prove the claim for N = 2 k > n . For n = 1, S is the directsum of two irreducible modules S ± , on which the endomorphism f acts via ±
1. The claimfollows for End U ( S ⊗ n ) old from Proposition 5.1 by induction assumption on n − U ( N ) ( S ⊗ n ) onto End U ( N − ( S ⊗ n ), given by restriction from S ⊗ n to S ⊗ n . Indeed, thesimple module of End U ( N ) ( S ⊗ n ) consisting of highest weight vectors of weight ( n , λ ′ ) coincideswith the simple module of End U ( N − ( S ⊗ n ) consisting of highest weight vectors of weight λ ′ .Its kernel is End U ( N ) ( S ⊗ nold ). By induction assumption, End U ( N − ( S ⊗ n ) ∼ = End U ( N ) ( S ⊗ nnew ) isgenerated by End U ( N − ( S ⊗ ). But the latter is just the restriction of End U ( N ) ( S ⊗ ) to S ⊗ .Hence End U ( N ) ( S ⊗ ) also generates End U ( N ) ( S ⊗ nnew ). The proof for N odd goes the same way,with the case N = 3 already proved in Proposition 2.4.5.3. Second fundamental theorem.
Recall that the algebra U ′ q o n was defined by addingan additional generator F to U ′ q so n with the relations F = 1, F B = − B F and F B i = B i F for i >
1. Also recall that the finite dimensional representations of the group O ( n ) are labeledby all Young diagrams whose first two columns contain at most n boxes, see [29] or also e.g.[27]. Theorem 5.3. (Second Fundamental Theorem) (a) If N is odd and U = U q so N , we havea surjective map U ′− q so n → End U ( S ⊗ n ) defined by B i C i , where C i is defined as inSection 2.4, using the map C defined in 3.9. Its image is the direct sum of all non-classicalrepresentations of U ′− q so n with highest weights µ such that µ ≤ N/ and in which all B i shave eigenvalues contained in { ( − j ( q N − j − q j − N ) / ( q − q − ) , ≤ j < N/ } .(b) If N is even and U = U q so N ⋊ Z/ , we have a surjective map U ′− q so n → End U ( S ⊗ n ) defined by B i C i , using the map C defined in 3.6. Its image is the direct sum of all classicalrepresentations of U ′ q so n with highest weights µ such that µ ≤ N/ , with µ i ∈ Z .(c) If N is even and U = U q so N , we have a surjective map U ′− q o n → End U ( S ⊗ n ) definedby B i C i and by F f ⊗ n − . Its image is the direct sum of irreducible representationsof U ′ q o n which specialize to representations of O ( N ) labeled by Young diagrams µ whose firsttwo columns contain ≤ n boxes and such that µ ≤ N/ for q = 1 . P roof.
It was shown in Proposition 4.2 that B i C i defines a representation of U ′− q so n .Substituting q by q for N even, we get the representations as stated. The eigenvalues of C were computed in Lemma 1.2 for q = 1. For general q , the eigenvalues can be computed by thesame method (see [27], Lemma 4.2 and Proposition 4.3) or by a deformation argument, usingthe explicit classification of irreducible representations of U ′ q so . From this also follows that C , respectively C and f ⊗ U = U q so k generate End U ( S ⊗ ). The surjectivity statementnow follows from Theorem 5.2. The explicit combinatorics will be studied in the next section.5.4. Combinatorics and representations of U ′ q so n and U ′ q o n . Our duality result impliesthat we can associate to each irreducible representation V λ of, say, U = U q so N ⋊ Z / S ⊗ n an irreducible representation W λ c of U ′ q so n such that the multiplicity of V λ in S ⊗ n is given by the dimension of W λ c . The fact that such multiplicities are given bydimension formulas (modified for N odd) has been known for a long time, see e.g. [3]. TheYoung diagram (or weight) λ c can be obtained from the Young diagram λ as its complementin a rectangle with side lengths N/ n/
2, reflected at the line y = x , to get it intousual Young diagram position. As our weights also involve half integers, and there are someadditional subtleties for N even, we will spell this out in more detail in this section, eventhough it is not new (see e.g. [3], [10] or [27]).Let us briefly describe the irreducible representations of P in ( N ). If N is even, the ir-reducible representations of O ( N ) are labeled by Young diagrams, whose first two columnscontain at most N boxes. The irreducible representations of P in ( N ) which do not factor over O ( N ) are given by N/ λ i ) with λ ≥ λ ≥ ... ≥ λ N/ > λ i ≡ / Z .We also view them as Young diagrams with an additional half box of width 1/2 and height1 at the end. We can now give a precse description of the diagram λ c which has alreadyappeared before, e.g. in [3]. Definition 5.4.
Let R be a rectangle of height N/ n/ N = 2 k and let λ be a Young diagram labeling an irreducible representation of P in ( N ) appearing in S ⊗ n . Then its n -complement is the dominant integral weight λ c of so n such that λ cj is equal to the number of boxes in the j -th coulmn from the right in R \ λ . If λ does not fit into R , i.e. if the first column of λ contains more than N/ λ cn/ = n/ − λ ′ <
0, where λ ′ is the number of boxes in the first column of λ .(b) If N = 2 k is even and λ a dominant integral weight labeling an irreducible so N moduleappearing in S ⊗ n , we associate to it the Young diagram λ c whose j -th column contains n/ − λ N/ − j boxes.(c) For N odd, and λ an integral dominant weight labeling a representation of so N appearingin S ⊗ n , we define the integral dominant weight λ c labeling a non-classical representation of U ′ q so n by λ cj to be equal to the number of boxes in the j -th column from the right of R \ λ . UALITIES FOR SPIN REPRESENTATIONS 17
Proposition 5.5. (see e.g. [3] , [10] ) Let S , U , U ′ q so n etc be as at the beginning of this section.Then the module S ⊗ n has a multiplicity one decomposition S ⊗ n ∼ = M λ V λ ⊗ W λ c , where λ ranges over the equivalence classes of irreducible U modules V λ which appear in S ⊗ n ,and λ c is the label of the simple U ′ q so n resp U ′ q o n -module W λ c associated to it in Def 5.4. P roof.
We give a proof for the case N odd by induction on n . We define λ r = λ r ( n ) by λ ri = n/ − λ i . This describes the number of boxes in the i -th row of the (unreflected)complement of λ in R . Let W λ d be the U ′ q so n module associated to λ (e.g. we could take themodule of highest weight vectors in S ⊗ n with weight λ ). It follows from the tensor productrules and induction assumption that W λ d ∼ = M W µ c ( n − as U ′ q so n − -modules, where the summation goes over all highest weights µ with V µ ⊂ S ⊗ n − such that V λ ⊂ V µ ⊗ S . This implies that λ = µ + ω for some weight ω of S . As ω = ε − P j ǫ i j for some subset { i j } ⊂ { , , ..., k } , it follows that µ r ( n − = ( n − ε − µ = nε − λ − X j ǫ i j = λ r ( n ) − X j ǫ i j , i.e. the rows of µ r ( n − and λ r ( n ) differ by at most 1. Recall that λ c ( n ) is obtained from λ r ( n ) by reflecting the latter at the line y = x , i.e. its j -th column contains λ r ( n )( N +1) / − j boxes. Itis probably easiest seen geometrically that the condition 0 ≤ λ ri − µ ri ≤
1, 1 ≤ i ≤ N , isequivalent to the branching conditions 2.1 and 2.2 for µ c and λ c . Hence the simple U ′ q so n module W λ d coincides with the simple U ′ q so n module W λ c as a U ′ q so n − -module. It is easyto see from 2.1 and 2.2 that non-isomorphic simple U ′ q so n -modules are still non-isomorphic ifviewed as U ′ q so n − modules. Hence W λ d ∼ = W λ c .The other cases can be shown similarly. Another method would be to calculate the characterof O ( N ) × SO ( n ) ⊂ O ( N n ) on the spin module of O ( N n ). This was essentially done in [10],with some minor errors. Hopefully the statement of the pairings is correct in this paper.
Remark . It was already observed in [3] that for N odd the multiplicity of V λ in S ⊗ k +1 co-incides with the dimension of the irreducible Sp (2 k ) module W λ c − ε , where ε = ( , , ..., ) ∈ R k . No such classical interpretation seems to be available for multiplicities in even tensorpowers of S . Corollary 5.7.
Let q not be a root of unity. Then every finite-dimensional irreduciblerepresentation of U ′ q so n appears in some tensor power S ⊗ n for some U = U q so N or U = U q so N ⋊ Z/ . Moreover, every irreducible finite dimensional representation of O ( n ) extendsto an irreducible representation of U ′ q o n . P roof
This was shown for classical representations with integer highest weights in [27] andalso in this paper. It can be shown the same way for classical representations with half integerhighest weights for N odd, using the bigger module ˜ S and [27] Lemma 4.2 and Proposition4.3. Existence of non-classical representations and of representations of U ′ q o n as stated followsfrom Proposition 5.5.5.5. Results for more general rings.
It is well-known that the Drinfeld-Jimbo quantumgroups can be defined over the ring Z [ q, q − ], see [18]. It is not hard to check that the sameis true for the subalgebra U ′ q so n ⊂ U q sl n . Also observe that the representations of U ′ q so n inTheorem 5.3 can be defined over the ring Z [ q, q − ] in type D (for N even) and over the ring Z [ q, q − , [2] − ] for N odd. Indeed, one checks easily that the matrix coefficients of the maps C in 3.6 and 3.9 with respect to the basis x ( m ) are in the rings as claimed. We can deducethe following result from this and Theorem 5.3: Proposition 5.8.
The representations of the algebras in Theorem 5.3 on S ⊗ n are well-definedalso over the ring Z [ q, q − ] for N even. For N odd, they are well-defined over the ring Z [ q, q − , [2] − ] . The statements of Theorem 5.3 also hold in these cases, except possibly thesurjectivity statements. Results for fusion categories.
For q = ± U q g is not semisimple. But one can define an important semisimplequotient tensor category F of the subcategory of tilting modules of Rep ( U q so N ), see [1]. Itonly has finitely many simple objects up to isomorphism. We only give some very basic factsabout it here, see e.g. [2] for more details.We assume q to be a primitive 2 ℓ -th (for N even) respectively a primitive 4 ℓ -th root ofunity (for N odd), with ℓ ≥ N in both cases. The simple objects of F are labeled by the ℓ -admissible integral dominant weights λ of so N , which are defined by λ + λ + N − ≤ ℓ. Theses categories also appear as level ℓ + 2 − N representations of loop groups connected with SO ( N ). The image of the spinor representation S is again simple and nonzero in the quotient F and will be denoted by the same letter. Tensoring a simpe object V λ by S is given by thesame rule 5.1, where we remove all representations which are not labeled by an ℓ -admissibleweight. With a little care, all the arguments we used above will also go through for fusioncategories related to U q so N . Theorem 5.9.
The statements of Theorem 5.2 and 5.3 also hold in the context of fusioncategories F as defined in this section. In particular, End F ( S ⊗ n ) is given by a representationof U ′ q o n for N even, and by a representation of U ′− q so n for N odd. P roof.
It is known that S is a tilting module, and that S ⊗ n can be written as a direct sumof indecomposable tilting modules T λ with highest weight λ . The quotient of S ⊗ n which liesin F is isomorphic to the summand consisting of those T λ for which λ is ℓ -admissible. In thiscase T λ ∼ = V λ is simple. These basic facts allow us to proceed by induction on n and on N UALITIES FOR SPIN REPRESENTATIONS 19 as in the proofs for Theorem 5.2 and 5.3. As there, one shows that End( S ⊗ ) is generatedby C (for N odd) and by C and F (for N even), using the fact that the eigenvalues of C for the different subrepresentations of S ⊗ are distinct as in the generic case (which wouldnot necessarily be true if q had a smaller degree). Our conditions also ensure that the q -dimensions of all simple objects in F are nonzero. This is all needed to show that also thestatement about S ⊗ nold in the proof of Theorem 5.2 holds in this context. We can now use thesame argument as in the proof of Theorem 5.2 to show that End F ( S ⊗ nnew ) will be isomorphicto a quotient of End F ( U ( N − ( S ( N − ⊗ n ), where F ( U ( N − U q so N − at the same root of unity.6. Related results and applications
Connection with results in [27] and Hasegawa duality.
This paper is closely re-lated to the paper [27] which in turn was inspired by the paper [10] by Hasegawa. The keyobservation there was (as far as this paper is concerned) the fact that the commuting actionsof O ( N ) and SO ( n ) on C N ⊗ C n extend to commuting actions of the corresponding spin groupson the Clifford algebra Cl ( N n ) ∼ = Cl ( N ) ⊗ n (isomorphism of vector spaces). This suggestedcommuting actions of these groups and the corresponding quantum groups on the n -th tensorpower of the spin representation of O ( N ). It was indeed shown in [27] for N even that thereexist actions of U = U q so N ⋊ Z / U ′ q so n on S ⊗ n which are each others commutants.As the Clifford algebra is not simple for N odd, no such simple statement could be shownin that case. The best one could do was to prove a duality statement between the action of U q so N and the subalgebra of U ′ q so n generated by the elements B i , 1 ≤ i < n (our q herecorresponds to q in the parametrization in [27] for N odd). This subalgebra does not seem toallow a convenient algebraic description on its own.In the current paper we do not apply Hasegawa’s results at all. Using the operator C , wedirectly show that the centralizers can be described in terms of the coideal algebra U ′− q so n .This does not change much our previous description in terms of U ′ q so n for N even, see Lemma2.5. But for N odd, it gives the desired duality result between actions of U q so N and U ′− q so n ,which now acts via its non-classical representations on S ⊗ n . Because of the latter, the resultseems to be new even in the classical case q = 1. In particular, there does not seem to beany indication of the algebra U ′− q so n in the context of Hasegawa duality or Howe duality (seenext section).6.2. Connections to q -Howe duality. Similarly as for Clifford algebras in Hasegawa du-ality, one obtains commuting actions of groups, say Gl ( N ) and Gl ( n ) on V ( C N ⊗ C n ) ∼ =( V C N ) ⊗ n . For Lie type A , it has been shown by Cautis, Kamnitzer and Morrison [6] thatthere similarly exist commuting actions of the quantum groups U q sl N and U q sl n on ( V C N ) ⊗ n .The situation is more complicated for other Lie types, such as for the commuting actions of so N and so n as well as of sp N and sp n on the exterior algebra V ( C N ⊗ C n ) ∼ = ( V C N ) ⊗ n . Itwas shown by Sartori and Tubbenhauer [21] that one obtains commuting actions of U q so N and U ′ q so n and of U q sp N and U ′ q sp n , where U ′ q so n is as in this paper, and U ′ q sp n ⊂ U q sl n similarly is a coideal subalgebra. In the orthogonal case, at least for N even, this result wouldalso follow from the results in [27], as S ⊗ ∼ = V C N as P in ( N )-modules. The methods in [21]are completely different from the ones in [27] and in this paper.6.3. Connections to the q -Clifford algebra in [7] . A q -Clifford algebra has also beendefined by Ding and Frenkel in [7]. Setting ˆ ψ k +1 − i = Ω i − ψ i and ˆ ψ † k +1 − i = Ω i − ψ † i , one canshow (see [7], Proposition 5.3.1) that the elements ˆ ψ i and ˆ ψ † i satisfy the relations of their q -Clifford algebra. Similarly, if one defines ˆ ψ i, − and ˆ ψ † i, − as before with Ω i − replaced byΩ − i − , we obtain the relations of their Clifford algebra with q replaced by q − . In particularthe element C would then have the somewhat more appealing form C = k X i =1 ˆ ψ i ⊗ ˆ ψ † i, − + ˆ ψ † i ⊗ ˆ ψ i, − in the even-dimensional case. The elements ˆ ψ i , ˆ ψ i, − etc already appeared in Section 4 as c i, ± and d i, ± . In particular, one can see the parameter q appear explicitly in the relations there,see e.g. 4.3.6.4. Existence of representations of U ′ q so n and U ′ q o n . The irreducible representations of U ′ q so n for q not a root of unity were constructed by Klimyk and his coauthors, and in specialcases also by different groups, see [12] and the references there. Another approach was givenby the author in [27]. In all of these cases, the constructions of the representations weresomewhat involved. Our duality result provides yet another construction of all irreduciblerepresentations of U ′ q so n and of many irreducible representations of U ′ q o n if q is not a root ofunity, see corollary 5.7. We also obtain some new nontrivial representations at roots of unityin the fusion cateogry setting. In our set-up, it suffices to find the linear map C . After thatthe representations are obtained in a comparatively painless way using our duality result.6.5. New vertex models.
Results in this paper were used by D. Gepner in the recentpreprint [9] to construct new vertex models in connection with spin representations and tostudy their algebraic properties.
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