On a Certain Subalgebra of U q (ŝl 2) Related to the Degenerate q-onsager Algebra

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1 Symmetry, Integrablty and Geometry: Methods and Applcatons SIGMA 11 (2015), 007, 13 pages On a Certan Subalgebra of U q (ŝl 2) Related to the Degenerate q-onsager Algebra Tomoya HATTAI and Tatsuro ITO Ida Hghschool, 1-1, Nonoe, Suzu, Ishkawa , Japan E-mal: tmyhtt@m2.shkawa-c.ed.jp School of Mathematcal Scences, Anhu Unversty, 111 Julong Road, Hefe , Chna E-mal: tto@staff.kanazawa-u.ac.jp Receved September 30, 2014, n fnal form January 15, 2015; Publshed onlne January 19, Abstract. In [Kyushu J. Math. 64 (2010), ], t s dscussed that a certan subalgebra of the quantum affne algebra U q (ŝl 2) controls the second knd TD-algebra of type I (the degenerate q-onsager algebra). The subalgebra, whch we denote by U q(ŝl 2), s generated by e + 0, e± 1, k±1 ( = 0, 1) wth e 0 mssng from the Chevalley generators e±, k±1 ( = 0, 1) of U q (ŝl 2). In ths paper, we determne the fnte-dmensonal rreducble representatons of U q(ŝl 2). Intertwners are also determned. Key words: degenerate q-onsager algebra; quantum affne algebra; TD-algebra; augmented TD-algebra; TD-par 2010 Mathematcs Subject Classfcaton: 17B37; 05E30 1 Introducton Throughout ths paper, the ground feld s C and q stands for a nonzero scalar that s not a root of unty. The symbols ε, ε stand for an nteger chosen from {0, 1}. Let A q = A (ε,ε ) q denote the assocatve algebra wth 1 generated by z, z subject to the defnng relatons [4] (TD) {[ z, [z, [z, z ] ( ] q ] q 1 = ε q 2 q 2) 2 [z, z ], [ z, [z, [z ], z] q ] q 1 = ε ( q 2 q 2) 2 [z, z], where [X, Y ] = XY Y X, [X, Y ] q = qxy q 1 Y X. Ths paper deals wth a subalgebra of the quantum affne algebra U q (ŝl 2) that s closely related to A q n the case of (ε, ε ) = (1, 0). If (ε, ε ) = (0, 0), A q s somorphc to the postve part of U q (ŝl 2). If (ε, ε ) = (1, 1), A q s called the q-onsager algebra. If (ε, ε ) = (1, 0), A q may well be called the degenerate q-onsager algebra. The algebra A q arses n the course of the classfcaton of TD-pars of type I, whch s a crtcally mportant step n the study of representatons of Terwllger algebras for P - and Q-polynomal assocaton schemes [3]. For ths reason, A q s called the TD-algebra of type I. Precsely speakng, the TD-algebra of type I s standardzed to be the algebra A q, where q s the man parameter for TD-pars of type I; so q 2 ±1 and q s allowed to be a root of unty. In our case where we assume q s not a root of unty, the classfcaton of the TD-pars of type I s equvalent to determnng the fnte-dmensonal rreducble representatons ρ : A q End(V ) wth the property that ρ(z), ρ(z ) are both dagonalzable. Such rreducble representatons Ths paper s a contrbuton to the Specal Issue on New Drectons n Le Theory. avalable at The full collecton s

2 2 T. Hatta and T. Ito of A q are determned n [4] va embeddngs of A q nto the augmented TD-algebra T q. (In the case of (ε, ε ) = (1, 1), the dagonalzablty condton of ρ(z), ρ(z ) can be dropped, because t turns out that ths condton always holds for every fnte-dmensonal rreducble representaton ρ of the q-onsager algebra A q.) T q s easer than A q to study representatons for, and each fnte-dmensonal rreducble representaton ρ : A q End(V ) wth ρ(z), ρ(z ) dagonalzable can be extended to a fnte-dmensonal rreducble representaton of T q va a certan embeddng of A q nto T q. The augmented TD-algebra T q = T (ε,ε ) q s the assocatve algebra wth 1 generated by x, y, k ±1 subject to the defnng relatons kk 1 = k 1 k = 1, (TD) 0 kxk 1 = q 2 x, (1) kyk 1 = q 2 y, and (TD) 1 {[ ] ( x, [x, [x, y]q ] q 1 = δ ε x 2 k 2 εk 2 x 2), [ ] ( y, [y, [y, x]q ] q 1 = δ ε k 2 y 2 + εy 2 k 2), (2) where δ = (q q 1 )(q 2 q 2 )(q 3 q 3 )q 4. The fnte-dmensonal rreducble representatons of T q are determned n [4] va embeddngs of T q nto the U q (sl 2 )-loop algebra U q (L(sl 2 )). Let e ±, k±1 ( = 0, 1) be the Chevalley generators of U q (L(sl 2 )). So the defnng relatons of U q (L(sl 2 )) are k 0 k 1 = k 1 k 0 = 1, k k 1 = k 1 k = 1, k e ± k 1 = q ±2 e ±, k e ± j k 1 = q 2 e ± j, j, [e+, e ] = k k 1 q q 1, [e+, e j ] = 0, j, [ e ±, [e ±, [e±, e± j ] q] q 1] = 0, j. (3) Note that f k 0 k 1 = k 1 k 0 = 1 s replaced by k 0 k 1 = k 1 k 0 n (3), we have the quantum affne algebra U q (ŝl 2): U q (L(sl 2 )) s the quotent algebra of U q (ŝl 2) by the two-sded deal generated by k 0 k 1 1. For a nonzero scalar s, defne the elements x(s), y(s), k(s) of U q (L(sl 2 )) by x(s) = q 1( q q 1) 2( se εs 1 e 1 k 1), y(s) = ε se 0 k 0 + s 1 e + 1, k(s) = sk 0. (4) Then the mappng ϕ s : T q U q (L(sl 2 )), x, y, k x(s), y(s), k(s), (5) gves an njectve algebra homomorphsm. If (ε, ε ) = (0, 0), the mage ϕ s (T q ) concdes wth the Borel subalgebra generated by e +, k±1 ( = 0, 1). If (ε, ε ) = (1, 0), the mage ϕ s (T q ) s properly contaned n the subalgebra generated by e + 0, e± 1, k±1 ( = 0, 1) wth e 0 mssng from the generators; we denote ths subalgebra by U q(l(sl 2 )). Through the natural homomorphsm U q (ŝl 2) U q (L(sl 2 )), pull back the subalgebra U q(l(sl 2 )) and denote the pre-mage by U q(ŝl 2): U q(ŝl 2) = e + 0, e± 1, k±1 = 0, 1 U q (ŝl 2). In [4], t s shown that n the case of (ε, ε ) = (1, 0), all the fnte-dmensonal rreducble representatons of T q are produced by tensor products of evaluaton modules for U q(l(sl 2 ))

3 On a Certan Subalgebra of U q (ŝl 2) Related to the Degenerate q-onsager Algebra 3 va the embeddng ϕ s of T q nto U q(l(sl 2 )). Usng ths fact and the Drnfel d polynomals, we show n ths paper that there are no other fnte-dmensonal rreducble representatons of U q(l(sl 2 )) and hence of U q(ŝl 2) than those afforded by tensor products of evaluaton modules, f we apply sutable automorphsms of U q(l(sl 2 )), U q(ŝl 2) to adjust the types of the representatons to be (1, 1). Here we note that the evaluaton parameters are allowed to be zero for U q(l(sl 2 )), U q(ŝl 2). Detals wll be dscussed n Sectons 2 and 3, where the somorphsm classes of fnte-dmensonal rreducble representatons of U q(ŝl 2) are also determned. In Secton 4, ntertwners wll be determned for fnte-dmensonal rreducble U q(ŝl 2)- modules. In our approach, Drnfel d polynomals are the key tool for the classfcaton of fntedmensonal rreducble representatons of U q (ŝl 2), U q(ŝl 2), although they are not the man subject of ths paper. They are defned n [4], and the pont s that they are drectly attached to T q -modules, not to U q (ŝl 2)- or U q(ŝl 2)-modules. (In the case of (ε, ε ) = (0, 0), they turn out to concde wth the orgnal ones up to the recprocal of the varable.) So n our approach to the case of (ε, ε ) = (0, 0), fnte-dmensonal rreducble representatons are naturally classfed frstly for the Borel subalgebra of U q (ŝl 2) and then for U q (ŝl 2) tself. Ths wll be brefly demonstrated n Secton 3 as a warm-up for the case of (ε, ε ) = (1, 0), thus gvng another proof to the classcal classfcaton theorem of Char Pressley [2] and to the man theorems (Theorems 1.16 and 1.17) of [1]. We now revew Drnfel d polynomals for T q -modules [4, p. 119]. Let V be a fnte-dmensonal T q -module. We assume the followng propertes for V : (D) 0 : k s dagonalzable on V wth V = d U, k U = sq 2 d, 0 d, for some nonzero constant s; (D) 1 : dm U 0 = 1. =0 By the relatons (TD) 0 : kk 1 = k 1 k = 1, kxk 1 = q 2 x, kyk 1 = q 2 y, t holds that xu U +1, yu U 1 (0 d), where U 1 = U d+1 = 0. So the one-dmensonal subspace U 0 s nvarant under y x and we have the sequence {σ } =0 of egenvalues σ of y x on U 0 : σ = y x U0. Notce that σ 0 = 1 and σ = 0, d + 1. The Drnfel d polynomal P V (λ) of the T q -module V s defned by P V (λ) = d ( 1) σ (q q 1 ) 2 ([]!) 2 =0 d ( λ εs 2 q 2(d j) ε s 2 q 2(d j)), j=+1 where [] = [] q = (q q )/(q q 1 ) and []! = [1][2] [] wth the understandng of [0]! = 1. Snce σ 0 = 1, P V (λ) s a monc polynomal of degree d. If V s an rreducble T q -module, t s known that V n fact satsfes the propertes (D) 0, (D) 1 [4, Lemma 1.2, Theorem 1.8], and these propertes provde a rather smple short proof for the njectve part of [4, Theorem 1.9],.e., for the fact that the somorphsm class of the rreducble T q -module V s determned by the tro ({σ } =0, s, d). If V s a tensor product of evaluaton modules for U q (L(sl 2 )) n the case of (ε, ε ) = (1, 1), (0, 0) or for U q(l(sl 2 )) n the case of (ε, ε ) = (1, 0), we regard V as a T q -module va the embeddng ϕ s of (5). Then t s apparent that the T q -module V satsfes the propertes (D) 0, (D) 1. Moreover t s known that a product formula holds for the Drnfel d polynomal P V (λ) and t turns out that P V (λ) does not depend on the parameter s of the embeddng ϕ s [4, Theorem 5.2]. The surjectve part of [4, Theorem 1.9] follows from the structure of the zeros of the Drnfel d polynomal for such a tensor product of evaluaton modules regarded as a T q -module va the embeddng ϕ s.

4 4 T. Hatta and T. Ito 2 Fnte-dmensonal rreducble representatons of U q (ŝl 2) The subalgebra U q(ŝl 2) of the quantum affne algebra U q (ŝl 2) s generated by e + 0, e± 1, k±1 ( = 0, 1), e 0 mssng from the generators, and has by the trangular decomposton of U q(ŝl 2) the defnng relatons k 0 k 1 = k 1 k 0, k k 1 = k 1 k = 1, k 0 e + 0 k 1 0 = q 2 e + 0, k 1e ± 1 k 1 1 = q ±2 e ± 1, k 1 e + 0 k 1 1 = q 2 e + 0, k 0e ± 1 k 1 0 = q 2 e ± 1, [e+ 1, e 1 ] = k 1 k1 1 q q 1, (6) [e + 0, e 1 ] = 0, [ e +, [e +, [e+, e+ j ] ] q] q 1 = 0, j. Note that f k 0 k 1 = k 1 k 0 s replaced by k 0 k 1 = k 1 k 0 = 1 n (6), we have the defnng relatons for U q(l(sl 2 )). Let V be a fnte-dmensonal rreducble U q(ŝl 2)-module. Let us frst observe that the U q(ŝl 2)- module V s obtaned from a U q(l(sl 2 ))-module by applyng an automorphsm of U q(ŝl 2) as follows. Snce the element k 0 k 1 belongs to the centre of U q(ŝl 2), k 0 k 1 acts on V as a scalar s by Schur s lemma. Snce k 0 k 1 s nvertble, the scalar s s nonzero: k 0 k 1 V = s C. Observe that U q(ŝl 2) admts an automorphsm that sends k 0 to s 1 k 0 and preserves k 1. Hence we may assume k 0 k 1 V = 1. Then we can regard V as a U q(l(sl 2 ))-module. Let V be a fnte-dmensonal rreducble U q(l(sl 2 ))-module. For a scalar θ, set V (θ) = {v V k 0 v = θv}. So f V (θ) 0, θ s an egenvalue of k 0 and V (θ) s the correspondng egenspace of k 0. For an egenvalue θ and an egenvector v V (θ), t holds that e + 0 v V (q2 θ) by the relaton k 0 e + 0 = q2 e + 0 k 0 and e ± 1 v V (q 2 θ) by k 0 e ± 1 = q 2 e ± 1 k 0. We have e + 0 V (θ) V ( q 2 θ ), e ± 1 V (θ) V ( q 2 θ ). (7) If dm V = 1, then e + 0 V = 0, e± 1 V = 0 by (7) and k 0 V = ±1 by [e + 1, e 1 ] = (k 1 k1 1 )/(q q 1 ) = (k0 1 k 0)/(q q 1 ). Such a U q(l(sl 2 ))-module V s sad to be trval. Assume dm V 2. Choose an egenvalue θ of k 0 on V. Then V (q ±2 θ) s nvarant under the actons of the generators Z by (7), and so we have V = Z V (q ±2 θ) by the rreducblty of the U q(l(sl 2 ))-module V. Snce V s fnte-dmensonal, there exsts a postve nteger d and a nonzero scalar s 0 such that the egenspace decomposton of k 0 s V = d V ( s 0 q 2 d). =0 (8) We want to show that s 0 = ±1 holds n (8). Consder the subalgebra of U q(l(sl 2 )) generated by e ± 1, k±1 1 and denote t by U : U = e ± 1, k 1 ±1. Regard V as a U-module. Snce U s somorphc to the quantum algebra U q(sl 2 ), V s a drect sum of rreducble U-modules, and for each rreducble U-submodule W of V, the egenvalues of k 1 = k0 1 on W are ether {q 2 l 0 l} or { q 2 l 0 l} for some nonnegatve nteger l. Ths mples that () s 0 = ±q m for some m Z and () f θ s an egenvalue of k 0, so s θ 1. It follows from () that V = d =0 V (±q 2 d+m ), and so by (), we obtan m = 0,.e., s 0 = ±1. Observe that U q(l(sl 2 )) admts an automorphsm that sends k to k ( = 0, 1) and e + 1 to e + 1. Hence we may assume s 0 = 1 n (8). Note that n ths case, k 1 has the egenvalues {s 1 q 2 d 0 d} wth s 1 = 1. Such an rreducble module or the rreducble representaton

5 On a Certan Subalgebra of U q (ŝl 2) Related to the Degenerate q-onsager Algebra 5 afforded by such s sad to be of type (1, 1), ndcatng (s 0, s 1 ) = (1, 1). We conclude that the determnaton of fnte-dmensonal rreducble representatons for U q(ŝl 2) s, va automorphsms, reduced to that of type (1, 1) for U q(l(sl 2 )). In the rest of ths secton, we shall ntroduce evaluaton modules for U q(l(sl 2 )) and state our man theorem that every fnte-dmensonal rreducble representaton of type (1, 1) of U q(l(sl 2 )) s afforded by a tensor product of them. For a C and l Z 0, let V (l, a) denote the (l + 1)- dmensonal vector space wth a bass v 0, v 1,..., v l. Usng (6), t can be routnely verfed that U q(l(sl 2 )) acts on V (l, a) by k 0 v = q 2 l v, k 1 v = q l 2 v, e + 0 v = aq[ + 1]v +1, e + 1 v = [l + 1]v 1, e 1 v = [ + 1]v +1, (9) where v 1 = v l+1 = 0 and [t] = [t] q = (q t q t )/(q q 1 ). Ths U q(l(sl 2 ))-module V (l, a) s rreducble and called an evaluaton module. The bass v 0, v 1,..., v l of the U q(l(sl 2 ))-module V (l, a) s called a standard bass. The vector v 0 s called the hghest weght vector. Note that the evaluaton parameter a s allowed to be zero. Also note that f l = 0, V (l, a) s the trval module. We denote the evaluaton module V (l, 0) by V (l), allowng l = 0, and use the notaton V (l, a) only for an evaluaton module wth a 0 and l 1. The U q (sl 2 )-loop algebra U q (L(sl 2 )) has the coproduct : U q (L(sl 2 )) U q (L(sl 2 )) U q (L(sl 2 )) defned by (k ±1 ) = k ±1 k ±1, (e + ) = k e + + e + 1, (e k ) = k e k + e k 1. (10) The subalgebra U q(l(sl 2 )) s closed under. Thus gven a set of evaluaton modules V (l), V (l, a ) (1 n) for U q(l(sl 2 )), the tensor product V (l) V (l 1, a 1 ) V (l n, a n ) (11) becomes a U q(l(sl 2 ))-module va. Note that by the coassocatvty of, the way of puttng parentheses n the tensor product of (11) does not affect the somorphsm class as a U q(l(sl 2 ))- module. Also note that f l = 0 n (11), then V (0) s the trval module and the tensor product of (11) s somorphc to V (l 1, a 1 ) V (l n, a n ) as U q(l(sl 2 ))-modules. Fnally we allow n = 0, n whch case we understand that the tensor product of (11) means V (l). Wth the evaluaton module V (l, a), we assocate the set S(l, a) of scalars aq l+1, aq l+3,..., aq l 1 : S(l, a) = { aq 2 l+1 0 l 1 }. The set S(l, a) s called a q-strng of length l. Two q-strngs S(l, a), S(l, a ) are sad to be n general poston f ether () the unon S(l, a) S(l, a ) s not a q-strng, or () one of S(l, a), S(l, a ) ncludes the other. Below s the man theorem of ths paper. It classfes the somorphsm classes of the fntedmensonal rreducble U q(l(sl 2 ))-modules of type (1, 1). Theorem 1. The followng (), (), (), (v) hold. () A tensor product V = V (l) V (l 1, a 1 ) V (l n, a n ) of evaluaton modules s rreducble as a U q(l(sl 2 ))-module f and only f S(l, a ), S(l j, a j ) are n general poston for all, j {1, 2,..., n}. In ths case, V s of type (1, 1).

6 6 T. Hatta and T. Ito () Consder two tensor products V = V (l) V (l 1, a 1 ) V (l n, a n ), V = V (l ) V (l 1, a 1 ) V (l m, a m) of evaluaton modules and assume that they are both rreducble as a U q(l(sl 2 ))-module. Then V, V are somorphc as U q(l(sl 2 ))-modules f and only f l = l, n = m and (l, a ) = (l, a ) for all, 1 n, wth a sutable reorderng of the evaluaton modules V (l 1, a 1 ),..., V (l n, a n ). () Every non-trval fnte-dmensonal rreducble U q(l(sl 2 ))-module of type (1, 1) s somorphc to some tensor product V (l) V (l 1, a 1 ) V (l n, a n ) of evaluaton modules. (v) If a tensor product V = V (l) V (l 1, a 1 ) V (l n, a n ) of evaluaton modules s rreducble as a U q(l(sl 2 ))-module, then any change of the orderngs of the evaluaton modules V (l), V (l 1, a 1 ),..., V (l n, a n ) for the tensor product does not change the somorphsm class of the U q(l(sl 2 ))-module V. 3 Proof of Theorem 1(), (), () Dscard the evaluaton module V (l) from the statement of Theorem 1 and replace U q(l(sl 2 )) by U q (L(sl 2 )) or by B, where B s the Borel subalgebra of U q (L(sl 2 )) generated by e +, k±1 ( = 0, 1). Then t precsely descrbes the classfcaton of the somorphsm classes of fntedmensonal rreducble modules of type (1, 1) for U q (L(sl 2 )) [2] or for B [1]. There s a one-toone correspondence of fnte-dmensonal rreducble modules of type (1, 1) between U q (L(sl 2 )) and B: every fnte-dmensonal rreducble U q (L(sl 2 ))-module of type (1, 1) s rreducble as a B- module and every fnte-dmensonal rreducble B-module of type (1, 1) s unquely extended to a U q (L(sl 2 ))-module. Ths sort of correspondence of fnte-dmensonal rreducble modules exsts between U q(l(sl 2 )) and T q va the embeddng ϕ s of (5), where T q s the augmented TDalgebra wth (ε, ε ) = (1, 0). Namely, we shall show that () every fnte-dmensonal rreducble U q(l(sl 2 ))-module of type (1, 1) s rreducble as a T q -module va certan embeddng ϕ s of (5), and () every fnte-dmensonal rreducble T q -module s unquely extended to a U q(l(sl 2 ))- module of type (1, 1) va the embeddng ϕ s of (5) wth s unquely determned. Snce fntedmensonal rreducble T q -modules are classfed n [4], ths gves a proof of Theorem 1. Apart from the Drnfel d polynomals, the key to our understandng of the correspondence s the followng two lemmas about U q (sl 2 )-modules. Let U denote the quantum algebra U q (sl 2 ): U s the assocatve algebra wth 1 generated by X ±, K ±1 subject to the defnng relatons KK 1 = K 1 K = 1, KX ± K 1 = q ±2 X ±, [X +, X ] = K K 1. (12) q q 1 Lemma 1 ([4, Lemma 7.5]). Let V be a fnte-dmensonal U-module that has the followng weght-space (K-egenspace) decomposton: V = d U, K U = q 2 d, 0 d. =0 Let W be a subspace of V as a vector space. Assume that W s nvarant under the actons of X + and K: X + W W, KW W. If t holds that dm(w U ) = dm(w U d ), 0 d, then X W W,.e., W s a U-submodule.

7 On a Certan Subalgebra of U q (ŝl 2) Related to the Degenerate q-onsager Algebra 7 Lemma 2. If V s a fnte-dmensonal U-module, the acton of X on V s unquely determned by those of X +, K ±1 on V. Proof. The clam holds f V s rreducble as a U-module. By the sem-smplcty of U, t holds generally. As a warm-up for the proof of Theorem 1, we shall demonstrate how to use these lemmas n the case of the correspondng theorem [2] for U q (L(sl 2 )). We want, and t s enough, to show part () of the theorem for U q (L(sl 2 )) by usng the classfcaton of fnte-dmensonal rreducble B-modules. Ths s because the parts (), (), (v) are well-known n advance of [2], whle the fnte-dmensonal rreducble B-modules are classfed n [4] rather straghtforward by the product formula of Drnfel d polynomals wthout usng the part () n queston. Let V be a fnte-dmensonal rreducble U q (L(sl 2 ))-module of type (1, 1). Then V has the weght-space decomposton V = d U, k 0 U = q 2 d, 0 d. =0 Regard V as a B-module. Let W be a mnmal B-submodule of V. Note that W s rreducble as a B-module. We want to show W = V,.e., V s rreducble as a B-module. Snce the mappng (e + 0 )d 2 : U U d s a bjecton and W U s mapped nto W U d by (e + 0 )d 2, we have dm(w U ) dm(w U d ), 0 [d/2]. Smlarly from the bjecton (e + 1 )d 2 : U d U, we get dm(w U d ) dm(w U ). Thus t holds that dm(w U ) = dm(w U d ), 0 d. Consder the algebra homomorphsm from U to U q (L(sl 2 )) that sends X +, X, K ±1 to e + 0, e 0, k 0 ±1, respectvely. Regard V as a U-module va ths algebra homomorphsm. Then X+ W W, KW W. Snce dm(w U ) = dm(w U d ), 0 d, we have by Lemma 1 that X W W,.e., e 0 W W. Smlarly, Lemma 1 can be appled to the U-module V va the algebra homomorphsm from U to U q (L(sl 2 )) that sends X +, X, K ±1 to e + 1, e 1, k±1 1, respectvely, n whch case the weght-space decomposton of the U-module V s V = d U d, K Ud = q 2 d, 0 d. Consequently, we get X W W,.e., e 1 W W. Thus W s U q (L(sl 2 ))-nvarant and we have W = V by the rreducblty of the U q (L(sl 2 ))-module V. We conclude that every fnte-dmensonal rreducble U q (L(sl 2 ))-module of type (1, 1) s rreducble as a B-module. Now consder the class of fnte-dmensonal rreducble B-modules V, where V runs through all tensor products of evaluaton modules that are rreducble as a U q (L(sl 2 ))-module: V = V (l 1, a 1 ) V (l n, a n ). Then t turns out that the Drnfel d polynomals P V (λ) of the rreducble B-modules V exhaust all that are possble for fnte-dmensonal rreducble B-modules of type (1, 1), as shown n [4, Theorem 5.2] by the product formula P V (λ) = n P V (l,a )(λ), P V (l,a )(λ) = =1 ζ S(l,a ) (λ + ζ). Snce the Drnfel d polynomal P V (λ) determnes the somorphsm class of the B-module V of type (1, 1) [4, the njectvty part of Theorem 1.9 ], there are no other fnte-dmensonal =0

8 8 T. Hatta and T. Ito rreducble B-modules of type (1, 1). Ths means that every fnte-dmensonal rreducble B- module of type (1, 1) comes from some tensor product of evaluaton modules for U q (L(sl 2 )). Let V be a fnte-dmensonal rreducble U q (L(sl 2 ))-module of type (1, 1). Then V s rreducble as a B-module and so there exsts an rreducble U q (L(sl 2 ))-module V = V (l 1, a 1 ) V (l n, a n ) such that V, V are somorphc as B-modules. By Lemma 2, V, V are somorphc as U q (L(sl 2 ))-modules. Ths completes the proof of part () of the theorem for U q (L(sl 2 )). The proof of Theorem 1 can be gven by an argument very smlar to the one we have seen above for the case of U q (L(sl 2 )). We prepare two more lemmas for the case of U q(l(sl 2 )) to make the pont clearer. Set (ε, ε ) = (1, 0) and let T q be the augmented TD-algebra defned by (TD) 0, (TD) 1 n (1), (2). For s C, let ϕ s be the embeddng of T q nto U q(l(sl 2 )) gven by (4), (5). Lemma 3. Let V 1, V 2 be fnte-dmensonal rreducble U q(l(sl 2 ))-modules. If V 1, V 2 are somorphc as ϕ s (T q )-modules for some s C, then V 1, V 2 are somorphc as U q(l(sl 2 ))-modules. Proof. By (4), ϕ s (T q ) s generated by se s 1 e 1 k 1, e + 1 and k±1 ( = 0, 1). Snce e ± 1, k±1 1 s somorphc to the quantum algebra U q (sl 2 ), the acton of e 1 on V, = 1, 2, s unquely determned by those of e + 1, k±1 1 ϕ s (T q ) by Lemma 2. Apparently the acton of e + 0 on V, = 1, 2, s unquely determned by those of se s 1 e 1 k 1, e 1, k 1, and hence by that of ϕ s (T q ). So the acton of U q(l(sl 2 )) on V, = 1, 2, s unquely determned by that of ϕ s (T q ). Lemma 4. Let V be a fnte-dmensonal rreducble U q(l(sl 2 ))-module of type (1, 1). Then there exsts a fnte set Λ of nonzero scalars such that V s rreducble as a ϕ s (T q )-module for each s C \ Λ. Proof. For s C, regard V as a ϕ s (T q )-module. Let W be a mnmal ϕ s (T q )-submodule of V. It s enough to show that W = V holds f s avods fntely many scalars. By (8) wth s 0 = 1, the egenspace decomposton of k 1 = k0 1 on V s V = d U d, k 1 Ud = q 2 d, 0 d. The subalgebra e ± 1, k±1 1 of U q(l(sl 2 )) s somorphc to the quantum algebra U = U q (sl 2 ) n (12) va the correspondence of e ± 1, k±1 1 to X ±, K ±1. The element (e + 1 )d 2 maps U d onto U bjectvely, 0 [d/2]. Also (e 1 k 1) d 2 maps U onto U d bjectvely, 0 [d/2]. The element (e + 1 )d 2 belongs to ϕ s (T q ). So (e + 1 )d 2 W W. Snce the mappng (e + 1 )d 2 : U d U s a bjecton, we have dm(w U d ) dm(w U ), 0 [d/2]. The element (e 1 k 1) d 2 does not belong to ϕ s (T q ), but (e s 2 e 1 k 1) d 2 does. By (7), (e s 2 e 1 k 1) d 2 maps U to U d, 0 [d/2]. We want to show t s a bjecton f s avods fntely many scalars. Identfy U d wth U as vector spaces by the bjecton (e + 1 )d 2 between them. Then t makes sense to consder the determnant of a lnear map from U to U d. Set t = s 2 and expand (e te 1 k 1) d 2 as t d 2 (e 1 k 1) d 2 + lower terms n t. Each term of the expanson gves a lnear map from U to U d. So the determnant of (e te 1 k 1) d 2 U equals t (d 2) dm U det(e 1 k 1) d 2 U + lower terms n t, (13) and ths s a polynomal n t of degree (d 2) dm U, snce det(e 1 k 1) d 2 U 0. Let Λ be the set of nonzero s such that t = s 2 s a root of the polynomal n (13). Then f s C \ Λ, (e s 2 e 1 k 1) d 2 maps U to U d bjectvely. Set Λ = [d/2] =0 Λ. Choose s C \ Λ. Then the mappng (e s 2 e 1 k 1) d 2 : U U d s a bjecton for 0 [d/2]. Snce e s 2 e 1 k 1 belongs to ϕ s (T q ), we have (e =0

9 On a Certan Subalgebra of U q (ŝl 2) Related to the Degenerate q-onsager Algebra 9 s 2 e 1 k 1) d 2 W W and so dm(w U ) dm(w U d ). Snce we have already shown dm(w U d ) dm(w U ), we obtan dm(w U ) = dm(w U d ), 0 [d/2]. Therefore by Lemma 1, we have e 1 W W. Snce (e+ 0 + s 2 e 1 k 1)W W, the ncluson e + 0 W W follows from e 1 W W and so W s U q(l(sl 2 ))-nvarant. Thus W = V holds by the rreducblty of V as a U q(l(sl 2 ))-module. Proof of Theorem 1. We use the classfcaton of fnte-dmensonal rreducble T q -modules n the case of (ε, ε ) = (1, 0) [4, Theorem 1.18]: () A tensor product V = V (l) V (l 1, a 1 ) V (l n, a n ) of evaluaton modules s rreducble as a ϕ s (T q )-module f and only f s 2 / S(l, a ) for all {1,..., n} and S(l, a ), S(l j, a j ) are n general poston for all, j {1,..., n}. () Consder two tensor products V = V (l) V (l 1, a 1 ) V (l n, a n ), V = V (l ) V (l 1, a 1 ) V (l m, a m) of evaluaton modules and assume that they are both rreducble as a ϕ s (T q )-module. Then V, V are somorphc as ϕ s (T q )-modules f and only f l = l, n = m and (l, a ) = (l, a ) for all {1,..., n} wth a sutable reorderng of the evaluaton modules V (l 1, a 1 ),..., V (l n, a n ). () Every fnte-dmensonal rreducble T q -module V, dm V 2, s somorphc to a T q -module V = V (l) V (l 1, a 1 ) V (l n, a n ) on whch T q acts va some embeddng ϕ s : T q U q(l(sl 2 )). Part () of Theorem 1 follows mmedately from the part () above, due to Lemma 4. Part () of Theorem 1 follows mmedately from the part () above, the f part due to Lemma 3 (and Lemma 4) and the only f part due to Lemma 4. We want to show part () of Theorem 1. Let V be a fnte-dmensonal rreducble U q(l(sl 2 ))- module of type (1, 1). By Lemma 4, there exsts a nonzero scalar s such that V s rreducble as a ϕ s (T q )-module. By the part () above, for the proof of whch Drnfel d polynomals play the key role, the T q -module V va ϕ s s somorphc to some T q -module V = V (l) V (l 1, a 1 ) V (l n, a n ) va some embeddng ϕ s of T q nto U q(l(sl 2 )). Snce k 0 has the same egenvalues on V, V, we have s = s and so V, V are somorphc as ϕ s (T q )-modules. By Lemma 3, V, V are somorphc as U q(l(sl 2 ))-modules. Part (v) wll be shown n the next secton. 4 Intertwners: Proof of Theorem 1(v) In ths secton, we show that for l, m Z >0, a C, there exsts an ntertwner between the U q(l(sl 2 ))-modules V (l, a) V (m), V (m) V (l, a),.e., a nonzero lnear map R from V (l, a) V (m) to V (m) V (l, a) such that R (ξ) = (ξ)r, ξ U q(l(sl 2 )), (14) where s the coproduct from (10). If such an ntertwner R exsts, then t s routnely concluded that V (l, a) V (m) s somorphc to V (m) V (l, a) as U q(l(sl 2 ))-modules and any other ntertwner s a scalar multple of R, snce V (m) V (l, a) s rreducble as a U q(l(sl 2 ))-module by Theorem 1. Usng the theory of Drnfel d polynomals [4] for the augmented TD-algebra T q = T (ε,ε ) q wth (ε, ε ) = (1, 0), we shall frstly show that V (l, a) V (m) s somorphc to V (m) V (l, a) as U q(l(sl 2 ))-modules. Ths proves Theorem 1(v), snce t s well-known [2, Theorem 4.11] that V (l, a ) V (l j, a j ) and V (l j, a j ) V (l, a ) are somorphc as U q (ŝl 2)-modules, f S(l, a ) and S(l j, a j ) are n general poston. Fnally we shall construct an ntertwner explctly. Let us denote the U q(l(sl 2 ))-modules V (l, a) V (m), V (m) V (l, a) by V, V : V = V (l, a) V (m), V = V (m) V (l, a).

10 10 T. Hatta and T. Ito Recall we assume that the ntegers l, m and the scalar a are nonzero. Let us denote a standard bass of the U q(l(sl 2 ))-module V (l, a) (resp. V (m)) by v 0, v 1,..., v l (resp. v 0, v 1,..., v m) n the sense of (9). Recall V (m) s an abbrevaton of V (m, 0) and the acton of U q(l(sl 2 )) on V, V are va the coproduct of (10). Let U denote the subalgebra of U q(l(sl 2 )) generated by e ± 1, k± 1. The subalgebra U s somorphc to the quantum algebra U q (sl 2 ). Let V (n) denote the rreducble U-module of dmenson n + 1: V (n) has a standard bass x 0, x 1,..., x n on whch U acts as k 1 x = q n 2 x, e + 1 x = [n + 1]x 1, e 1 x = [ + 1]x +1, where [t] = [t] q = (q t q t )/(q q 1 ), x 1 = x n+1 = 0. We call x n (resp. x 0 ) the lowest (hghest) weght vector: k 1 x n = q n x n, e 1 x n = 0 (k 1 x 0 = q n x 0, e + 1 x 0 = 0). Note that V (l, a) s somorphc to V (l) as U-modules. By the Clebsch Gordan formula, V = V (l, a) V (m) s decomposed nto the drect sum of U-submodules Ṽ (n), l m n l+m, n l+m mod 2, where Ṽ (n) s the unque rreducble U-submodule of V somorphc to V (n). Wth n = l + m 2ν, we have V = V (l, a) V (m) = mn{l,m} ν=0 Let x n denote a lowest weght vector of the U-module Ṽ (n). So (k 1 ) x n = q n x n, (e 1 ) x n = 0. Ṽ (l + m 2ν). (15) Snce V has a bass {v l v m j 0 l, 0 j m} and k 1 acts on t by (k 1 )(v l v m j ) = q (l+m)+2(+j) v l v m j, the lowest weght vector x n of Ṽ (n) can be expressed as x n = +j=ν c j v l v m j, n = l + m 2ν. Solvng (e 1 ) x n = 0 for the coeffcents c j, we obtan c j m 2j+2 [l ν + j] = q c j 1 [m j + 1] and so wth a sutable choce of c 0 x n = ν ( 1) j q j(m j+1) [l ν + j]![m j]!v l ν+j v m j, (16) j=0 where n = l + m 2ν and [t]! = [t][t 1] [1]. Lemma 5. (e + 0 ) x n = aq x n+2. Proof. By (10), we have (e + 0 ) = e k 0 e + 0. By (9), the element e+ 0 vanshes on V (m) and acts on V (l, a) as aqe 1. Snce e 1 v l ν+j = [l (ν 1) + j]v l (ν 1)+j, the result follows from (16), usng v l+1 = 0. Corollary 1. Any nonzero U q(l(sl 2 ))-submodule of V (l, a) V (m) contans x l+m, the lowest weght vector of the U-module V (l, a) V (m). We are ready to prove the followng

11 On a Certan Subalgebra of U q (ŝl 2) Related to the Degenerate q-onsager Algebra 11 Theorem 2. The U q(l(sl 2 ))-modules V (l, a) V (m), V (m) V (l, a) are somorphc for every l, m Z >0, a C. Proof. Let T q = T (ε,ε ) q be the augmented TD-algebra wth (ε, ε ) = (1, 0). Let ϕ s : T q U q(l(sl 2 )) denote the embeddng of T q nto U q(l(sl 2 )) gven n (5). By [4, Theorem 5.2], the Drnfel d polynomal P V (λ) of the ϕ s (T q )-module V = V (l, a) V (m) turns out to be l 1 P V (λ) = λ m ( λ + aq 2 l+1 ). =0 (Note that the parameter s of the embeddng ϕ s does not appear n P V (λ). So the polynomal P V (λ) can be called the Drnfel d polynomal attached to the U q(l(sl 2 ))-module V.) Let W be a mnmal U q(l(sl 2 ))-submodule of V = V (l, a) V (m); notce that we have shown the rreducblty of the U q(l(sl 2 ))-module V = V (m) V (l, a) n Theorem 1 but not yet of V = V (l, a) V (m). By Corollary 1, W contans the lowest and hence hghest weght vectors of V. In partcular, the rreducble U q(l(sl 2 ))-module W s of type (1, 1). By Lemma 4, there exsts a fnte set Λ of nonzero scalars such that W s rreducble as a ϕ s (T q )-module for any s C \ Λ. By the defnton, the Drnfel d polynomal P W (λ) of the rreducble ϕ s (T q )- module W concdes wth P V (λ): P W (λ) = P V (λ). By Theorem 1, V = V (m) V (l, a) s rreducble as a U q(l(sl 2 ))-module. So by Lemma 4, there exsts a fnte set Λ of nonzero scalars such that V s rreducble as a ϕ s (T q )-module for any s C \ Λ. By [4, Theorem 5.2], the Drnfel d polynomal P V (λ) of the rreducble ϕ s (T q )-module V concdes wth P V (λ): P V (λ) = P V (λ). Both of the rreducble ϕ s (T q )-modules W, V have type s, dameter d = l + m and the Drnfel d polynomal P V (λ). By [4, Theorem 1.9 ], W and V are somorphc as ϕ s (T q )-modules. By Lemma 3, W and V are somorphc as U q(l(sl 2 ))-modules. In partcular, dm W = dm V. Snce dm V = dm V, we have W = V,.e., V and V are somorphc as U q(l(sl 2 ))-modules. Fnally we want to construct an ntertwner R between the rreducble U q(l(sl 2 ))-modules V, V. Regard V = V (m) V (l, a) as a U-module. By the Clebsch Gordan formula, we have the drect sum decomposton V = V (m) V (l, a) = mn{l,m} ν=0 Ṽ (l + m 2ν), (17) where Ṽ (n) s the unque rreducble U-submodule of V somorphc to V (n), n = l + m 2ν. Let x n be a lowest weght vector of the U-module Ṽ (n). By (16), we have x n = ν ( 1) j q j(l j+1) [m ν + j]![l j]!v m ν+j v l j (18) j=0 up to a scalar multple, where n = l + m 2ν. It can be easly checked as n Lemma 5 that the lowest weght vectors x n, n = l + m 2ν, 0 ν mn{l, m}, are related by (e 1 1) x n = x n+2, (19) where V = V (m) V (l, a) s regarded as a (U U)-module n the natural way.

12 12 T. Hatta and T. Ito Lemma 6. (e + 0 ) x n = aq q n+2 x n+2. Proof. We have (e + 0 ) x n = aq(k1 1 e 1 ) x n, snce (e + 0 ) = e k 0 e + 0, and e+ 0 vanshes on V (m) and acts on V (l, a) as aqe 1. Express k 1 1 e 1 as k 1 1 e 1 = (k 1 1 1)(1 e 1 ) = (k1 1 1)( (e 1 ) e 1 k 1 1 ) = (k 1 1 1) (e 1 ) k 1 1 e 1 k 1 1 = (k1 1 1) (e 1 ) q2 (e 1 1) (k 1 1 ). Snce (e 1 ) x n = 0, (k1 1 ) x n = q n x n, the result follows from (19). There exsts a unque lnear map R n : V = V (l, a) V (m) Ṽ (n) that commutes wth the acton of U and sends x n to x n. The lnear map R n vanshes on Ṽ (t) for t n and affords an somorphsm between Ṽ (n) and Ṽ (n) as U-modules. If R s an ntertwner n the sense of (14), then R can be expressed as R = mn{l,m} ν=0 α ν R l+m 2ν, (20) regardng R as an ntertwner for the U-modules V, V. By (14), we have R (e + 0 ) = (e+ 0 )R. (21) Apply (21) to the lowest weght vector x n n (16). By Lemma 5, (e + 0 ) x n = aq x n+2 and so wth n = l + m 2ν, we have R (e + 0 ) x n = aqα ν 1 x n+2. (22) On the other hand, R x n = α ν x n, n = l + m 2ν, and so by Lemma 6, we have (e + 0 )R x n = aqα ν q n+2 x n+2. (23) By (22), (23), we have α ν /α ν 1 = q n 2 = q l m+2(ν 1) and so α ν = ( 1) ν q ν(l+m ν+1) (24) by choosng α 0 = 1. An ntertwner exsts by Theorem 2. If t exsts, t has to be n the form of (20), (24). Thus we obtan the followng. Theorem 3. The lnear map R = mn{l,m} ν=0 ( 1) ν q ν(l+m ν+1) R l+m 2ν s an ntertwner between the U q(l(sl 2 ))-modules V (l, a) V (m), V (m) V (l, a). Any other ntertwner s a scalar multple of R. Remark 1. Let R(a, b) be an ntertwner between the rreducble U q(l(sl 2 ))-modules V = V (l, a) V (m, b), V = V (m, b) V (l, a), where a 0, b 0: R(a, b) : V = V (l, a) V (m, b) V = V (m, b) V (l, a). As n (20), we wrte R(a, b) = mn{l,m} ν=0 α ν R l+m 2ν.

13 On a Certan Subalgebra of U q (ŝl 2) Related to the Degenerate q-onsager Algebra 13 Recall R n s the lnear map from V to V that commutes wth the acton of U = e ± 1, k± 1 and sends x n to x n, where x n, x n are the lowest weght vectors of Ṽ (n), Ṽ (n) from (15), (17) that are explctly gven by (16), (18) and satsfy (e 1 1) x n = x n+2, (e 1 1) x n = x n+2 as n (19). Snce the U q(l(sl 2 ))-modules V, V can be extended to U q (ŝl 2)-modules, we have by [2, Theorem 5.4] α ν = ν 1 j=0 a bq l+m 2j, (25) b aql+m 2j where we choose α 0 = 1. Note that the denomnator and the numerator of (25) are non-zero, snce V, V are assumed to be rreducble and so S(l, a), S(m, b) are n general poston. The ntertwner R(a, b) wth a 0, b 0 s derved from the unversal R-matrx for the quantum affne algebra U q (ŝl 2) [5]. If we put b = 0 n (25), then the spectral parameter u dsappears, where q 2u = a/b, and we get (24). In ths sense, the ntertwner R of Theorem 3 s related to the unversal R-matrx for U q (ŝl 2), but we cannot expect that R comes from t drectly, because the U q(l(sl 2 ))-modules V = V (l, a) V (m), V = V (m) V (l, a) cannot be extended to U q (ŝl 2)-modules. In order to derve both of the ntertwners R(a, b), R from a unversal R-matrx drectly, we need to construct t for the subalgebra U q(ŝl 2) of U q (ŝl 2). Acknowledgements The research of the second author s supported by Scence Foundaton of Anhu Unversty (Grant No. J ). References [1] Benkart G., Terwllger P., Irreducble modules for the quantum affne algebra U q(ŝl2) and ts Borel subalgebra, J. Algebra 282 (2004), , math.qa/ [2] Char V., Pressley A., Quantum affne algebras, Comm. Math. Phys. 142 (1991), [3] Ito T., Tanabe K., Terwllger P., Some algebra related to P - and Q-polynomal assocaton schemes, n Codes and Assocaton Schemes (Pscataway, NJ, 1999), DIMACS Ser. Dscrete Math. Theoret. Comput. Sc., Vol. 56, Amer. Math. Soc., Provdence, RI, 2001, , math.co/ [4] Ito T., Terwllger P., The augmented trdagonal algebra, Kyushu J. Math. 64 (2010), , arxv: [5] Tolstoy V.N., Khoroshkn S.M., Unversal R-matrx for quantzed nontwsted affne Le algebras, Funct. Anal. Appl. 26 (1992),