Antiholomorphic Representations for Orthogonal and Symplectic Quantum Groups

JOURNAL OF ALGEBRA ARTICLE NO.

184, 71]101 Ž1996.

0250

Antiholomorphic Representations for Orthogonal and Symplectic Quantum Groups ˇ ´ˇ ,† Pavel ˇ Stovıcek* Arnold Sommerfeld Institute, TU Clausthal, Leibnizstr. 10, D-38678 Clausthal-Zellerfeld, Germany Communicated by Corrado de Concini Received May 25, 1995

The coadjoint orbits for the series Bl , Cl , and D l are considered in the case when the base point is a multiple of a fundamental weight. A quantization of the big cell is suggested by means of introducing a )-algebra generated by holomorphic coordinate functions. Starting from this algebraic structure the irreducible representations of the deformed universal enveloping algebra are derived as acting in the vector space of polynomials in quantum coordinate functions. Q 1996 Academic Press, Inc.

1. INTRODUCTION Recently remarkable attention has been paid to quantum homogeneous spaces. Particularly interesting is the relation to the representation theory in the spirit of the classical method of orbits due to Kirillov-Kostant. We quote rather randomly and incompletely w10, 9, 16, 13, 14, 1]3x, etc. The majority of the quoted papers are concerned with a deformation of the algebra of holomorphic functions. The quantization of a general coadjoint orbit as a complex manifold, valid for any compact group from the series A l , Bl , Cl , and D l , has been described in w8x in terms of quantum holomorphic coordinate functions living on the big cell. In the same paper, it was shown with the help of quantum coherent states that any irreducible finite-dimensional representation of the deformed enveloping algebra Uh *Humboldt Fellow. † On leave of absence from: Department of Mathematics and Doppler Institute, Faculty of Nuclear Science, CTU, Prague, Czech Republic. 71 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

72

PAVEL ˇˇ STOVICEK ´ˇ

admits an Žanti-. holomorphic realization when acting in a space of polynomials in non-commutative variables, the quantum coordinate functions. The goal of the present paper is to extend the results of the paper w15x, where the series A l has been treated, also to orthogonal and symplectic groups. For the unitary groups SUŽ N ., the coadjoint orbits were considered and quantized with the base point being a multiple of a fundamental weight. These homogeneous spaces coincide exactly with the Grassmann manifolds. Moreover, antiholomorphic representations of Uh in the Borel]Weil spirit were derived from the structure of the quantized orbit. Here we focus on the series Bl , Cl , and D l , again in the case when the base point is a multiple of a fundamental weight. This means an attempt to suggest a quantization of these homogeneous spaces as real analytic manifolds. Thus the result should be an algebra C generated jointly by the holomorphic and antiholomorphic quantum coordinate functions z jk and zUst living on the big cell. However, the main point consists in deriving representations of Uh from the algebraic structure of C . Since we restrict ourselves to a special sort of orbit the obtained representations are characterized by a lowest weight l being an integer multiple of a fundamental weight, yl g Zq v r . While the general strategy is quite parallel to that of the paper w15x, the commutation relations derived for orthogonal and symplectic groups turn out to be considerably more complex and less transparent. Apparently this is not the first case when the series Bl , Cl , and D l are much more difficult to approach than the series A l . Perhaps one fact standing behind this rather unpleasant observation is a property of the R-matrices. While in the A l case the R-matrix obeys the Hecke condition 2 Ž R12 P . y Ž q y qy1 . R12 P y I s 0,

or equivalently, y1 R12 y Ry1 . P, 21 s Ž q y q

the R-matrices for the series Bl , Cl , D l obey a more complicated relation Ž2.8. below. Unfortunately, this is why the construction of the algebra C was not clarified entirely and has to some extent a speculative character. However, the prescription for the representation of Uh acting in the vector space of polynomials in quantum coordinate functions zUjk is derived quite unambiguously and verified rigorously. In principle, Section 4, concerned with the quantum parameterization of the big cell, can be skipped. But starting directly from the defining relations for the representation could seem then rather obscure.

SYMPLECTIC QUANTUM GROUPS

73

The paper is organized as follows. In Section 2, the notation is introduced and a summary of basic relations is given. Section 3 starts from a quantum version of the orthogonal transformation of matrices. Furthermore, quantum Žanti-. holomorphic coordinate functions on the big cell are introduced. A parameterization of the big cell as a real analytic manifold is suggested in Section 4. The quantization means a specification of commutation relations between the coordinate functions. Section 5 is devoted to verification of the rules according to which the algebra generated by the quantum antiholomorphic coordinate functions becomes a left Uh-module. These rules are suggested by the Žpartially informal. considerations of Section 4. The module depends on one free parameter and its values are specified in Section 6 so that one obtains finite-dimensional irreducible representations of Uh . Section 7 contains a few final remarks about some points suggested in Section 4 and not clarified in full detail.

2. PRELIMINARIES The deformation parameter is q s eyh , h ) 0, and the quantum numbers are defined as w x x [ Ž q x y qyx .rŽ q y qy1 .. The rank of the group is denoted by l. Let N [ 2 l q 1 for the series Bl and N [ 2 l in the case of D l and Cl . The standard basis in C N is denoted by  eyl , . . . , ey1 , e0 , e1 , . . . ,e l 4 for the series Bl and by  eylq1r2 , . . . , ey1r2 , e1r2 , . . . , e ly1r2 4 for the series Cl and D l . Set also

e [ 1 Ž [ y1 . for the series Bl and D l Ž resp. Cl . ,

Ž 2.1.

« j [ 1 Ž [ sgn Ž j . . for the series Bl and D l Ž resp. Cl . .

Ž 2.2.

The R-matrix obeying the Yang]Baxter ŽYB. equation R12 R13 R 23 s R 23 R13 R12 in C N m C N is given by w4, 6, 11x R jk , st [ d js d k t q Ž q y q sgnŽ kyt . . d jt d k s y « k « t Ž qysgnŽ kyt . y qy1 . qyr kqr tqd k 0 d t 0 d jqk , 0 d sqt , 0 Ž 2.3.

r k [ yk q e 12 sgn k.

Ž 2.4.

Remark. Of course, the term d k 0 d t 0 appearing in the exponent in the third summand plays a role only for the series Bl . Further we introduce the N = N matrices

r [ diag Ž ryŽ Ny1.r2 , ryŽ Ny3.r2 , . . . , rŽ Ny1.r2 . ,

Ž 2.5.

C [ C 0 q r s qyr C 0 ,

Ž 2.6.

where C jk0 [ « j d jqk , 0 .

PAVEL ˇˇ STOVICEK ´ˇ

74

Thus it holds true that C t s C* s e C 0 qyr s e q r C 0 , Cy1 s e C and C t C s q 2 r . Let K be an N 2 = N 2 matrix with the entries K jk , st [ C jkt Cy1 st .

Ž 2.7.

The symbol P ' P12 s P21 stands for the flip operator. A survey follows giving the basic relations valid for the matrices R, K, P, and C. Quite essential is the first one, y1 R12 y Ry1 . Ž P y K 12 . . 21 s Ž q y q

Ž 2.8.

Furthermore, t R 21 ' PR12 P s R12 , t

y1 2 Cy1 2 R 12 C 2 s Ž R 12 . ,

Ry1 q s R q y1 ,

Ž Cy1 1 .

t

Ž 2.9. t

1 R12 C1t s Ž Ry1 12 . ,

C1C2 R12 C1C2 s R 21 , C1C2 K 12 s

C1t C2t

t K 21 ' PK 12 P s K 12 ,

Ž 2.10. Ž 2.11.

K 12 s PK 12 ,

Ž 2.12.

K qy1 s K qt .

Ž 2.13.

One can also show that RPK s e qyN q e K

Ž 2.14.

y1 Ny e and hence Ry1 PK 12 . For any N 2 = N 2 respec12 K 12 s K 21 R 21 s e q tively N = N matrix X Žthe dimension is clear from the context. we have

K 12 X 12 K 12 s tr Ž K 12 X 12 . K 12 ,

Ž 2.15.

2r

tr Ž K 12 PX1 . s e tr Ž q X . .

Ž 2.16.

In the last relation, the trace on the LHS is taken in C N m C N while on the RHS it is taken in C N. Consequently, K 12 PX1 K 12 s e tr Ž q 2 r X . K 12 .

Ž 2.17.

It holds true that t K 12 X 1 s K 12 Cy1 2 X 2 C2 ,

t

t t K 12 X 2 s K 12 Ž Cy1 1 . X 1 C1 ;

Ž 2.18.

particularly, K 12 Ry1 31 s K 12 R 32 ,

K 12 Ry1 23 s K 12 R 13 .

Ž 2.19.

75

SYMPLECTIC QUANTUM GROUPS

Notice also that K 12 K 32 s K 12 P13 ,

K 12 K 13 s K 12 P23 .

Next let us introduce a family of orthogonal projectors in C N. The symbol EŽ s . stands for the matrix Ždiagonal. expressed in the basis  e k 4 and corresponding to the projector onto span e j ; j F s4 ; particularly EŽyŽ N q 1.r2. s 0, EŽŽ N y 1.r2. s I. We have K 12 E Ž s . 1 s K 12 Ž I y E Ž ys y 1 . 2 . ,

Ž 2.20.

s q t - 0 « K 12 E Ž s . 1 E Ž t . 2 s 0.

Ž 2.21.

Furthermore, for all s, E Ž s . 1 R12 Ž I y E Ž s . 1 . s 0,

Ž I y E Ž s . 2 . R12 E Ž s . 2 s 0. Ž 2.22.

One can derive other relations like R 21ŽI y EŽ s . 2 . s ŽI y EŽ s . 2 . R 21ŽI y EŽ s . 2 ., etc. Let us fix for the rest of the paper an index r ) 0 Ž r g  1, . . . , l 4 , in the Bl case, and r g  12 , . . . , l y 12 4 , for the series Cl and D l .. One can show that

Ž I y E Ž yr . 1 . R12 E Ž yr . 1 E Ž yr . 2 s 0.

Ž 2.23.

It is straightforward to verify that y2 rq1q e E Ž yr . 2 PK 12 , Ry1 12 E Ž yr . 2 K 12 s e q

Ž 2.24.

R 21 Ž I y E Ž yr . 2 . K 12 s e qy2 rq1q e Ž I y E Ž yr . 2 . PK 12 y Ž q y qy1 . E Ž yr . 1 PK 12 .

Ž 2.25.

The last relation implies

Ž I y E Ž yr . 1 . Ž I y E Ž yr . 2 . R 21 Ž I y E Ž yr . 1 . Ž I y E Ž yr . 2 . K 12 s e qy2 rq1q e Ž I y E Ž yr . 1 . Ž I y E Ž yr . 2 . PK 12 .

Ž 2.26.

The goal of this paper is to determine a relation between the coadjoint orbits and the representations of the deformed enveloping algebra Uh . Let Hj , X j", j s 1, . . . , l, be the Chevalley generators of Uh for a simple Lie algebra of rank l. The enumeration of roots in the Dynkin diagram is chosen so that the first root is the shortest one or the longest one, in the case of Bl or Cl , respectively, and nodes 1 and 2 are connected with node 3 in the D l case. The commutation relations defining Uh can be found in various papers w5, 4, 11x. Uh is isomorphic as a )-algebra to the algebra Ad of quantum

PAVEL ˇˇ STOVICEK ´ˇ

76

functions living on the dual solvable group w11x. The generators of Ad can be arranged into an upper triangular N = N matrix L s Ž a jk . with positive Žin a convenient sense. entries on the diagonal, a j j ) 0, and obeying Ł a j j s 1. The commutation relations can be written as w11, 7x LU1 Ry1L 2 s L 2 Ry1LU1 ,

RL 1 L 2 s L 2 L 1 R,

Ž 2.27.

jointly with the orthogonality condition, CLt Cy1 s Cy1Lt C s Ly1 .

Ž 2.28.

The isomorphism between Uh and Ad is given explicitly in terms of generators w11x, q H j s Ž a jy1, jy1 .

y1

ajj ,

j s 1, . . . , l,

Ž q y qy1 . Xyj s qy1r2 Ž a jy1, jy1 a j j . y1 r2 a jy1, j ,

j s 1, . . . , l,

Ž 2.29a.

for the Bl case Ž a 00 s 1.; 2

q H 1 s Ž a 1r2 , 1r2 . , q H j s Ž a jy3r2, jy3r2 .

Ž q y qy1 . Xy1 s Ž q q qy1 .

y1 r2

y1

a jy1r2, jy1r2 ,

j s 2, . . . , l,

Ž 2.29b.

qy1ay1r2, 1r2 ,

Ž q y qy1 . Xyj s qy1r2 Ž a jy3r2, jy3r2 a jy1r2, jy1r2 . y1 r2 a jy3r2, jy1r2 , j s 2, . . . , l, for the Cl case; q H 1 s a 1r2, 1r2 a 3r2, 3r2 , q H j s Ž a jy3r2, jy3r2 .

y1

a jy1r2, jy1r2 ,

1r2 y1 r2 Ž q y qy1 . Xy1 s qy1r2a 1r2, 1r2 a 3r2, 3r2 ay1r2, 3r2 ,

j s 2, . . . , l,

Ž 2.29c.

Ž q y qy1 . Xyj s qy1r2 Ž a jy3r2, jy3r2 a jy1r2 , jy1r2 . y1 r2 a jy3r2, jy1r2 , j s 2, . . . , l, for the D l case Žlet us note that ay1r2, 1r2 s 0.; and y Xq j s Ž X j . *,

in all three cases.

; j,

Ž 2.30.

77

SYMPLECTIC QUANTUM GROUPS

Instead of Ad we shall deal with the )-algebra A˜d whose generators are arranged into a ‘‘positive’’ matrix M [ L*L ;

hence M* s M.

Ž 2.31.

The generators of the algebra Ad can be recovered by decomposing M into a product of a lower triangular matrix time an upper triangular matrix. With more details, this relationship has been described in w15x. From the commutation relations Ž2.27. one can deduce that y1 y1 y1 M2 Ry1 12 M1 R 21 s R 12 M1 R 21 M2 .

Ž 2.32.

Concerning the orthogonality condition, notice first that Ž2.27., Ž2.10., and U y1 y1 U y1 Ž2.28. imply C2 LU1 Lt2 Cy1 . Consequently, using 2 s L 1 L 2 s RL 2 L 1 R Ž2.14., M jk s

y1 U y1 Ý Ž Cy1 2 R 12 L 2 L 1 R 12 C 2 . jk , s s

s

s e q Ny e

y1 U Ý Ž Cy1 2 R 12 L 2 L 1 C 2 . jk , s s

s

s e q Ny e

Ý

t

y1 y1 U t Cy1 2 R 12 L 2 Ž C 2 L 2 C 2 . C 2

s

s e q Ny e

jk , s s

y1 t Ý Ž Cy1 2 R 12 M2 C 2 . jk , s s .

s

This equality can be rewritten as y1 Ny e R12 My1 M1 K 12 s R12 My1 2 R 12 K 12 s e q 2 PK 12 .

Ž 2.33.

Summing up, we have DEFINITION 2.1. The algebra A˜d is generated by the entries of the N = N matrix M and is determined by the relations Ž2.32., Ž2.33.. The )-involution in A˜d is given by M* s M ŽŽ M*. jk [ Ž Mk j .*.. Let us show that the element tr Ž q 2 r M . belongs to the center of A˜d , i.e., tr Ž q 2 r M . M s M tr Ž q 2 r M . .

Ž 2.34.

y1 . t 2 M2t M1 s Ž Ry1 . t 2 . Owing to Actually, Ž2.32. implies Ž Ry1 12 12 M1 R 21 M2 R 21 Ž2.10. and Ž2.19., one gets y1 y1 y1 K 12 M3t M2 s K 12 Cy1 3 R 13 C 3 Ž R 23 M2 R 32 M3 R 32 . y1 t t3 t s K 12 Ry1 23 M2 R 32 M3 R 32 C 3 R 13 Ž C 3 .

ž

t3

y1 t 3

/

,

PAVEL ˇˇ STOVICEK ´ˇ

78 and hence

y1 y1 K 12 M3 M2 s K 12 Ry1 23 M2 R 32 M3 R 32 R 13 .

Having multiplied this equation by K 34 P34 from the left and by K 34 from the right, one can use Ž2.17. and repeatedly Ž2.19. to get y1 y1 e tr Ž q 2 r M . K 12 M2 K 34 s K 12 K 34 P34 Ry1 23 M2 R 32 M3 R 32 R 13 K 34 y1 y1 s K 12 K 34 P34 Ry1 23 M2 R 32 M3 R 14 R 42 K 34 y1 y1 y1 s K 12 K 34 Ry1 23 R 24 P34 M2 R 32 R 42 M3 K 34 . y1 y1 y1 Since K 34 Ry1 23 R 24 s K 34 , K 34 R 42 R 32 s K 34 , we obtain

tr Ž q 2 r M . K 12 M2 K 34 s K 12 M2 tr Ž q 2 r M . K 34 . Actually much more is known. The elements trŽ q 2 r M j ., j s 1, . . . , l, generate the center of the algebra A˜d w11x. An immediate consequence of Ž2.33. is that tr Ž q 2 r M . s tr Ž q 2 r My1 . .

Ž 2.35.

It is enough to apply K 12 P from the left and to take into account Ž2.17. and Ž2.14.. Notice also that it holds tr Ž q 2 r . s e q w N y e x .

Ž 2.36.

This equality follows also from Ž2.17., with X s I, and from Ž2.8. and Ž2.14. as y1 Ž q y qy1 . K 12 PK 12 s yK 12 P Ž R12 y Ry1 . K 12 . 21 . q Ž q y q

3. QUANTUM ANTIHOLOMORPHIC COORDINATE FUNCTIONS ON THE BIG CELL Let us start from an observation which is easy to verify and is quite essential for the rest of the paper. LEMMA 3.1. To any N = N matrix X there exists a matrix Y of the same dimension such that K 12 X 2 PR12 s K 12 Y2 .

Ž 3.1.

SYMPLECTIC QUANTUM GROUPS

79

The same matrix Y obeys also Y2 K 21 s R 21 PX 2 K 21 . Moreo¨ er, X * s X implies Y * s Y. Thus the entries of Y depend linearly on X, namely Yjk s e« j « k qyr jyr kyd jk Xyk , yj q Ž q y qysgnŽ jqk . . qyd j0 d k 0 X jk y e Ž q y qy1 . d jk

Ý

q 2 rs Xss .

Ž 3.2.

s)yj

The term d j0 d k 0 appearing in an exponent can be again omitted for the series Cl and D l . The relation Ž3.1. can be inverted, K 12 X 2 s K 12 Y2 PRy1 21 , and it holds true that X 2 K 21 s Ry1 12 PY2 K 21 . As a first application, let us show that the orthogonality condition Ž2.33. follows partially from Ž2.32.. By exchanging the indices, one can rewrite y1 Ž2.32. as M2 Ry1 12 M1 R 12 s R 21 M1 R 21 M2 . Subtracting these two equations Ž . and using 2.8 we get y1 M2 Ry1 12 M1 K 12 s K 21 M1 R 21 M2 .

Ž 3.3.

˜ be the matrix defined by Let M y1 ˜2 [ K 12 M2 PRy1 K 12 M 21 s K 12 M2 R 12 P.

Ž 3.4.

Multiply Ž3.3. by K 12 from the left. Notice that, by Ž2.17., K 12 K 21 s ˜ and K 12 PK 12 P s e trŽ q 2 r . K 12 P. Taking into account the definition of M Ž2.17. also on the LHS, one obtains

˜ . K 12 s tr Ž q 2 r . K 12 M˜2 M2 , tr Ž q 2 r MM ˜ s cI, MM

˜ . rtr Ž q 2 r . . where c s tr Ž q 2 r MM

Ž 3.5. Ž 3.6.

˜ Ž . Since it is true that Ry1 12 M1 K 12 s PM1 K 12 , the LHS of 3.5 also equals ˜ . K 12 . For M* ˜ s M, ˜ it holds that c* s c and thus MM ˜ s trŽ q 2 r MM ˜ ˜ ˜ Ž MM .* s MM. Consequently, cM s MMM s Mc and c belongs to the center of A˜d . Whence the orthogonality condition Ž 2.33. is equi¨ alent to the equality c s e q Ny e .

PAVEL ˇˇ STOVICEK ´ˇ

80

Recall that an index r has been fixed as a positive integer Žhalf-integer . not exceeding l Žresp. l y 12 . for the series Bl Ž Cl and D l .. The classical coadjoint orbit can be identified with an orbit of the corresponding simple compact group when acting on the set of positive N = N matrices as ŽU, M . ¬ U*MU. The positive matrix M is also required to obey C 0 M t C 0 s My1. The base point of the orbit is M0 [ diagŽ jy1 , . . . , jy1 , 1, . . . , 1, j , . . . , j ., for some j ) 0. The eigen-values j and jy1 have the same multiplicity Ž N y 2 r q 1.r2. Here we are doing one exception: in the D l case, with r s 3r2, the orbit corresponds to a multiple of the weight v 1 q v 2 rather than to a multiple of the fundamental weight v 2 Žcf. Sect. 6.. The algebra Ch generated by the quantum ‘‘holomorphic coordinate functions’’ on the big cell has been introduced in w8x. Let us recall the definition. Let Z be an N = N matrix split into blocks as I1 Zs 0 0



Z1 I2 0

Z2 Z3 , I3

0

Ž 3.7.

with the diagonal unit blocks I 1 and I 3 having the same dimension 1 1 Ž . Ž . Ž 2 N y 2 r q 1 = 2 N y 2 r q 1 and with I 2 having the dimension 2 r y 2 2 . Ž . 1 = 2 r y 1 . Introduce an N = N matrix Q by Q jk , st s R jk , st ,

if j, s F yr or yr - j, s - r or r F j, s,

s 0,

otherwise.

Ž 3.8.

Notice that after replacing j, s by k, t everywhere in the condition following ‘‘if’’ on the first line, we get the same matrix Q. The commutation relation defining the algebra Ch reads as y1 R12 Qy1 12 Z1 Q12 Z2 s Q 21 Z2 Q 21 Z1 R 12 .

Ž 3.9.

In addition, Z is required to obey the ‘‘orthogonality’’ condition

d jk s

t y1 Ý Ž Z2 Cy1 2 Q12 Z2 Q12 C 2 . k j, s s .

s y1 y1 y1 Since Cy1 and owing to Ž2.12., this equality can be 1 C 2 Q12 s Q 21 C1 C 2 rewritten as t y1 K 12 s Z1 Q12 Cy1 1 Z1 C1 Q12 K 12 .

Ž 3.10.

The relations adjoint to Ž3.9. and Ž3.10. define another algebra Cah generated by the quantum ‘‘antiholomorphic coordinate functions’’ zUjk .

81

SYMPLECTIC QUANTUM GROUPS

The relations Ž3.9., Ž3.10. can be simplified. To this end we set Z [ Ž Z1 , Z 2 . ,

Z[

ž

0 0

Z . 0

/

Ž 3.11.

Thus Z s Ž z jk . is a 12 Ž N y 2 r q 1. = 12 Ž N q 2 r y 1. block with the indices restricted by j F yr, yr - k, and Z is an N = N matrix; the dimensions of the blocks are determined implicitly. Set Ey[ E Ž yr . ,

E 0 [ E Ž r y 1 . y E Ž yr . ,

Eq[ I y E Ž r y 1 . . Ž 3.12.

Thus I s Eyq E 0 q Eq is a resolution of unity. We also have Z s Ey Z ŽI y Ey. . The matrix Q can be written as Žcf. Ž2.22.. Q12 s E2yR 12 E2yq E20 R12 E20 q E2qR 12 E2q s R12 E2yq E20 R12 E20 q E2qR 12 .

Ž 3.13.

A similar equality holds true also provided the projectors Ey, E 0 , Eq are applied in the first factor of the tensor product rather than in the second one. Particularly, y E1yR 12 Qy1 12 s E1 .

Ž 3.14.

y y Ž . Furthermore, we have Q qy1 s Qy1 q . Now, let us multiply 3.9 by E1 E2 y y from the left. Using Ž3.13., Ž3.14., Ž2.22. and noticing that E Z s E q Z, one can derive easily y1 y y Ž E1yq Z 1 . R12 Ž E2y q Z 2 . Ry1 12 s R 12 Ž E2 q Z 2 . R 21 Ž E1 q Z 1 . . Ž 3.15 .

Ž . Similarly, by multiplying Ž3.10. by E1y Ey 2 from the left and using 2.21 , the "1 "1 y Ž . Ž . equality Ey Q s R E , 2.24 , and 2.18 , one obtains 2 12 12 2

Ž E2yq Z 2 . R 21 Ž E1y q Z 1 . K 12 s 0.

Ž 3.16.

In fact, one can show that LEMMA 3.2. The relation Ž3.16. follows from Ž3.15.. Actually, interchange the sides and the indices 1, 2 in Ž3.15. to get

Ž E1yq Z 1 . R12 Ž E2yq Z 2 . R 21 s R12 Ž E2y q Z 2 . R 21 Ž E1yq Z 1 . . Subtracting this equation from Ž3.15. and using Ž2.8., one arrives at K 21 Ž E1yq Z 1 . R12 Ž E2y q Z 2 . s Ž E2y q Z 2 . R 21 Ž E1yq Z 1 . K 12 .

PAVEL ˇˇ STOVICEK ´ˇ

82

Now it is enough to multiply the last equality by E1yEy 2 from the left. Summing up we have DEFINITION 3.3. The algebra Ch is generated by the entries of matrix Z s Ž z jk . introduced in Ž3.11. and is determined by the relation Ž3.15.. The algebra Cah is generated by the entries of the matrix Z* s Ž zUk j . and determined by the adjoint relation to Ž3.15.. Let Y be a matrix defined by the relation Y1 K 12 [ e qy2 rq1q e R12 P Ž E1yq Z 1 . K 12 .

Ž 3.17.

One can deduce from Ž3.2. that Y s Eqq V ,

where V [

ž

0 0

V 0

/

Ž 3.18.

and the ranges of indices in the 12 Ž N q 2 r y 1. = 12 Ž N y 2 r q 1. block V s Ž ¨ jk . are given by j - r, r F k. One can write explicitly ¨ jk s qy2 rq1q e« j « k qyr jyr k zyk , yj q ex jFyr Ž q y qysgnŽ jqk . . z jk ,

Ž

.

Ž 3.19.

where x jFyr s 1 Ž0. provided the condition j F yr is Žis not. satisfied. Thus the orthogonality condition Ž3.16. means that LEMMA 3.4. The matrix V s Ž ¨ jk . defined by Ž3.19. fulfills

Ž I Z . V s 0.

ž / I

Ž 3.20.

Notation. The following shorthand notation will be used mainly in the proofs. Set

g [ q y qy1 ,

p [ e qy2 rq1q e .

Ž 3.21.

X [ Eyq Z*,

Y [ Eqq V*.

Ž 3.22.

Furthermore,

So X 1 R12 X 2 R 21 s R12 X 2 R 21 X 1 , K 21 Y 1 s pK 21 X 1 P12 R 21 and Y 1 K 12 s pR12 P12 X 1 K 12 . Notice that X 2 s X and Y 2 s Y Žsince Ž Z*. 2 s 0 s Ž V*. 2 .. Moreover, Y X s 0 ŽLemma 3.4. and X Y s 0 Ževident.. In addition to the equality K 21 X 1 R12 X 2 s 0 ŽLemma 3.2. we have also X 1 R12 X 2 K 21 s 0, y as a consequence of E1yEy 2 K 12 s 0 and X s X E .

Ž 3.23.

SYMPLECTIC QUANTUM GROUPS

83

LEMMA 3.5. The matrix V defined in Ž3.17., Ž3.18. obeys U U U q y1 y1 q y1 q Ž E1qq VU1 . Ry1 21 Ž E2 q V 2 . R 12 s R 21 Ž E2 q V 2 . R 12 Ž E1 q V 1 . Ž 3.24.

and consequently U q Ž E1qq VU1 . Ry1 21 Ž E2 q V 2 . K 21 s 0.

Ž 3.25.

Moreo¨ er, it holds true that R 21 Ž E1yq ZU1 . R12 Ž E2qq VU2 . s Ž E2qq VU2 . R 21 Ž E1yq ZU1 . R12 Ž 3.26. and also Ž equi¨ alently . U U U q y1 y1 q y1 y Ž E1y q ZU1 . Ry1 21 Ž E2 q V 2 . R 12 s R 21 Ž E2 q V 2 . R 12 Ž E1 q Z 1 . . Ž 3.27.

Remark. The algebra Cah is also generated by the entries of the matrix V * s Ž ¨ Uk j . and is determined by the relation Ž3.24.. Proof. The equality K 13 K 24 X 3 R 34 X 4 R 43 s K 13 K 24 R 34 X 4 R 43 X 3 leads to Žusing also Ž2.19.. y1 y1 K 13 K 24 Y 3 P13 Ry1 31 Y 4 P24 R 42 R 32 R 43 y1 y1 s K 13 K 24 Y 4 P24 Ry1 42 Y 3 P13 R 31 R 41 R 12 .

Using the YB equation twice on both sides one obtains K 13 K 24 Y 3 Y 4 R 23 R 21 s K 13 K 24 Y 4 Y 3 R14 R 34 . The relation Ž2.19. then gives y1 K 13 K 24 R 43 Y 3 Ry1 43 Y 4 s K 13 K 24 Y 4 R 34 Y 3 R 34 .

To show Ž3.26. one can proceed similarly starting from y1 K 31 X 2 R 21 X 1 Ry1 21 s K 31 R 12 X 1 R 12 X 2 . y1 Inserting R12 s Ry1 21 q g P12 y g K 12 resp. R 21 s R 12 q g P12 y g K 21 on the LHS resp. the RHS of the equality y1 X 1 R12 Y 2 Ry1 12 s R 21 Y 2 R 21 X 1 ,

one obtains easily Ž3.27..

PAVEL ˇˇ STOVICEK ´ˇ

84

4. PARAMETERIZATION OF THE QUANTUM ORBIT The character of this section is to some extent speculative. However, one can extract from the procedure presented here a prescription for an action of A˜d on Cah . Its rigorous verification is the goal of the next section. Introduce N = N matrices Q " by Qy[

I Z*

ž /Ž

I q ZZ* .

y1

ŽI Z . ,

Qq[

V I

ž /Ž

I q V *V .

y1

Ž V * I. . Ž 4.1.

They possess projector-like properties: Ž Q ". 2 s Q " and Ž Q ". * s Q ". Moreover, in virtue of Ž3.20., Qy Qqs Qq Qys 0. Relate to a triplet of parameters j 0 , jq, jy the matrix M [ j 0 I q Ž jqy j 0 . Qqq Ž jyy j 0 . Qy.

Ž 4.2.

DEFINITION 4.1. Denote by C the )-algebra generated by z jk , zUjk and determined by Ž3.15., y1 yZ Ž I Z . 2 Ry1 12 M 1 R 21

ž / I

s 0,

Ž 4.3.

2

and by

˜ s e q Ny e My1 , M

˜ 2 [ K 12 M 2 Ry1 where K 12 M 12 P.

Ž 4.4.

Finally, the element m [ tr Ž q 2 r M .

Ž 4.5.

is required to be central in A˜d . Of course, the adjoint relations should be fulfilled as well. Remark. The structure of the relation Ž4.3. is not transparent enough and so it is not clear whether C / 0, i.e., whether the unit is not contained in the ideal generated by the defining relations. Nevertheless, in this section we shall proceed optimistically as if 1 g C . The equality Ž4.3. means exactly that y1 y1 y Qy 2 R 12 M 1 R 21 Ž I y Q 2 . s 0.

Ž 4.6.

SYMPLECTIC QUANTUM GROUPS

85

From the Hermitian property of Qy and M and from Ž2.9. one deduces that this is the same as y1 y1 y1 y1 y Qy 2 R 12 M 1 R 21 s R 12 M 1 R 21 Q 2 .

Ž 4.7.

It holds again true that Žin the same way as in Ž2.35.. tr Ž q 2 r M . s tr Ž q 2 r My1 . .

Ž 4.8.

t "[ tr Ž q 2 r Q " . .

Ž 4.9.

Set

The element m is central in A˜d and, by Ž4.2., m s j 0 tr Ž q 2 r . q Ž jqy j 0 . tqq Ž jyy j 0 . ty .

Ž 4.10.

Now, inserting the expression Ž4.2. for M and the expression y1 y1 y1 y1 My1 s jy1 0 I q Ž jq y j 0 . Qqq Ž jy y j 0 . Qy

Ž 4.11.

for My1 into the equality Ž4.8. one gets a linear dependence between tq and ty. It follows that, in the generic case, both tq and ty are expressible in terms of m, and thus tq, ty are central elements in A˜d as well. Furthermore, the substitution of Ž4.2. and Ž4.11. for M and My1 , respectively, into Ž4.4. leads to the equality K 12 Ž q Ny ej 0y1 y ej 0 PRy1 21 . Ny e y1 q K 12 Qq Ž jqy1 y jy1 2 Žq 0 . y e Ž jqy j 0 . PR 21 . Ny e y1 q K 12 Qy Ž jyy1 y jy1 2 Žq 0 . y e Ž jyy j 0 . PR 21 . s 0. Ž 4.12 . y1 . Provided the matrix q Ny e Ž jy1 I y e Ž jqy j 0 . PRy1 q y j0 21 is regular one q y can express Q in terms of Q . The solution can be found with the help of the following ansatz: y y K 12 Qq 2 s m K 12 q h K 12 Q 2 q z K 12 Q 2 PR 12 .

Ž 4.13.

y1 Ny e y1 y m K 12 q h K 12 Qy K 12 Qq 2 PR 21 s e q 2 PR 21 q z K 12 Q 2 .

Ž 4.14.

Whence,

Owing to Ž2.8., y y1 y1 K 12 Qy . K 12 Qy2 y e Ž q y qy1 . ty K 12 . 2 PR 12 s K 12 Q 2 PR 21 q Ž q y q

Ž 4.15.

PAVEL ˇˇ STOVICEK ´ˇ

86

Inserting Ž4.13., Ž4.14. into Ž4.12., taking into account Ž4.15., and compary y1 ing the coefficients at the terms K 12 , K 12 Qy 2 , and K 12 Q 2 PR 21 , one arrives at a system of equations

jy1 0 yj 0 q

Ž jqy1 y jy1 0 . y Ž jq y j 0 .

y1 m y e Ž qyqy1 . Ž jq y jy1 0 . ty z s0,

Ž 4.16a. y1 y1 y1 jy y jy1 0 q Ž jq y j 0 . h y1 q ye q Ny e Ž jqy j 0 . q Ž q y qy1 . Ž jy1 q y jy . z s 0, Ž 4.16b . y1 jyy j 0 q Ž jqy j 0 . h y e q Ny e Ž jq y jy1 0 . z s 0.

Ž 4.16c.

In the generic case, the unique solution Ž m , h , z . determines the RHS in Ž4.13.. We claim that there exists a )-algebra morphism c : A˜d ª C prescribed by its ¨ alues on the generators, namely c Ž M . s M . To verify this statement one has to check the definition of A˜d , i.e., Ž2.32., Ž2.33., and the definition of C , i.e., Ž4.3., Ž4.4., giving also Ž4.7.. The rest is a consequence of Ž4.13. and the following proposition. PROPOSITION 4.2. Let X and N be N = N matrices with entries from some associati¨ e algebra and assume that they obey the equality y1 y1 y1 X 1 Ry1 21 N 2 R 12 s R 21 N 2 R 12 X 1 .

Ž 4.17.

Then the matrix Y defined by the relation K 12 Y2 [ K 12 X 2 PR12 fulfills an analogous equality, y1 y1 y1 Y1 Ry1 21 N 2 R 12 s R 21 N 2 R 12 Y1 .

Proof. The symbol tr1Ž?. stands for the trace taken in the first factor of a tensor product. Notice also the if A12 , B123 are two matrices obeying A12 B345 s B345 A12 Žthe entries commute. then t

t tr1 Ž K 12 At233 B123 . s tr1 Ž B123 At232 K 21 . t s tr1 Ž B123 P12 C1C2 A 23 K 21 . .

Ž 4.18.

87

SYMPLECTIC QUANTUM GROUPS

In the last equality we have used Ž2.18. and Ž2.12.. Furthermore, Ž4.17. can be rewritten as

y1 X 1 N 2t s Ž Ry1 21 N 2 R 12 X 1 R 12 .

t2

Ž Ry1 21 .

t 2 y1

.

Now, taking into account that tr1 K 12 s I 2 and applying successively the definition of Y, Ž4.17. and Ž2.10., Ž4.18., then Ž2.11., YB equation and Ž2.19., the definition of Y and YB equation, Ž2.19. and Ž2.12., and finally Ž2.18., we get

Ž Y2 Ry1 32 N 3 .

t3

s tr1 K 12 X 2 P12 R12 N 3t Ž Ry1 32 .

ž s tr ž K 1

12

t3

y1 Ž Ry1 32 N 3 R 23 X 2 R 23 .

/

t3

t C3t Ry1 32 P12 R 12 R 32 Ž C 3 .

y1 y1 y1 s tr1 Ž Cy1 3 R 23 R 21 R 13 C1 C 2 C 3 R 32 N 3 R 23 X 2 R 23 K 21 . y1 y1 s tr1 Ž C1C2 Ry1 31 R 12 N 3 R 23 R 13 X 2 K 21 . y1 y1 s tr1 Ž C1C2 Ry1 31 N 3 R 13 R 23 P12 Y2 K 21 . y1 y1 s tr1 Ž C1 Ry1 31 N 3 R 13 Y1 R 13 C1 K 21 . t

2 y1 s tr1 Ž Ry1 32 N 3 R 23 Y2 R 23 . K 21

ž

y1

/

t

t

t

t

t

/

t

3 y1 s Ž Ry1 B 32 N 3 R 23 Y2 R 23 . .

The construction of representations is based on the same idea as in the case of the series A l w15x. Owing to the morphism c : A˜d ª C , C is a left A˜d-module. Let I be the left ideal in C generated by the ‘‘holomorphic’’ I be the factor module. As a next step, one elements z jk and let CrI I with Cah as a vector space Žrecall that Cah is a unital should identify CrI subalgebra in C generated only by zUjk . and consider the cyclic submodule M with the cyclic vector 1 g Cah . Though the structure of relations defining the algebra C has not been clarified in full detail yet we shall I ' Cah as a hypothesis. It has been also accept the identification CrI confirmed by explicit calculations in the lowest rank cases. Nevertheless, further, we shall arrive at a quite unambiguous prescription for the action of A˜d Žand hence of Uh . on M ŽProposition 5.4.. In what follows, the central dot ‘‘?’’ stands for this action.

PAVEL ˇˇ STOVICEK ´ˇ

88 LEMMA 4.3.

I ' Cah it holds true that Assuming CrI

Ž I q ZZ* .

y1

?1se

Ž1 y m. z

qy2 rq1q e I.

Ž 4.19.

Consequently, ty? 1 s e

Ž1 y m. z

q Ž Ny2 rq1.r2 Ž N y 2 r q 1 . r2

Ž 4.20.

Žw x x s Ž q x y qyx .rŽ q y qy1 ... Ž . Ž . Proof. Temporarily set X st [ ŽI q ZZ*.y1 st ? 1. From 4.1 , 4.13 , and Ž2.3., one finds that for j, k G r, y1

Ž I q V *V . jk

y yr kyr j s Ž m I q h Qy . jk q e« j « k z Qyk q z Ž q y qysgnŽ jqk . . Qy , yj q jk

y ezd jk Ý Ž q sgnŽ kq s . y qy1 . q 2 rs Qy ss .

Ž 4.21.

s

y Ž .y1 Notice that Qy st ? 1 s 0, for t ) yr, Q st s I q ZZ* st , for s, t F yr, y1 and ŽI q V *V . ? 1 s I. Thus Ž4.21. applied on the unit yields Žwe set s s yk, t s yj; s, t F yr .

d st s md st q e« s « t z q r sqr t X st y ezd st

Ž q sgnŽ sys. y qy1 . q 2 r

s

Ý

sF s Fyr

Xss .

Ž 4.22. It follows that X st s 0, for s / t. Set x s [ q 2 r s X s s . One deduces from Ž4.22. that 1 s m q ez x s y ez

Ý

sF s Fyr

Ž q sgnŽ sys. y qy1 . x s .

The last equality amounts to a recurrent relation for x s leading immediately to the solution. As a consequence one finds that Qy? 1 s e

1ym

z

qy2 rq1q e

ž

I Z*

0 , 0

/

Ž 4.23.

and, owing to Ž4.13. and Ž3.17., Ž3.18., Qq? 1 s m I q e

Ž1 y m.h z

qy2 rq1q e

ž

I Z*

0 0 q Ž1 y m. 0 V*

/

ž

0 I

/

Ž 4.24.

89

SYMPLECTIC QUANTUM GROUPS

Žhere the dimensions of the zero and unit blocks vary and are determined implicitly.. Thus, in virtue of Ž4.2., one can evaluate M ? 1. Notice that M j j ? 1 s jq, for j G r. One expects that, moreover, M j j ? 1 s 1, for yr y1 j - r, and M j j ? 1 s jq , for j F yr. It is straightforward to check that this is actually the case provided

m s Ž 1 y j 0 . r Ž jqy j 0 . ,

j 0 s q Ny 2 rq1 .

Ž 4.25.

Here we have used the equations Ž4.16b., Ž4.16c. to express h and z . 5. Cah AS A LEFT A˜d-MODULE As a first step we shall rewrite the relation Ž4.3.. LEMMA 5.1. The relation adjoint to Ž4.3. is equi¨ alent to U y1 U y1 y y y1 I y Ž I y Ey 2 . R 12 Z 2 R 12 M 1 R 21 Ž E2 q Z 2 . s E2 M 1 R 21 . Ž 5.1 . U y1 . Furthermore, the matrix I y ŽI y Ey 2 R 12 Z 2 R 12 is in¨ ertible and it holds true that U y1 I y Ž I y Ey 2 . R 12 Z 2 R 12

y1

Ey 2

s R12 Ž E2yq ZU2 . R 21 y Ž q y qy1 . Ž E1y q ZU1 . R12 P12 Ž E1y q ZU1 . q Ž q y qy1 . e qy2 rq1q e R12 Ž E2y q ZU2 . K 21 P12 Ž E2y q ZU2 . R 21 .

Ž 5.2. Proof. The adjoint to Ž4.3. means that U y1 y Ž I y E2yy ZU2 . Ry1 12 M 1 R 21 Ž E2 q Z 2 . s 0

or, equivalently, U y1 U y y1 y1 y Ž I y E2yy ZU2 . Ry1 12 M 1 R 21 Z 2 s y Ž I y E2 y Z 2 . R 12 M 1 R 21 E2 .

. Multiplying this equality from the left by ŽI y Ey 2 R 12 or, in the opposite y1 . Ž Ž .. direction, by ŽI y Ey R and noting that cf. 2.22 2 12 "1 .1 s I y E2y Ž I y E2y . R12 Ž I y E2y . R12 y1 U . y1 U and ŽI y Ey 2 R 21 Z 2 s R 21 Z 2 one finds that it is equivalent to U y1 y1 U I y Ž I y Ey 2 . R 12 Z 2 R 12 M 1 R 21 Z 2 y1 y s y Ž I y E2y . Ž I y R12 ZU2 Ry1 12 . M 1 R 21 E2 .

From the last equality one can easily derive Ž5.1..

PAVEL ˇˇ STOVICEK ´ˇ

90

U y1 4 . x s 0 and so I y ŽI y It is easy to show that wŽI y Ey 2 R 12 Z 2 R 12 is invertible. This is clear also from the following calculation. Using Ž Z*. 2 s 0 one finds that U y1 . Ey 2 R 12 Z 2 R 12

k

ky1 U y1 U y1 y Ey Ž I y Ey Ž I y Ey 2 . R 12 Z 2 R 12 2 s Ž y1 . 2 . Ž R 12 Z 2 R 12 E2 .

k

Ž 5.3.

and consequently U y1 I y Ž I y Ey 2 . R 12 Z 2 R 12

y1

Ey 2

U y1 y U y1 y y s Ey 2 q Ž I y E2 . R 12 Z 2 R 12 E2 Ž I q R 12 Z 2 R 12 E2 . U U y1 y y1 y s R12 Ž Ey 2 q Z 2 . R 12 E2 Ž I q R 12 Z 2 R 12 E2 .

y1

y1

.

Ž 5.4.

Applying Ž5.3. in the reversed direction to the expression ŽI q y .y1 R12 ZU2 Ry1 occurring after the last equality sign in Ž5.4. one derives 12 E2 that U y1 I y Ž I y Ey 2 . R 12 Z 2 R 12

y1

Ey 2

U U y1 y1 y s R12 Ž Ey 2 q Z 2 . R 12 I y Ž I y E2 . R 12 Z 2 R 12

y1

Ey 2 . Ž 5.5 .

Replace Ry1 standing between the brackets on the RHS of Ž5.5. by 12 U R 21 y g P12 q g K 21. The result is Žusing E1y Ey 2 R 12 Z 2 s 0 in the second summand. y R12 Ž E2yq ZU2 . R 21 y g R12 Ž E2y q ZU2 . P12 Ž I q R12 ZU2 Ry1 12 . E2 U U y1 y q g R12 Ž Ey 2 q Z 2 . K 21 I y Ž I y E2 . R 12 Z 2 R 12

y1

Ey 2 .

Owing to Ž3.15. the second summands equals yg Ž E1y q ZU1 . R12 P12 = Ž E1y q ZU1 .. To complete the verification of Ž5.2. it remains to show that U y1 K 21 I y Ž I y Ey 2 . R 12 Z 2 R 12

y1

U y Ey 2 s pK 21 P12 Ž E2 q Z 2 . R 21 .

In virtue of Ž5.4. this is the same as U U y1 y y y K 21 R12 Ž E2y q ZU2 . Ry1 12 E2 s pK 21 P12 Ž E2 q Z 2 . R 21 Ž I q R 12 Z 2 R 12 E2 . . y Ž Since pK 21 P12 Ey cf. Ž2.24.. the last equality can be rewrit2 R 21 s K 21 E2 ten as y K 21 Ž I y E2y . R12 Ž E2yq ZU2 . Ry1 12 E2 y s pK 21 P12 ZU2 R 21 Ž I q R12 ZU2 Ry1 12 . E2 .

Ž 5.6.

91

SYMPLECTIC QUANTUM GROUPS

Using Ž2.25., ŽI y Ey. Z* s Z* and Ž2.24. one finds that the LHS of Ž5.6. equals y y U y1 y pK 21 P12 ZU2 Ry1 12 E2 y g K 21 P12 E1 Z 2 R 12 E2 U y1 y y y s pK 21 P12 ZU2 Ry1 12 E2 y g pK 21 E1 R 12 Z 2 R 12 E2 .

Ž . Substitute Ry1 12 q g P12 y g K 21 for R 21 on the RHS of 5.6 to get U U y1 y y pK 21 P12 ZU2 Ry1 12 E2 q g pK 21 Z 1 Ž I q R 12 Z 2 R 12 . E2 .

The second summand in the last expression can be simplified using Ž3.16. as U y y1 y g pK 21 Ž ZU1 y E1yR 12 ZU2 Ry1 12 y Z 1 R 12 E2 R 12 . E2 y s yg pK 21 E1yR 12 ZU2 Ry1 12 E2 .

This verifies Ž5.6. and hence Ž5.2. as well. We shall need the following identity. LEMMA 5.2.

It holds true that

U y1 I y Ž I y Ey 2 . R 12 Z 2 R 12

y1

U y1 y Ey 2 R 23 I y Ž I y E3 . R 13 Z 3 R 13

U y1 s R 23 I y Ž I y Ey 3 . R 13 Z 3 R 13

y1

Ey 3 R 32

y1

U y1 y =Ey 3 R 32 I y Ž I y E2 . R 12 Z 2 R 12

y1

Ey 2 .

Proof. Because of Lemma 5.1 and using the notation introduced in Section 3 we have to show that

Ž R12 X 2 R 21 y g X 1 R12 P12 X 1 q g pR12 X 2 K 21 P12 X 2 R 21 . R 23 = Ž R13 X 3 R 31 y g X 1 R13 P13 X 1 q g pR13 X 3 K 31 P13 X 3 R 31 . R 32 s R 23 Ž R13 X 3 R 31 y g X 1 R13 P13 X 1 q g pR13 X 3 K 31 P13 X 3 R 31 . =R 32 Ž R12 X 2 R 21 y g X 1 R12 P12 X 1 q g pR12 X 2 K 21 P12 X 2 R 21 . .

Ž 5.7. Unfortunately, I was not able to find a proof other than one based on a straightforward and rather tedious calculation. Below I confine myself to listing the intermediate equalities which altogether imply the identity Ž5.7.

PAVEL ˇˇ STOVICEK ´ˇ

92

and to giving hints just for several less obvious steps. It holds true that R12 X 2 R 21 R 23 R13 X 3 R 31 R 32 s R 23 R13 X 3 R 31 R 32 R12 X 2 R 21 ,

Ž 5.8.

R12 X 2 K 21 P12 X 2 R 21 R 23 X 1 R13 P13 X 1 R 32 s 0,

Ž 5.9a.

R 23 X 1 R13 P13 X 1 R 32 R12 X 2 K 21 P12 X 2 R 21 s 0,

Ž 5.9b.

X 1 R12 P12 X 1 R 23 R13 X 3 K 31 P13 X 3 R 31 R 32 s 0,

Ž 5.10a.

R 23 R13 X 3 K 31 P13 X 3 R 31 R 32 X 1 R12 P12 X 1 s 0,

Ž 5.10b.

X 1 R12 P12 X 1 R 23 R13 X 3 R 31 R 32 s R 23 R13 X 3 R 31 R 32 X 1 R12 P12 X 1 , Ž 5.11. R12 X 2 R 21 R 23 X 1 R13 P13 X 1 R 32 y R 23 X 1 R13 P13 X 1 R 32 R12 X 2 R 21 s g X 1 R12 R13 X 2 R 23 X 3 Ž R 21 P12 P13 y R 32 P13 P12 . ,

Ž 5.12a.

X 1 R12 P12 X 1 R 23 X 1 R13 P13 X 1 R 32 s X 1 R12 R13 X 2 R 23 X 3 R 21 P12 P13 ,

Ž 5.12b. R 23 X 1 R13 P13 X 1 R 32 X 1 R12 P12 X 1 s X 1 R12 R13 X 2 R 23 X 3 R 32 P13 P12 ,

Ž 5.12c. R12 X 2 K 21 P12 X 2 R 21 R 23 R13 X 3 R 31 R 32 s R 23 R13 X 3 R 31 R 32 R12 X 2 K 21 P12 X 2 R 21 .

Ž 5.13.

To obtain the last few equalities one can use that Y 2 s Y and so K 21 P12 X 2 R 21 R 23 R13 X 3 K 31 P13 s py2 K 21 Y 1 R 23 Y 1 K 13 s py1 K 21 R 23 R13 X 3 K 31 P13 s py1 K 21 X 3 K 31 P13 . Furthermore, observe that y y E2yEy 3 K 21 K 31 s E2 E 3 P 23 K 31 .

Ž 5.14.

In fact, one can readily check that even the equality K 21 K 31 s P23 K 31 holds true but Ž5.14. can be verified directly as follows y1 y y E2 E3 Ž Ry1 E2yEy 3 K 21 K 31 s g 12 y R 21 . K 31 y1 s gy1 E2yEy 3 Ž R 32 y R 23 . K 31

s E2yEy 3 P 23 K 31 .

93

SYMPLECTIC QUANTUM GROUPS

So finally we have R12 X 2 R 21 R 23 R13 X 3 K 31 P13 X 3 R 31 R 32 yR 23 R13 X 3 K 31 P13 X 3 R 31 R 32 R12 X 2 R 21 s yg P23 R13 X 3 K 31 X 2 X 1 R13 R12 P13 q g R 23 R13 X 3 K 31 X 2 X 1 R13 P12 P13 ,

Ž 5.15a.

pR12 X 2 K 21 P12 X 2 R 21 R 23 R13 X 3 K 31 P13 X 3 R 31 R 32 s P23 R13 X 3 K 31 X 2 X 1 R13 R12 P13 ,

Ž 5.15b.

pR 23 R13 X 3 K 31 P13 X 3 R 31 R 32 R12 X 2 K 21 P12 X 2 R 21 s R 23 R13 X 3 K 31 X 2 X 1 R13 P12 P13 .

Ž 5.15c.

The equalities Ž5.8., Ž5.9a., Ž5.9b., Ž5.11., Ž5.12a. ] Ž5.12c., Ž5.13., and Ž5.15a. ] Ž5.15c. imply Ž5.7.. Now we can equip Cah with the structure of a left A˜d-module. First we should specify the value of M ? 1. The result is suggested by Ž4.2., Ž4.23., and Ž4.24.. Accepting Ž4.25., the value will depend just on one parameter j ' jq/ 0. To extend the action to the whole algebra Cah one can use Lemma 5.1. Thus we postulate M ? 1 s I q Ž jy1 y 1 . Ž Eyq Z* . q Ž j y 1 . Ž Eqq V* . ,

Ž 5.16.

U y M1 Ry1 21 ? Ž E2 q Z 2 . f U y1 s I y Ž I y Ey 2 . R 12 Z 2 R 12

y1

y1 Ey ; f g Cah . 2 M1 R 21 ? f ,

Ž 5.17.

Before verifying that Ž5.16., Ž5.17. really define a left module structure let us show LEMMA 5.3. The relation Ž5.17. implies U q M1 Ry1 21 ? Ž E2 q V 2 . f U y1 q s Eq 2 I y R 12 V 2 R 12 Ž I y E2 .

y1

M1 Ry1 ; f g Cah . 21 ? f ,

Ž 5.18.

q. Ž The matrix I y R12 VU2 Ry1 is in¨ ertible and it holds true that 12 I y E2 U y1 q Eq 2 I y R 12 V 2 R 12 Ž I y E2 .

y1

U U q y1 y1 q s Ry1 . Ž E1q q VU1 . Ry1 21 Ž E2 q V 2 . R 12 q Ž q y q 21 P12 Ž E1 q V 1 . U U q q y1 y Ž q y qy1 . e q 2 ry1y e Ry1 21 Ž E2 q V 2 . K 21 P12 Ž E2 q V 2 . R 12 .

Ž 5.19.

PAVEL ˇˇ STOVICEK ´ˇ

94

q. Ž Proof. The matrix I y R12 VU2 Ry1 is again invertible since, as 12 I y E2 U y1 . is nilpotent. The one can easily check, the matrix R12 V 2 R12 ŽI y Eq 2 equality Ž5.19. can be verified in a way quite similar to that of relation Ž5.2. given in Lemma 5.1. We omit the details. Let us turn to the equality Ž5.18.. One can start from Žcf. Ž5.17.. y1 pM3 Ry1 23 ? X 2 fR 13 K 21 U y1 s p I y Ž I y Ey 2 . R 32 Z 2 R 32

y1

y1 y1 Ey 2 M3 ? fR 23 R 13 K 21 . Ž 5.20 .

The LHS equals y1 y1 y1 y1 M3 ? Ry1 23 R 13 R 12 Y 1 P12 K 21 f s R 12 M3 ? R 13 Y 1 R 13 P12 K 21

and so it is possible to rewrite Ž5.20. as M3 ? Ry1 23 Y 2 R 23 K 21 f U y1 s pP12 R12 I y Ž I y Ey 2 . R 32 Z 2 R 32

y1

Ey 2 K 21 M3 ? f . Ž 5.21 .

Next we take into account the identity Ž5.2.. The following equalities are straightforward to verify: y1 pP12 R12 R 32 X 2 R 23 K 21 s Ry1 23 Y 2 R 32 K 21 ,

pP12 R12 X 3 R 32 P23 X 3 K 21 s pX 3 P12 K 31 P23 X 3 , p 2 P12 R12 R 32 X 2 K 23 P23 X 2 R 23 K 21 s Y 3 Ry1 23 Y 2 K 31 K 21 . Now it is enough to take the trace in the first factor of Ž5.21. and to use tr1 K 12 s I 2 ,

tr1 Ž P12 K 31 . s K 32 ,

tr1 Ž K 31 K 21 . s P23 . Ž 5.22.

Observe also that y1 pX 3 K 32 P23 X 3 s py1 Ry1 23 Y 2 K 23 P23 Y 2 R 32 .

Ž 5.23.

This way one obtains M1 Ry1 21 ? Y 2 f y1 y1 y1 y1 s Ž Ry1 R 21 Y 2 K 21 P12 Y 2 Ry1 21 Y 2 R 12 q g Y 1 R 21 P12 Y 1 y g p 12 .

=M1 Ry1 21 ? f . The relation Ž5.18. then follows from Ž5.19..

95

SYMPLECTIC QUANTUM GROUPS

Now we can state PROPOSITION 5.4. The relations Ž5.16., Ž5.17. define unambiguously on Cah the structure of a left A˜d-module depending on one parameter j / 0. Proof. First we have to show that Ž5.16., Ž5.17. define correctly a linear mapping Cah ª MatŽ N . m Cah : f ¬ M ? f. Let Cah be the free algebra generated by the entries of the matrix Z* s Ž ZUjk . s Ž zUk j ., with yr - j, k F yr. Hence Cah is obtained from Cah by means of factorization by the two-sided ideal generated by the relation adjoint to Ž3.15. and zUk j are the factor images of zUk j . In the obvious notation, the matrix Z* is obtained from Z* when replacing all elements zUk j by zUk j . Set Žin this proof. X [ Eyq Z* and U y1 X 12 [ I y Ž I y Ey 2 . R 12 Z 2 R 12

y1

Ey 2 .

It is clear that when replacing the elements zUk j by zUk j everywhere in Ž5.16., Ž5.17. one obtains a well-defined linear mapping Cah ª Mat Ž N . m Cah : f ¬ M ? f .

Ž 5.24.

A straightforward calculation gives y1 M1 ? Ry1 21 R 31 X 2 R 23 X 3 R 32 y R 23 X 3 R 32 X 2 f

ž

/

y1 s X 12 R 23 X 13 R 32 y R 23 X 13 R 32 X 12 M1 ? Ry1 21 R 31 f ,

ž

/

; f g Cah .

This means that the linear mapping Ž5.24. factorizes from Cah to Cah if and only if the factor-image of the matrix X 12 R 23 X 13 R 32 y R 23 X 13 R 32 X 12 vanishes. But this is the content of Lemma 5.2. To show that Cah is really a left A˜d-module we have to verify the equality y1 y1 y1 M2 Ry1 12 M1 R 21 ? 1 s R 12 M1 R 21 M2 ? 1

Ž 5.25.

and the implication Žsome parentheses appear here and in what follows only for graphical reasons. y1 y1 y1 Ž M2 Ry1 12 M1 R 21 . R 31 R 32 ? f U y1 y1 y1 y1 y1 y1 y1 s Ž Ry1 12 M1 R 21 M2 . R 31 R 32 ? f« Ž M2 R 12 M1 R 21 . R 31 R 32 ? Z 3 f U y1 y1 y1 s Ž Ry1 ; f g Cah , Ž 5.26. 12 M1 R 21 M2 . R 31 R 32 ? Z 3 f ,

PAVEL ˇˇ STOVICEK ´ˇ

96

since then Ž5.25. and Ž5.26. jointly imply y1 y1 y1 M2 Ry1 12 M1 R 21 ? f s R 12 M1 R 21 M2 ? f ,

; f g Cah .

In this proof we set U y1 X 12 [ I y Ž I y Ey 2 . R 12 Z 2 R 12

y1

U y1 q Y12 [ Eq 2 I y R 12 V 2 R 12 Ž I y E2 .

Ey 2 , y1

.

Verification of Ž5.25.. Using Ž5.16., Ž5.17., and Ž5.18. one finds immediately that y1 y1 M2 Ry1 y 1 . X 21 q Ž j y 1 . Y21 . 12 M1 R 21 ? 1 s Ž I q Ž j y1 = Ž I q Ž jy1 y 1 . X 2 q Ž j y 1 . Y 2 . Ry1 12 R 21 , y1 y1 y1 Ry1 y 1 . X 12 q Ž j y 1 . Y12 . 12 M1 R 21 M2 ? 1 s R 12 Ž I q Ž j

= Ž I q Ž jy1 y 1 . X 1 q Ž j y 1 . Y 1 . Ry1 21 . This means that Ž5.25. holds for every j / 0 if and only if the following five equations are satisfied: y1 X 21 X 2 Ry1 12 s R 12 X 12 X 1 ,

Ž 5.27.

y1 X 21 Ž I y X 2 y Y 2 . Ry1 12 q Ž I y X 21 y Y21 . X 2 R 12 y1 s Ry1 Ž 5.28. 12 X 12 Ž I y X 1 y Y 1 . q R 12 Ž I y X 12 y Y12 . X 1 , y1 y1 Y21 X 2 Ry1 12 q Ž I y X 21 y Y21 . Ž I y X 2 y Y 2 . R 12 q X 21 Y 2 R 12 y1 y1 s Ry1 12 Y12 X 1 q R 12 Ž I y X 12 y Y12 . Ž I y X 1 y Y 1 . q R 12 X 12 Y 1 ,

Ž 5.29. y1 Y21 Ž I y X 2 y Y 2 . Ry1 12 q Ž I y X 21 y Y21 . Y 2 R 12 y1 s Ry1 12 Y12 Ž I y X 1 y Y 1 . q R 12 Ž I y X 12 y Y12 . Y 1 , y1 Y21 Y 2 Ry1 12 s R 12 Y12 Y 1 .

Ž 5.30. Ž 5.31.

The equations Ž5.27. ] Ž5.31. are not independent. In fact, it is enough to verify Ž5.27., Ž5.31. and the following two equations: y1 Ž X 21 qX 2 yX 21 Y 2 yY21 X 2 . Ry1 12 sR 12 Ž X 12 qX 1 yX 12 Y 1 yY12 X 1 . , Ž 5.32.

97

SYMPLECTIC QUANTUM GROUPS

y1 Ž Y21 q Y 2 yY21 X 2 yX 21 Y 2 . Ry1 12 sR 12 Ž Y12 qY 1 yY12 X 1 yX 12 Y 1 . . Ž 5.33.

This can be done by a straightforward calculation using Ž5.2. and Ž5.19.. We omit the details. Verification of Ž5.26.. Let us first show that y1 y1 y1 y1 y1 Ž I y X 3 . Ry1 23 R 13 Ž M2 R 12 M1 R 21 . R 31 R 32 ? X 3 f s 0.

Ž 5.34.

Actually, Ž5.17. means that Žcheck the proof of Lemma 5.1. y1 y1 M1 Ry1 21 ? X 2 f s R 12 X 2 R 12 M1 R 21 ? X 2 f

and so y1 y1 y1 Ž M2 Ry1 12 M1 R 21 . R 31 R 32 ? X 3 f y1 y1 y1 s M2 Ry1 12 R 32 M1 R 31 ? X 3 R 21 f y1 y1 y1 y1 s M2 ? Ž R13 Ry1 32 R 12 X 3 R 13 M1 R 31 ? X 3 R 21 f . y1 y1 y1 y1 y1 s R13 R 23 X 3 Ry1 23 M2 ? Ž R 32 X 3 R 12 R 13 M1 R 31 ? X 3 R 21 f . . y1 Now multiply both sides by ŽI y X 3 . Ry1 23 R 13 from the left and recall that ŽI y X . X s 0. Further one can proceed in a manner similar to the proof of Lemma 5.1. The relation Ž5.34. is equivalent to U y1 y1 y1 y1 y1 Ž I y E3yy ZU3 . Ry1 23 R 13 Ž M2 R 12 M1 R 21 . R 31 R 32 ? Z 3 f y1 y1 y1 y1 y1 y s y Ž I y E3yy ZU3 . Ry1 23 R 13 Ž M2 R 12 M1 R 21 . R 31 R 32 E3 ? f .

. Multiplying this equation by ŽI y Ey 3 R 13 R 23 from the left and taking into account that y1 y Ž I y E3y . R13 R 23 Ž I y E3y . Ry1 23 R 13 s I y E3 , y1 U y1 y1 U Ž I y E3y . Ry1 31 R 32 Z 3 s R 31 R 32 Z 3 ,

one arrives at U y1 y1 I y Ž I y Ey 3 . R 13 R 23 Z 3 R 23 R 13

U y1 y1 y1 Ž M2 Ry1 12 M1 R 21 . R 31 R 32 ? Z 3 f y1 s y Ž I y E3y . Ž I y R13 R 23 ZU3 Ry1 23 R 13 . y1 y1 y1 y = Ž M2 Ry1 Ž 5.35. 12 M1 R 21 . R 31 R 32 E3 ? f .

PAVEL ˇˇ STOVICEK ´ˇ

98 Quite similarly one obtains

U y1 y1 I y Ž I y Ey 3 . R 13 R 23 Z 3 R 23 R 13

U y1 y1 y1 Ž Ry1 12 M1 R 21 M2 . R 31 R 32 ? Z 3 f y1 syŽ IyE3y . Ž IyR13 R 23 ZU3 Ry1 23 R 13 . y1 y1 y1 y = Ž Ry1 Ž 5.36. 12 M1 R 21 M2 . R 31 R 32 E3 ? f .

The right-hand sides of Ž5.35. and Ž5.36. are equal by assumption and so, to complete the proof, it suffices to show that that the matrix I y ŽI y U y1 y1 . Ž . Ey 3 R 13 R 23 Z 3 R 23 R 13 is invertible. To this end, observe from 2.3 that the R matrix R12 is lower triangular provided the lexicographical ordering of the basis vectors in the tensor product has been chosen Ž R jk, st s 0 for j - s, and R jk, jt s d k t R jk, jt .. Hence in the tensor product C N m C N m C N , "1 "1 the matrix I y E3y is diagonal, the matrices R13 , R 23 are lower trianguU lar, and the matrix Z 3 is lower triangular with vanishing diagonal. ConseU y1 y1 . quently, the matrix ŽI y Ey 3 R 13 R 23 Z 3 R 23 R 13 is lower triangular with vanishing diagonal and hence nilpotent.

6. IRREDUCIBLE REPRESENTATIONS Set j s qy2 s . Observe that M ? 1 is a lower triangular matrix Žcf. Ž5.16... Since M s L*L and in view of the form of the isomorphism Ž2.29., Ž2.30., this means that Xy j ? 1 s 0, ; j, and so 1 is a lowest weight vector with a lowest weight l. Denote by Ms the cyclic A˜d-submodule in Cah with the cyclic vector 1. Owing to the isomorphism Ž2.29., Ž2.30., Ms can be also regarded as a Uh-module. It remains to determine the conditions implying that the module Ms is finite dimensional. This means to determine the parameter s . Recall that by the results due to Rosso w12x, Ms is determined unambiguously, up to equivalence, by the lowest weight l. In more detail, Mj j ? 1 s q 2 s ,

for j F yr ,

s 1, y2 s

sq

yr - j - r , ,

j G r.

Ž 6.1.

Consequently q H j ? 1 s qys , s 1,

for j s r , j / r,

Ž 6.2.

99

SYMPLECTIC QUANTUM GROUPS

in the case of Bl ; q H j ? 1 s qy2 s , s 1, q H j ? 1 s qy s ,

for j s 1, j ) 1,

5

1

for j s r q

2 1

s 1,

j/rq

q H j ? 1 s qy2 s ,

for j s 1,

if r s

1 2

,

¦

,

¥ if r ) 1 ,

§

2

,

2

Ž 6.3a.

Ž 6.3b.

in the case of Cl ;

s 1,

Ž 6.4a.

¥ if r s 3 ,

Ž 6.4b.

2

¦

for j s 1,

s qys ,

j s 2,

s 1,

j ) 2,

s 1,

1

,

j ) 1,

q H j ? 1 s qys ,

q H j ? 1 s qy s ,

5

if r s

§

for j s r q j/rq

1 2 1 2

2

¦

,

¥ if r ) 3 ,

§

,

2

Ž 6.4c.

in the case of D l . Let  v 1 , . . . , v l 4 ; h* be the set of fundamental weights corresponding to the set of simple roots P s  a 1 , . . . , a l 4 . Its values on the basis  H1 , . . . , Hl 4 of the Cartan algebra h are given by

v j Ž Hk . s 12 ² a k , a k :d jk . By our choice, specified in Section 2, we have the case Bl : ² a 1 , a 1 : s 1, ² a j , a j : s 2, j ) 1; the case Cl : ² a 1 , a 1 : s 4, ² a j , a j : s 2, j ) 1; the case D l : ² a j , a j : s 2, ; j.

PAVEL ˇˇ STOVICEK ´ˇ

100 Thus we have arrived at

Ms is a finite-dimensional irreducible Uh-module pro-

PROPOSITION 6.1. ¨ ided

sg

½

1 2

Zq , for r s 1; Zq , r ) 1;

then l s y2 sv 1 , l s ysv r ,

Ž 6.5.

then l s ysv rq1r2 ,

Ž 6.6.

in the case of Bl ;

s g Zq , for ; r ; in the case of Cl ;

¡Z

s g~

then l s y2 sv 1 ,

q,

for r s 12 ;

Zq ,

rs ;

l s ys Ž v 1 q v 2 . ,

q,

r) ;

l s ysv rq1r2 ,

1 2

¢Z

3 2 3 2

Ž 6.7.

in the case of D l . 7. CONCLUDING REMARKS Though the main goal, the explicit derivation of antiholomorphic representations in the Borel]Weil spirit, has been reached the presented results are not fully satisfactory. This concerns Section 4. The weakest point is that the commutation relations Ž4.3., Ž4.4. are apparently over-determined. In the ideal case one should extract from them a minimal set of commutation relations between Z1 and Z2U and show the rest to be a consequence; particularly this concerns the equality Ž4.4. or equivalently Ž4.13.. However, the semiclassical limit yields a formula which is not very encouraging in this respect. The semiclassical limit h ª 0 is based on the rules Ž q s eyh ; r is the classical r-matrix. R s I y i hr q O Ž h2 . ,

fg y gf s i h  f , g 4 q O Ž h 2 . .

In the classical case one replaces j 0 by 1, j " by j " 1, Qy class is formally parameterized in the same way as in the quantum case Ž4.1., and Qq class s t 0 . C 0 Ž Qy C . So class "1 .1 M class s I q Ž j " 1 y 1 . Qq y 1 . Qy class q Ž j class .

Applying the above rules to Ž4.3. one finds easily that

 M 1 ; Z2 4 s Ž I Z . 2 Ž r 21 M 1 q M 1 r 12 . yZ .

ž / I

2

SYMPLECTIC QUANTUM GROUPS

101

y 4 It is not difficult to obtain also the bracket  My1 1 ; Z 2 . Now, since Q class is expressible as a linear combination of the matrices I, M class , and My1 class , 4  U 4 one can calculate the bracket  Qy 1 ; Z 2 and hence Z1 ; Z 2 . There is no need to give the final expression but it is rather awkward. However, explicit calculations in the lowest rank cases, though not presented here, support the conjecture that the definition of the algebra C contains no contradiction. To get a more transparent picture, one should perhaps attempt to derive the quantum cell C as descendant from some simpler and more fundamental structure, like the Euclidean space, an idea pursued already in the paper w11x. Another approach notable in the current literature suggests that one can try to represent part of the algebra C with the help of quantum differential operators w1x.

REFERENCES 1. H. Awata, M. Nuomi, and S. Odake, Heisenberg realization for Uq Ž sl n . on the flag manifold, Lett. Math. Phys. 30 Ž1994., 35. 2. T. Brezinski ´ and S. Majid, Quantum group gauge theory on quantum spaces, Comm. Math. Phys. 157 Ž1993., 591. 3. L. Dabrowski and J. Sobczyk, Left regular representations and contractions of sl q Ž2. to e q Ž2., Lett. Math. Phys. 32 Ž1994., 249. 4. V. G. Drinfeld, Quantum groups, in ‘‘Proc. ICM Berkley, 1986,’’ p. 798, Amer. Math. Soc., Providence, 1987. 5. M. Jimbo, A q-difference analogue of UŽ g . and the Yang]Baxter equation, Lett. Math. Phys. 10 Ž1985., 63. 6. M. Jimbo, Quantum R-matrix for the generalized Toda system, Comm. Math. Phys. 102 Ž1986., 537. ˇ ´ˇ Quantum dressing orbits on compact groups, Comm. Math. 7. B. Jurco ˇ and P. ˇStovıcek, Phys. 152 Ž1993., 97. ˇ ´ˇ 8. B. Jurco Coherent states for quantum compact groups, ˇ and P. ˇStovıcek, CERNTH.7201r94, 1994, Comm. Math. Phys., to appear. 9. V. Lakshmibai and N. Reshetikhin, Quantum deformations of flag and Schubert schemes, C. R. Acad. Sci. Paris Ser. ´ I Math. 313 Ž1991., 121. 10. B. Parshall and J. Wang, ‘‘Quantum Linear Groups,’’ Amer. Math. Soc., Providence, 1991. 11. N. Yu. Reshetikhin, L. A. Takhtajan, and L. D. Faddeev, Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 Ž1990., 193. 12. M. Rosso, Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra, Comm. Math. Phys. 117 Ž1988., 581. 13. Y. Soibelman, On quantum flag manifolds, RIMS 780 Ž1991.. 14. Y. Soibelman, Orbit method for algebras of functions on quantum groups and coherent states, Internat. Math. Res. Notes 5 Ž1993., 151. ˇ ´ˇ Quantum Grassmann manifolds, Comm. Math. Phys. 158 Ž1993., 135. 15. P. ˇ Stovıcek, 16. E. Taft and J. Towber, Quantum deformation of flag schemes and Grassmann schemes. I. A q-deformation of the shape-algebra for GLŽ n., J. Algebra 142 Ž1991..