The Universal R-matrix for 𝔘h𝔟+

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 28 June 2011

The Universal R-matrix of 𝔘h𝔟+

1.1 Recall that D(𝔘h𝔟+) is the Hopf algebra given by generators X,H,H,Y with multiplication and comultiplication given by [ H , X ]=2X, [ H , Y ]= - h 2 Y, Δ(H)= H1+1H, Δ(X)= X e h 4 H + e - h 4 H X, Δ(H)= H1+1H, Δ(Y)= Y1+e-2HY. Let E= e h 4 H X . Then Δ(E)= 1E+ e h 2 H ,and E e h 2 H = e h 2 (H-2) e h 4 H X = e-h e h 2 H E, since XH=(H-2)X.

2.2 Recall that there is a grading on 𝔘h𝔟+ given by setting deg(H)=0 and deg(X)=1. This grading induces a grading on (𝔘h𝔟+) given by deg(H)=0 and deg(Y)=1. The inner product between 𝔘h𝔟+ and (𝔘h𝔟+) respects this grading. It follows that (H)r , Es = H , Δ(r) (E)s =0 for all r,s1 since the left hand component of the inner product has degree zero and the right hand component of the inner product has degree s. Using this observation we have that (H)r Yk , HrEk = (H)r Yk , Δ(H)r Δ(Ek) = (H)r Yk , (H1+1H) r ( 1E+ E e h 2 H ) k = r! (H)r Hk , (Hr1) (1Ek) = Yk , Ek . Let us evaluate this inner product Yk , Ek = YY , ( 1E+1E e h 2 H e h 2 H ) k = YY , πSk (e-h) l(π) E e h 2 H E ( e h 2 H ) 2 E ( e h 2 H ) k-1 E = YY , πSk (e-h) l(π) EEE = pSk (e-h) l(π) . Let q=eh2 and recall that πSk ql(π) = [ k ; q ]! where [ k ; q ]= (qk-1) (q-1) and [ k ; q ]!= [ k ; q ] [ k-1 ; q ] [ 1 ; q ] . Thus Yk , Ek = [ k ; q-2 ]!, and (H)r Yk , HrEk = r! [ k ; q-2 ]!. It follows easil by degree counts and arguments as above that (H)r Yk , HsEl =0, if rs or kl. Thus, the bases 1 r! [ k ; q-2 ] (H)rYk and HsEl are dual bases in (𝔘h𝔟+) and D ( 𝔘h𝔟+ ) respectively.

2.3 It follows that the universal R-matrix for D ( 𝔘h𝔟+ ) is given by R= r,k0 1 r! [ k ; q-2 ] HrEk (H)r Yk.

2.4 We shall write the universal R-matrix in terms of the generators Y, X,J1 and J2 so that we may project this R-matrix modulo the ideal in 𝔘h𝔟+ generated by J2 in order to get an R-matrix for 𝔘h𝔰𝔩(2) . Recall that in the notations of section 1 we had Y ˆ = YeH and Y= Y ˆ e h 2 - e - h 2 = Y ˆ q-q-1 . Since Y= H = ( H+h2 ) Y, it follows that Y e-H = e - (H+h2) Y , and we have Yk = ( Y e-H (q-q-1) ) k = (q-q-1)k Y e-H Y e-H Y e-H = (q-q-1)k e-kH (q-1) k 2 Y k. In a similar fashion, using the fact that E= e h 4 H X and XH=(H-2)X

we have that Ek = e h 4 H X e h 4 H X e h 4 H X = e k-1 2 h 2 e kh 4 H Xk. Now we may rewrite the R-matrix as follows R = r,k0 1 r! [ k ; q-2 ] HrEk (H)r Yk = r,k 1 r! [ k ; q-2 ] Hr q k-1 2 e kh 4 H Xk (H)r (q-q-1)k e-kH (q-1) k 2 Y k = r,k0 (q-q-1)k (q-1) k 2 q - k-1 2 r! [ k ; q-2 ] Hr e kh 4 H Xk (H)r e-kH Yk. Recall that H= h 4 (J1+J2) . Thus (H)r e-kH = ( h 4 (J1+J2) ) r e kh 4 (J1+J2) , Hr e kh 4 H = (J1-J2)r e kh 4 (J1-J2) . Now we may project modulo the ideal of 𝔘h𝔟+ generated by J2 and write the universal R-matrix for 𝔘h𝔰𝔩(2) in the form R = r,k0 (q-q-1)k (q-1) k 2 q - k-1 2 r! [ k ; q-2 ] Hr e kh 4 H Xk (H)r e-kH Yk = r,k0 (q-q-1)k (q-1) k 2 q - k-1 2 r! [ k ; q-2 ] J1r e kh 4 J1 Xk ( h 4 J1 ) r e - kh 4 J1 Yk = r,k0 (q-q-1)k (q-1) k 2 q - k-1 2 r! [ k ; q-2 ] exp ( h 4 (J1J1) + kh 4 ( J11+1J1 ) ) Xk Yk.

References

The fact that 𝔘h𝔰𝔩(2) is a quotient of the double of 𝔘h𝔟+ is stated in §13 of [D] along with the formula for the universal R-matrix of 𝔘h𝔰𝔩(2)

[Bou] N. Bourbaki, Lie groups and Lie algebras, Chapters I-III, Springer-Verlag, 1989.

[D] V.G. Drinfeld, Quantum Groups, Vol. 1 of Proccedings of the International Congress of Mathematicians (Berkeley, Calif., 1986). Amer. Math. Soc., Providence, RI, 1987, pp. 198–820. MR0934283

The theorem that a Lie bialgebra structure determinse a co-Poisson Hopf algebra structure on its enveloping algebra is due to Drinfel'd and appears as Theorem 1 in the following article.

[D1] V.G. Drinfel'd, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32 (1985), 254–258. MR0802128

Derivations of the double and the universal R-matrix for 𝔘h𝔰𝔩(n) appear in [R] and [B]. Our derivation of 𝔘h𝔰𝔩(2) follows the example given in [B].

[B] N. Burroughs, The universal R-matrix for Uqsl(3) and beyond!, Comm. Math. Phys. 127 (1990), 109–128. MR1036117

[R] M. Rosso, An analogue of P.B.W. theorem and the universal R-matrix for Uhsl(N+1) Uhsl(N+1)., Quantum groups (Clausthal, 1989), Comm. Math. Phys. 124 (1989), no. 2, 307–318. MR1012870

[DHL] H.-D. Doebner, Hennig, J. D. and W. Lücke, Mathematical guide to quantum groups, Quantum groups (Clausthal, 1989), Lecture Notes in Phys., 370, Springer, Berlin, 1990, pp. 29–63. MR1201823

There is a very readable and informative chapter on Lie algebra cohomolgy in the forthcoming book

[HGW] R. Howe, R. Goodman, and N. Wallach, Representations and Invariants of the Classical Groups, manuscript, 1993.

The following book is a standard introductory book on Lie algebras; a book which remains a standard reference. This book has been released for a very reasonable price by Dover publishers. There is a short description of Lie algebra cohomology in Chapt. III §11, pp.93-96.

[J] N. Jacobson, Lie algebras, Interscience Publishers, New York, 1962.

A very readable and complete text on Lie algebra cohomology is

[Kn] A. Knapp, Lie groups, Lie algebras and cohomology, Mathematical Notes 34, Princeton University Press, 1998. MR0938524

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