## The Universal $R-matrix of$${𝔘}_{h}{𝔟}^{+}$

1.1 Recall that $D\left({𝔘}_{h}{𝔟}^{+}\right)$ is the Hopf algebra given by generators $X,H,{H}^{\ast },Y$ with multiplication and comultiplication given by $[ H , X ]=2X, [ H∗ , Y ]= - h 2 Y, Δ(H)= H⊗1+1⊗H, Δ(X)= X⊗ e h 4 H + e - h 4 H ⊗X, Δ(H∗)= H∗⊗1+1⊗H∗, Δ(Y)= Y⊗1+e-2H∗⊗Y.$ Let $E={e}^{\frac{h}{4}H}X$. Then $Δ(E)= 1⊗E+ e h 2 H ,and E e h 2 H = e h 2 (H-2) e h 4 H X = e-h e h 2 H E,$ since $XH=\left(H-2\right)X$.

2.2 Recall that there is a grading on ${𝔘}_{h}{𝔟}^{+}$ given by setting $deg\left(H\right)=0$ and $deg\left(X\right)=1$. This grading induces a grading on ${\left({𝔘}_{h}{𝔟}^{+}\right)}^{\ast }$ given by $deg\left({H}^{\ast }\right)=0$ and $deg\left(Y\right)=1$. The inner product between ${𝔘}_{h}{𝔟}^{+}$ and ${\left({𝔘}_{h}{𝔟}^{+}\right)}^{\ast }$ respects this grading. It follows that $⟨ (H∗)r , Es ⟩= ⟨ H∗⊗⋯∗ , Δ(r) (E)s ⟩=0$ for all $r,s\ge 1$ 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 $⟨ (H∗)r Yk , HrEk ⟩ = ⟨ (H∗)r ⊗Yk , Δ(H∗)r Δ(Ek) ⟩ = ⟨ (H∗)r ⊗Yk , (H⊗1+1⊗H) r ( 1⊗E+ E⊗ e h 2 H ) k ⟩ = r! ⟨ (H∗)r ⊗Hk , (Hr⊗1) (1⊗Ek) ⟩ = ⟨ Yk , Ek ⟩ .$ Let us evaluate this inner product $⟨ Yk , Ek ⟩ = ⟨ Y⊗⋯⊗Y , ( 1⊗⋯E+1⊗⋯⊗E ⊗ e h 2 H ⊗⋯⊗ e h 2 H ) k ⟩ = ⟨ Y⊗⋯⊗Y , ∑ π∈Sk (e-h) l(π) E ⊗ e h 2 H E ⊗ ( e h 2 H ) 2 E ⊗⋯ ⊗ ( e h 2 H ) k-1 E ⟩ = ⟨ Y⊗⋯⊗Y , ∑ π∈Sk (e-h) l(π) E⊗E⊗⋯⊗E ⟩ = ∑ p∈Sk (e-h) l(π) .$ Let $q={e}^{h⁄2}$ and recall that $∑ π∈Sk ql(π) = [ k ; q ]!$ where $\left[k;q\right]=\left({q}^{k}-1\right)⁄\left(q-1\right)$ and $\left[k;q\right]!=\left[k;q\right]\left[k-1;q\right]\cdots \left[1;q\right]$. 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 Thus, the bases $1 r! [ k ; q-2 ] (H∗)rYk and HsEl$ are dual bases in ${\left({𝔘}_{h}{𝔟}^{+}\right)}^{\ast }$ and $D\left({𝔘}_{h}{𝔟}^{+}\right)$ respectively.

2.3 It follows that the universal $R$-matrix for $D\left({𝔘}_{h}{𝔟}^{+}\right)$ is given by $R= ∑ r,k≥0 1 r! [ k ; q-2 ] HrEk⊗ (H∗)r Yk.$

2.4 We shall write the universal $R$-matrix in terms of the generators $\stackrel{}{Y},X,{J}_{1}$ and ${J}_{2}$ so that we may project this $R$-matrix modulo the ideal in ${𝔘}_{h}{𝔟}^{+}$ generated by ${J}_{2}$ in order to get an $R$-matrix for ${𝔘}_{h}𝔰𝔩\left(2\right)$. 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 $\stackrel{}{Y}={H}^{\ast }=\left({H}^{\ast }+h⁄2\right)\stackrel{}{Y},$ it follows that $\stackrel{}{Y}{e}^{-{H}^{\ast }}={e}^{-\left({H}^{\ast }+h⁄2\right)}Y$, 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 $E={e}^{\frac{h}{4}H}X$ and $XH=\left(H-2\right)X$

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 $R = ∑ r,k≥0 1 r! [ k ; q-2 ] HrEk⊗ (H∗)r Yk = ∑ r,k 1 r! [ k ; q-2 ] Hr q k-1 2 e kh 4 H Xk⊗ (H∗)r (q-q-1)k e-kH∗ (q-1) k 2 Y ‾ k = ∑ r,k≥0 (q-q-1)k (q-1) k 2 q - k-1 2 r! [ k ; q-2 ] Hr e kh 4 H Xk⊗ (H∗)r e-kH∗ Y ‾ k.$ Recall that ${H}^{\ast }=\frac{h}{4}\left({J}_{1}+{J}_{2}\right)$. 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 ${𝔘}_{h}{𝔟}^{+}$ generated by ${J}_{2}$ and write the universal $R$-matrix for ${𝔘}_{h}𝔰𝔩\left(2\right)$ in the form $R = ∑ r,k≥0 (q-q-1)k (q-1) k 2 q - k-1 2 r! [ k ; q-2 ] Hr e kh 4 H Xk⊗ (H∗)r e-kH∗ Y ‾ k = ∑ r,k≥0 (q-q-1)k (q-1) k 2 q - k-1 2 r! [ k ; q-2 ] J1r e kh 4 J1 Xk ⊗ ( h 4 J1 ) r e - kh 4 J1 Y ‾ k = ∑ r,k≥0 (q-q-1)k (q-1) k 2 q - k-1 2 r! [ k ; q-2 ] exp ( h 4 (J1⊗J1) + kh 4 ( J1⊗1+1⊗J1 ) ) Xk⊗ Y ‾ k.$

## References

The fact that ${𝔘}_{h}𝔰𝔩\left(2\right)$ is a quotient of the double of ${𝔘}_{h}{𝔟}^{+}$ is stated in §13 of [D] along with the formula for the universal $R$-matrix of ${𝔘}_{h}𝔰𝔩\left(2\right)$

[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 ${𝔘}_{h}𝔰𝔩\left(n\right)$ appear in [R] and [B]. Our derivation of ${𝔘}_{h}𝔰𝔩\left(2\right)$ follows the example given in [B].

[B] N. Burroughs, The universal R-matrix for ${U}_{q}sl\left(3\right)$ 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 ${U}_{h}sl\left(N+1\right)$ 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