The Universal <math><mi>R</mi></math>-matrix

The Universal $R -matrix for$ $𝔘_{h} 𝔟^{+}$

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 $R -matrix of$ $𝔘_{h} 𝔟^{+}$

1.1 Recall that $D (𝔘_{h} 𝔟^{+})$ is the Hopf algebra given by generators $X, H, H^{*}, Y$ with multiplication and comultiplication given by $\begin{matrix} [H, X] = 2 X, \\ [H^{*}, Y] = - \frac{h}{2} Y, \\ Δ (H) = H \otimes 1 + 1 \otimes H, Δ (X) = X \otimes e^{\frac{h}{4} H} + e^{- \frac{h}{4} H} \otimes X, \\ Δ (H^{*}) = H^{*} \otimes 1 + 1 \otimes H^{*}, Δ (Y) = Y \otimes 1 + e^{- 2 H^{*}} \otimes Y . \end{matrix}$ Let $E = e^{\frac{h}{4} H} X$ . Then $\begin{matrix} Δ (E) = 1 \otimes E + e^{\frac{h}{2} H}, and \\ E e^{\frac{h}{2} H} = e^{\frac{h}{2} (H - 2)} e^{\frac{h}{4} H} X = e^{- h} e^{\frac{h}{2} H} E, \end{matrix}$ since $X H = (H - 2) X$ .

2.2 Recall that there is a grading on $𝔘_{h} 𝔟^{+}$ given by setting $deg (H) = 0$ and $deg (X) = 1$ . This grading induces a grading on ${(𝔘_{h} 𝔟^{+})}^{*}$ given by $deg (H^{*}) = 0$ and $deg (Y) = 1$ . The inner product between $𝔘_{h} 𝔟^{+}$ and ${(𝔘_{h} 𝔟^{+})}^{*}$ respects this grading. It follows that $⟨ {(H^{*})}^{r}, E^{s} ⟩ = ⟨ H^{*} \otimes \dots^{*}, Δ^{(r)} {(E)}^{s} ⟩ = 0$ for all $r, s \geq 1$ 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 $\begin{array}{rcl} ⟨ {(H^{*})}^{r} Y^{k}, H^{r} E^{k} ⟩ & = & ⟨ {(H^{*})}^{r} \otimes Y^{k}, Δ {(H^{*})}^{r} Δ (E^{k}) ⟩ \\ = & ⟨ {(H^{*})}^{r} \otimes Y^{k}, {(H \otimes 1 + 1 \otimes H)}^{r} {(1 \otimes E + E \otimes e^{\frac{h}{2} H})}^{k} ⟩ \\ = & r! ⟨ {(H^{*})}^{r} \otimes H^{k}, (H^{r} \otimes 1) (1 \otimes E^{k}) ⟩ \\ = & ⟨ Y^{k}, E^{k} ⟩ . \end{array}$ Let us evaluate this inner product $\begin{array}{rcl} ⟨ Y^{k}, E^{k} ⟩ & = & ⟨ Y \otimes \dots \otimes Y, {(1 \otimes \dots E + 1 \otimes \dots \otimes E \otimes e^{\frac{h}{2} H} \otimes \dots \otimes e^{\frac{h}{2} H})}^{k} ⟩ \\ = & ⟨ Y \otimes \dots \otimes Y, \sum_{π \in S_{k}} {(e^{- h})}^{l (π)} E \otimes e^{\frac{h}{2} H} E \otimes {(e^{\frac{h}{2} H})}^{2} E \otimes \dots \otimes {(e^{\frac{h}{2} H})}^{k - 1} E ⟩ \\ = & ⟨ Y \otimes \dots \otimes Y, \sum_{π \in S_{k}} {(e^{- h})}^{l (π)} E \otimes E \otimes \dots \otimes E ⟩ \\ = & \sum_{p \in S_{k}} {(e^{- h})}^{l (π)} . \end{array}$ Let $q = e^{h ⁄ 2}$ and recall that $\sum_{π \in S_{k}} q^{l (π)} = [k; q]!$ where $[k; q] = (q^{k} - 1) ⁄ (q - 1)$ and $[k; q]! = [k; q] [k - 1; q] \dots [1; q]$ . Thus $\begin{matrix} ⟨ Y^{k}, E^{k} ⟩ = [k; q^{- 2}]!, and \\ ⟨ {(H^{*})}^{r} Y^{k}, H^{r} E^{k} ⟩ = r! [k; q^{- 2}]! . \end{matrix}$ It follows easil by degree counts and arguments as above that $⟨ {(H^{*})}^{r} Y^{k}, H^{s} E^{l} ⟩ = 0, if r \neq s or k \neq l .$ Thus, the bases $\frac{1}{r! [k; q^{- 2}]} {(H^{*})}^{r} Y^{k} and H^{s} E^{l}$ are dual bases in ${(𝔘_{h} 𝔟^{+})}^{*}$ and $D (𝔘_{h} 𝔟^{+})$ respectively.

2.3 It follows that the universal $R$ -matrix for $D (𝔘_{h} 𝔟^{+})$ is given by $R = \sum_{r, k \geq 0} \frac{1}{r! [k; q^{- 2}]} H^{r} E^{k} \otimes {(H^{*})}^{r} Y^{k} .$

2.4 We shall write the universal $R$ -matrix in terms of the generators $\overline{Y}, X, J_{1}$ 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 $𝔘_{h} 𝔰 𝔩 (2)$ . Recall that in the notations of section 1 we had $\hat{Y} = Y e^{H^{*}} and \overline{Y} = \frac{\hat{Y}}{e^{\frac{h}{2}} - e^{- \frac{h}{2}}} = \frac{\hat{Y}}{q - q^{- 1}} .$ Since $\overline{Y} = H^{*} = (H^{*} + h ⁄ 2) \overline{Y},$ it follows that $\overline{Y} e^{- H^{*}} = e^{- (H^{*} + h ⁄ 2)} Y$ , and we have $\begin{array}{rcl} Y^{k} & = & {(\overline{Y} e^{- H^{*}} (q - q^{- 1}))}^{k} \\ = & {(q - q^{- 1})}^{k} \overline{Y} e^{- H^{*}} \overline{Y} e^{- H^{*}} \dots \overline{Y} e^{- H^{*}} \\ = & {(q - q^{- 1})}^{k} e^{- k H^{*}} {(q^{- 1})}^{(\binom{k}{2})} {\overline{Y}}^{k} . \end{array}$ In a similar fashion, using the fact that $E = e^{\frac{h}{4} H} X$ and $X H = (H - 2) X$

we have that

\begin{array}{rcl} E^{k} & = & e^{\frac{h}{4} H} X e^{\frac{h}{4} H} X \dots e^{\frac{h}{4} H} X \\ = & e^{(\binom{k - 1}{2}) \frac{h}{2}} e^{\frac{k h}{4} H} X^{k} . \end{array}

Now we may rewrite the

R

-matrix as follows

\begin{array}{rcl} R & = & \sum_{r, k \geq 0} \frac{1}{r! [k; q^{- 2}]} H^{r} E^{k} \otimes {(H^{*})}^{r} Y^{k} \\ = & \sum_{r, k} \frac{1}{r! [k; q^{- 2}]} H^{r} q^{(\binom{k - 1}{2})} e^{\frac{k h}{4} H} X^{k} \otimes {(H^{*})}^{r} {(q - q^{- 1})}^{k} e^{- k H^{*}} {(q^{- 1})}^{(\binom{k}{2})} {\overline{Y}}^{k} \\ = & \sum_{r, k \geq 0} \frac{{(q - q^{- 1})}^{k} {(q^{- 1})}^{(\binom{k}{2})} q^{- (\binom{k - 1}{2})}}{r! [k; q^{- 2}]} H^{r} e^{\frac{k h}{4} H} X^{k} \otimes {(H^{*})}^{r} e^{- k H^{*}} {\overline{Y}}^{k} . \end{array}

Recall that

H^{*} = \frac{h}{4} (J_{1} + J_{2})

. Thus

\begin{array}{rcl} {(H^{*})}^{r} e^{- k H^{*}} & = & {(\frac{h}{4} (J_{1} + J_{2}))}^{r} e^{\frac{k h}{4} (J_{1} + J_{2})}, \\ H^{r} e^{\frac{k h}{4} H} & = & {(J_{1} - J_{2})}^{r} e^{\frac{k h}{4} (J_{1} - J_{2})} . \end{array}

Now we may project modulo the ideal of

𝔘_{h} 𝔟^{+}

generated by

J_{2}

and write the universal

R

-matrix for

𝔘_{h} 𝔰 𝔩 (2)

in the form

\begin{array}{rcl} R & = & \sum_{r, k \geq 0} \frac{{(q - q^{- 1})}^{k} {(q^{- 1})}^{(\binom{k}{2})} q^{- (\binom{k - 1}{2})}}{r! [k; q^{- 2}]} H^{r} e^{\frac{k h}{4} H} X^{k} \otimes {(H^{*})}^{r} e^{- k H^{*}} {\overline{Y}}^{k} \\ = & \sum_{r, k \geq 0} \frac{{(q - q^{- 1})}^{k} {(q^{- 1})}^{(\binom{k}{2})} q^{- (\binom{k - 1}{2})}}{r! [k; q^{- 2}]} J_{1}^{r} e^{\frac{k h}{4} J_{1}} X^{k} \otimes {(\frac{h}{4} J_{1})}^{r} e^{- \frac{k h}{4} J_{1}} {\overline{Y}}^{k} \\ = & \sum_{r, k \geq 0} \frac{{(q - q^{- 1})}^{k} {(q^{- 1})}^{(\binom{k}{2})} q^{- (\binom{k - 1}{2})}}{r! [k; q^{- 2}]} exp (\frac{h}{4} (J_{1} \otimes J_{1}) + \frac{k h}{4} (J_{1} \otimes 1 + 1 \otimes J_{1})) X^{k} \otimes {\overline{Y}}^{k} . \end{array}

References

The fact that

𝔘_{h} 𝔰 𝔩 (2)

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} 𝔰 𝔩 (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 $𝔘_{h} 𝔰 𝔩 (n)$ appear in [R] and [B]. Our derivation of $𝔘_{h} 𝔰 𝔩 (2)$ follows the example given in [B].

[B] N. Burroughs, The universal R-matrix for $U_{q} s l (3)$ 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 $U_{h} s l (N + 1)$ 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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The Universal R-matrix for 𝔘h𝔟+

The Universal R-matrix of 𝔘h𝔟+

References

The Universal $R -matrix for$ $𝔘_{h} 𝔟^{+}$

The Universal $R -matrix of$ $𝔘_{h} 𝔟^{+}$