RTT algebras

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

Last updates: 25 November 2011

Bethe subalgebras

Let $U$ be a Hopf algebra with an element $ℛ\in U\otimes U$ such that $(\Delta \otimes \mathrm{id})\left(ℛ\right)={ℛ}_{13}{ℛ}_{23}$. The dual of $U$, $$U^{*}=\{\ell :U\to ℂ|\ell \text{is linear}\},\text{has product given by}({\ell }_1{\ell }_2\left)\right(a)=({\ell }_1\otimes {\ell }_2\left)\Delta \right(a).$$ The function $$\begin{array}{cccc}\Psi :& U^{*}& ⟶& U\\ & \ell & ⟼& (\ell \otimes \mathrm{id})\left(ℛ\right)\end{array}$$ is an algebra homomorphism since $$\begin{array}{rl}\Psi \left({\ell }_1{\ell }_2\right)& =({\ell }_1{\ell }_2\otimes \mathrm{id})\left(ℛ\right)=({\ell }_1\otimes {\ell }_2\otimes \mathrm{id})(\Delta \otimes \mathrm{id})\left(ℛ\right)\\ & =({\ell }_1\otimes {\ell }_2\otimes \mathrm{id})\left({ℛ}_{13}{ℛ}_{23}\right)=({\ell }_1\otimes \mathrm{id})\left(ℛ\right)({\ell }_2\otimes \mathrm{id})\left(ℛ\right)=\Psi \left({\ell }_1\right)\Psi \left({\ell }_2\right).\end{array}$$ Since $C_0=\{\ell \in U^{*}|\ell \left(xy\right)=\ell \left(yx\right)\}$ is a commutative subalgebra of $U^{*}$, the Bethe subalgebra of $U$, $$\Psi \left(C_0\right)=\left\{\right(\ell \otimes \mathrm{id}\left)\right(ℛ\left)\right|\ell \in C_0\left)\right\},\text{where}{C}_0=\{\ell \in U^{*}|\ell \left(xy\right)=\ell \left(yx\right)\},$$ is a "large" commutative subalgebra of $U$.

RTT algebras

Let $U$ be a Hopf algebra with an invertible element $$ℛ=\sum _r{a}_r\otimes b_r\in U\otimes U\text{such that}ℛ\Delta \left(a\right){ℛ}^{-1}={\Delta }^{\mathrm{op}}\left(a\right),$$ for $a\in U$. The dual $U^{*}$ of $U$ is a Hopf algebra. Fix a positive integer $n$ and an index set $\hat{T}$. Let $$\{{\rho }^{\lambda }:U\to M_n\left(ℂ\right)|\lambda \in \hat{T}\}$$ be a set of representations of $U$. Their matrix entries $${\rho }_{ij}^{\lambda }:U\to ℂ\text{are elements of}{U}^{*}.$$

On the ${\rho }_{ij}^{\lambda }$ the coproduct $\Delta :U^{*}\to U^{*}\otimes U^{*}$ has values $$\Delta \left({\rho }_{ij}^{\lambda }\right)=\sum _{k=1}^n{\rho }_{ik}^{\lambda }\otimes {\rho }_{kj}^{\lambda },\text{since}{\rho }_{ij}^{\lambda }\left(u_1{u}_2\right)=\sum _{k=1}^n{\rho }_{ik}^{\lambda }\left(u_1\right){\rho }_{kj}^{\lambda }\left(u_2\right),$$ for $u_1,u_2\in U$. Let $$ℛ(\lambda ,\mu )=({\rho }^{\lambda }\otimes {\rho }^{\mu })\left(ℛ\right)\text{and}T\left(\lambda \right)=\left({\rho }_{ij}^{\lambda }\right)$$ so that $T\left(\lambda \right)$ is a matrix in $M_n\left(U^{*}\right)$. Then $$T\left(\lambda \right)\otimes \mathrm{id}=\sum _{i,j,k}{t}_{ij}^{\lambda }(E_{ij}\otimes E_{kk}),\mathrm{id}\otimes T\left(\mu \right)=\sum _{i,k,l}{t}_{kl}^{\mu }(E_{ii}\otimes E_{kl}),$$ and $$ℛ(\lambda ,\mu )=\sum _{i,j,k,l}{\rho }_{ij}^{\lambda }\left(a_r\right){\rho }_{kl}^{\mu }\left(b_r\right)(E_{ij}\otimes E_{kl}).$$

Since $$ℛ(\lambda ,\mu )\left(T\right(\lambda )\otimes \mathrm{id})(\mathrm{id}\otimes T(\mu \left)\right)=\sum _{i,j,k,l,x,y}{\rho }_{ix}^{\lambda }\left(a_r\right)t_{xj}^{\lambda }{\rho }_{ky}^{\mu }\left(b_r\right)t_{yl}^{\mu }(E_{il}\otimes E_{kl}),$$ and $$(\mathrm{id}\otimes T(\mu \left)\right)\left(T\right(\lambda )\otimes \mathrm{id})ℛ(\lambda ,\mu )=\sum _{i,j,k,l,\alpha ,\beta }{t}_{k\beta }^{\mu }{t}_{i\alpha }^{\lambda }{\rho }_{\alpha j}^{\lambda }\left(a_s\right){\rho }_{\beta l}^{\lambda }\left(b_s\right),$$ the equation $$(\mathrm{id}\otimes T(\mu \left)\right)\left(T\right(\lambda )\otimes \mathrm{id})ℛ(\lambda ,\mu )=ℛ(\lambda ,\mu )\left(T\right(\lambda )\otimes \mathrm{id})(\mathrm{id}\otimes T(\mu \left)\right)$$ is a concise way of encoding the relations $$\begin{array}{rcl}\left(\sum _{x,y}{\rho }_{ix}^{\lambda }\left(a_r\right){\rho }_{ky}^{\mu }\left(b_r\right){\rho }_{xj}^{\lambda }{\rho }_{yl}^{\mu }\right)\left(a\right)& =& \sum _{x,y,a}{\rho }_{ix}^{\lambda }\left(a_r\right){\rho }_{ky}^{\mu }\left(b_r\right){\rho }_{xj}^{\lambda }\left(a_{\left(1\right)}\right){\rho }_{yl}^{\mu }\left(a_{\left(2\right)}\right)\\ & =& \sum _a{\rho }_{ij}^{\lambda }\left(a_r{a}_{\left(1\right)}\right){\rho }_{kl}^{\mu }\left(b_r{a}_{\left(2\right)}\right)\\ & =& ({\rho }_{ij}^{\lambda }\otimes {\rho }_{kl}^{\mu })\left(ℛ\Delta \left(a\right)\right)\\ & =& ({\rho }_{ij}^{\lambda }\otimes {\rho }_{kl}^{\mu })\left({\Delta }^{\mathrm{op}}\left(a\right)ℛ\right)\\ & =& \sum _a{\rho }_{ij}^{\lambda }\left(a_{\left(2\right)}{a}_s\right){\rho }_{kl}^{\mu }\left(a_{\left(1\right)}{b}_s\right)\\ & =& \sum _{\alpha ,\beta }{\rho }_{i\alpha }^{\lambda }\left(a_{\left(2\right)}\right){\rho }_{\alpha j}^{\lambda }\left(a_s\right){\rho }_{k\beta }^{\mu }\left(a_{\left(1\right)}\right){\rho }_{\beta l}^{\mu }\left(b_s\right)\\ & =& \left(\sum _{\alpha ,\beta }{\rho }_{k\beta }^{\mu }{\rho }_{i\alpha }^{\lambda }{\rho }_{\alpha j}^{\lambda }\left(a_s\right){\rho }_{\beta l}^{\mu }\left(b_s\right)\right)\left(a\right)\end{array}$$ which are satisfied by the ${\rho }_{ij}^{\lambda }$ in $U^{*}$.

Let $B$ be the Hopf algebra given by $$\text{generators}{t}_{ij}^{\lambda },1\le i,j\le n,\lambda \in \hat{T},$$ and relations $$(\mathrm{id}\otimes T(\mu \left)\right)\left(T\right(\lambda )\otimes \mathrm{id})ℛ(\lambda ,\mu )=ℛ(\lambda ,\mu )\left(T\right(\lambda )\otimes \mathrm{id})(\mathrm{id}\otimes T(\mu \left)\right)$$ with comultiplication given by $$\Delta \left(t_{ij}^{\lambda }\right)=\sum _{k=1}^n{t}_{ik}^{\lambda }\otimes t_{kl}^{\lambda }.$$ Then the map $$\begin{array}{rcl}B& \to & U^{*}\\ t_{ij}^{\lambda }& \mapsto & {\rho }_{ij}^{\lambda }\end{array}$$ is a Hopf algebra homomorphism.

We really want a map $B\to U$, not $B\to U^{*}$. But it is "easy" to make maps $U^{*}\to U$. For example, one can construct a map $U^{*}\to U$ by $$l\mapsto (\mathrm{id}\otimes l)\left(ℛ\right)\text{or}l\mapsto (\mathrm{id}\otimes l)\left({ℛ}_{21}^{-1}\right)\text{or}l\mapsto (\mathrm{id}\otimes l)\left({ℛ}_{21}ℛ\right).$$ In the case of the Yangian or $U_q𝔤$ the composition $\Phi :B\to U^{*}\to U$ is surjective and $\ker \Phi$ is generated by the elements of the center of $B$.

RTT and the quantum double

Let $A$ be a Hopf algebra, $$D\left(A\right)=A\otimes A^{*\mathrm{coop}}\text{the Drinfeld double.}$$ Let $$\left\{e_s\right\}\text{be a basis of}A,\left\{e^s\right\}\text{the dual basis in}{A}^{*}\text{so that}ℛ=\sum _s{e}_s\otimes e^s.$$ Let $$\left\{f_s^t\right\}\text{be the basis of}{D\left(A\right)}^{*}\text{dual to the basis}\left\{e_s{e}^t\right\}\text{of}D\left(A\right).$$ Let $$𝒯=\sum _{s,t}{e}_s{e}^t\otimes f_t^s\in D\left(A\right)\otimes {D\left(A\right)}^{*}$$ $${ℒ}^{+}=\sum _s{e}^s\otimes e_s={ℛ}_{21}\text{and}{ℒ}^{-}=\sum _{s}S\left(e_s\right)\otimes e^s={ℛ}^{-1}.$$ Let $\pi :D\left(A\right)\to M_n\left(ℂ\right)$ be a representation of $D\left(A\right)$ and $$R=(\pi \otimes \pi )\left(ℛ\right),T=(\pi \otimes \mathrm{id})\left(𝒯\right),L^{\pm }=(\pi \otimes \mathrm{id})\left({ℒ}^{\pm }\right)$$ so that $L^{+}=(\pi \otimes \mathrm{id})\left({ℛ}_{21}\right)$ and $L^{-}=(\pi \otimes \mathrm{id})\left({ℛ}^{-1}\right)$.

Notes and References

These notes are an attempt to work out and exposit a portion of the material in [D, last paragraph of section 10, section 11, and the last paragraph of section 12]. These notes are retyped and extended version of part of the notes at http://researchers.ms.unimelb.edu.au/~aram@unimelb/Notes2005/ygnsBonn2.08.05.pdf which were written in collaboration with N. Rojkovskaia. The section § 3 RTT algebras and the quantum double is taken from [RTF, § 3].

References

[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

[Dr] V.G. Drinfelʹd, Almost cocommutative Hopf algebras, (Russian) Algebra i Analiz 1 (1989), 30–46; translation in Leningrad Math. J. 1 (1990), 321–342. MR1025154

[LR] R. Leduc and A. Ram, A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras: The Brauer, Birman-Wenzl and type A Iwahori-Hecke algebras, Adv. Math. 125 (1997), 1-94. MR1427801.

[RTF] N. Yu. Reshetikhin, L.A. Takhtadzhyan and L.D. Faddeev, Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 193-225.

[Re] N. Yu. Reshetikhin, Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I, LOMI preprint no. E–4–87, (1987).

page history