## Bethe subalgebras

Let $U$ be a Hopf algebra with an element $ℛ\in U\otimes U$ such that $\left(\Delta \otimes \mathrm{id}\right)\left(ℛ\right)={ℛ}_{13}{ℛ}_{23}$. The dual of $U$, $U*= {ℓ:U→ℂ | ℓis linear}, has product given by (ℓ1ℓ2) (a)=(ℓ1 ⊗ℓ2)Δ(a) .$ The function $Ψ: U* ⟶ U ℓ ⟼ (ℓ⊗id)(ℛ)$ is an algebra homomorphism since $Ψ(ℓ1 ℓ2) = (ℓ1 ℓ2⊗id) (ℛ) = (ℓ1⊗ ℓ2⊗id) (Δ⊗id) (ℛ) = (ℓ1⊗ ℓ2⊗id) (ℛ13ℛ23) = (ℓ1⊗id)(ℛ) (ℓ2⊗id)(ℛ) = Ψ(ℓ1) Ψ(ℓ2).$ Since ${C}_{0}=\left\{\ell \in {U}^{*}\phantom{\rule{.2em}{0ex}}|\phantom{\rule{.2em}{0ex}}\ell \left(xy\right)=\ell \left(yx\right)\right\}$ is a commutative subalgebra of ${U}^{*}$, the Bethe subalgebra of $U$, $Ψ(C0) = {(ℓ⊗id) (ℛ) | ℓ∈C0) }, where C0 = {ℓ∈U* | ℓ(xy) = ℓ(yx)},$ is a "large" commutative subalgebra of $U$.

## RTT algebras

Let $U$ be a Hopf algebra with an invertible element $ℛ=∑r ar⊗br ∈U⊗U such that ℛΔ(a) ℛ-1 =Δop(a),$ for $a\in U$. The dual ${U}^{*}$ of $U$ is a Hopf algebra. Fix a positive integer $n$ and an index set $\stackrel{^}{T}$. Let ${ ρλ:U→ Mn(ℂ) | λ∈T^ }$ be a set of representations of $U$. Their matrix entries $ρijλ :U→ℂ are elements of U*.$

On the ${\rho }_{ij}^{\lambda }$ the coproduct $\Delta :{U}^{*}\to {U}^{*}\otimes {U}^{*}$ has values $Δ ( ρijλ ) = ∑k=1n ρikλ ⊗ ρkjλ ,since ρijλ (u1u2) = ∑k=1n ρikλ (u1) ρkjλ (u2) ,$ for ${u}_{1},{u}_{2}\in U$. Let $ℛ(λ,μ) = (ρλ⊗ρμ ) (ℛ) and T(λ) =( ρijλ )$ so that $T\left(\lambda \right)$ is a matrix in ${M}_{n}\left({U}^{*}\right)$. Then $T(λ)⊗id = ∑i,j,k tijλ (Eij⊗ Ekk), id⊗T(μ) = ∑i,k,l tklμ (Eii⊗ Ekl),$ and $ℛ(λ,μ) = ∑i,j,k,l ρijλ (ar) ρklμ (br) (Eij⊗ Ekl).$

Since $ℛ(λ,μ) (T(λ)⊗id) (id⊗T(μ)) = ∑i,j,k,l ,x,y ρixλ (ar) txjλ ρkyμ (br) tylμ (Eil⊗ Ekl),$ and $(id⊗T(μ)) (T(λ)⊗id) ℛ(λ,μ) = ∑i,j,k,l ,α,β tkβμ tiαλ ραjλ (as) ρβlλ (bs),$ the equation $(id⊗T(μ)) (T(λ)⊗id) ℛ(λ,μ) = ℛ(λ,μ) (T(λ)⊗id) (id⊗T(μ))$ is a concise way of encoding the relations $( ∑x,y ρixλ (ar) ρkyμ (br) ρxjλ ρylμ ) (a) = ∑x,y,a ρixλ (ar) ρkyμ (br) ρxjλ (a(1)) ρylμ (a(2)) = ∑a ρijλ (ar a(1) ) ρklμ (br a(2) ) = ( ρijλ ⊗ ρklμ ) (ℛΔ(a) ) = ( ρijλ ⊗ ρklμ ) ( Δop(a) ℛ) = ∑a ρijλ ( a(2) as ) ρklμ ( a(1) bs ) = ∑α,β ρiαλ ( a(2) ) ραjλ (as) ρkβμ ( a(1) ) ρβlμ (bs) = ( ∑α,β ρkβμ ρiαλ ραjλ (as) ρβlμ (bs) ) (a)$ which are satisfied by the ${\rho }_{ij}^{\lambda }$ in ${U}^{*}$.

Let $B$ be the Hopf algebra given by $generators tijλ , 1≤i,j≤n, λ∈T^,$ and relations $(id⊗T(μ)) (T(λ)⊗id) ℛ(λ,μ) = ℛ(λ,μ) (T(λ)⊗id) (id⊗T(μ))$ with comultiplication given by $Δ( tijλ ) = ∑k=1n tikλ ⊗ tklλ .$ Then the map $B → U* tijλ ↦ ρijλ$ 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 $l↦(id⊗l) (ℛ) or l↦(id⊗l) (ℛ21-1) or l↦(id⊗l) (ℛ21ℛ) .$ In the case of the Yangian or ${U}_{q}𝔤$ the composition $\Phi :B\to {U}^{*}\to U$ is surjective and $\mathrm{ker}\Phi$ is generated by the elements of the center of $B$.

## RTT and the quantum double

Let $A$ be a Hopf algebra, $D(A) =A⊗A*coop the Drinfeld double.$ Let ${es} be a basis of A, {es} the dual basis in A* so that ℛ= ∑s es⊗ es .$ Let ${fst} be the basis of D(A)* dual to the basis {eset} of D(A).$ Let $𝒯= ∑s,t eset⊗ fts ∈D(A) ⊗D(A)*$ $ℒ+=∑s es⊗es =ℛ21 and ℒ-=∑s S(es)⊗es =ℛ-1 .$ Let $\pi :D\left(A\right)\to {M}_{n}\left(ℂ\right)$ be a representation of $D\left(A\right)$ and $R=(π⊗π) (ℛ), T=(π⊗id) (𝒯), L± =(π⊗id)( ℒ±)$ so that ${L}^{+}=\left(\pi \otimes \mathrm{id}\right)\left({ℛ}_{21}\right)$ and ${L}^{-}=\left(\pi \otimes \mathrm{id}\right)\left({ℛ}^{-1}\right)$.

## 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).