## The Hopf algebras ${\stackrel{˜}{U}}^{\ge 0}$ and ${\stackrel{˜}{U}}^{\le 0}$

1.1 Let $I$ be an infinite set and let $Q={\sum }_{i\in I}ℤi$ be the free abelian group generated by the set $I$. Let ${Q}^{\vee }$ be the free abelian group generated by the set ${I}^{\vee }$. Let $⟨,⟩$ be a $ℤ$ valued bilineare pairing between $Q$ and ${Q}^{\vee }$.

1.2 Let ${\stackrel{˜}{U}}^{\ge 0}$ be the Hopf algebra over $ℚ\left(q\right)$ with generators $Ei, i∈I, Kμ, μ∈Q,$ and relations $K0=1, KλKμ= Kλ+μ, KμEi= q ⟨ μ , i′ ⟩ EiKμ, i∈I,μ∈Q.$ and with coproduct given by $ΔKμ= Kμ⊗Kμ, ΔEi= Ei⊗1+Ki⊗Ei.$

1.3 ${\stackrel{˜}{U}}^{\le 0}$ be the Hopf algebra over $ℚ\left(q\right)$ with generators $Fi,i∈I, K μ ∗ , μ∈Q,$ and relations $K 0 ∗ =1, K λ ∗ K μ ∗ = K λ+μ ∗ , K μ ∗ Fi= q ⟨ -μ , i′ ⟩ Fi K μ ∗ , i∈I,μ∈Q,$ and with coproduct given by $Δ K μ ∗ = K μ ∗ ⊗ K μ ∗ , ΔFi= Fi⊗ K -i ∗ + 1⊗Fi.$

1.4 $Δx= ∑ x x(1) K ∣x(2)∣ ⊗ x(2), Δy= ∑ y y(1)⊗ K -∣y(1)∣ ∗ ⊗ y(2).$

1.5 $ΔE(p)= ∑ t+t′=p qtt′ Ei(t) Ktt′⊗ Ei(t′), ΔF(p)= ∑ t+t′=p qtt′ Fi(t) Kt′i⊗ Fi(t′)$

1.6 $ϵKμ=1, ϵEi=0, ϵ K μ ∗ = 1, ϵFi=0, SKμ= K-μ, SEi=? S K μ ∗ = K -μ ∗ , SFi=?? S-1 Kμ=K-μ, S-1 Ei= -EiK-i, S K μ ∗ = K -μ ∗ , SFi= - K i ∗ Fi.$

## The form

2.1 We would like to identify ${\stackrel{˜}{U}}^{\le 0}$ with the Hopf algbra ${\left({\stackrel{˜}{U}}^{\ge 0}\right)}^{\ast coop}$

There is a unique bilinear pairing $U ˜ ≥0 × U ˜ ≤0 → ℚq$ given by $⟨ Ei , Fj ⟩ = -δij q-q-1 , ⟨ x , y1y2 ⟩ = ⟨ Δx , y1⊗y2 ⟩, ⟨ x1⊗x2 , Δy ⟩= ⟨ x1x2 , y ⟩.$

 Proof. The following relations are forced by the conditions. (1a)   $⟨{K}_{\mu },{1}^{\ast }⟩=1$, $⟨{E}_{i},{1}^{\ast }⟩=0?$ (1b)   $⟨1,{K}_{\mu }^{\ast }⟩=1$, $⟨1,{F}_{i}⟩=0?$ (1c)   $⟨{K}_{\mu },{K}_{\lambda }^{\ast }⟩=⟨{K}_{\mu },{K}_{-\lambda }^{\ast }⟩\ne 0$. (1d)   $⟨{E}_{i},{K}_{\mu }^{\ast }⟩=0$, $⟨{K}_{\mu },{F}_{i}⟩=0$. (2)   If $x\in {\stackrel{˜}{U}}^{\ge 0}$ and $y\in {\stackrel{˜}{U}}^{\ge 0}$ are homogenous and $|x|\ne |y|$ then $⟨x,y⟩=0$. (3)   If $x\in {\stackrel{˜}{U}}^{\ge 0}$ and $y\in {\stackrel{˜}{U}}^{\le 0}$ then $⟨ x Kμ , y ⟩= ⟨ x , y K μ ∗ ⟩= ⟨ x , y ⟩.$ (4)   If $x\in {\stackrel{˜}{U}}^{\ge 0}$ and $y\in {\stackrel{˜}{U}}^{\le 0}$ then $⟨ Kμx , y ⟩= ⟨ x , y ⟩ ⟨ Kμ , K -|x| ∗ ⟩, ⟨ x , K μ ∗ y ⟩= ⟨ x , y ⟩ ⟨ K|x| , K μ ∗ ⟩.$ (5)   $⟨{K}_{\mu },{K}_{\lambda }^{\ast }={q}^{-⟨\lambda ,\mu ⟩}⟩$. (6)   If $x,z\in {\stackrel{˜}{U}}^{\ge 0}$ and $y,w\in {\stackrel{˜}{U}}^{\le 0}$ then $⟨ xz , yw ⟩ = ∑ x,yz,w ⟨ x(1) , y(2) ⟩ ⟨ x(2) , w(2) ⟩ ⟨ z(1) , y(1) ⟩ ⟨ z(2) , w(1) ⟩ q- ⟨ |x(1)| , |z(1)| ⟩ q- ⟨ |x(2)| , |z(2)| ⟩ q ⟨ |x(2)| , |z(1)| ⟩ .$
$\square$

## The radical of the form $⟨,⟩$

3.1 Let ${ℐ}^{+}$ and ${ℐ}^{-}$ be the left and right radicals of the form $⟨,⟩$ respectively: Then

(1)   ${ℐ}^{+}\cap {\stackrel{˜}{U}}^{0}=0$.
(2)   ${ℐ}^{+}$ is a graded vector space.
(3)   ${ℐ}^{+}$ is an ideal.
(4)   ${ℐ}^{+}$ is a coideal.

 Proof. (1)   Since $⟨{K}_{\mu },{1}^{\ast }⟩=1$, ${K}_{\mu }\ne {ℐ}^{+}$. (2)   Suppose that $x\in {\stackrel{˜}{U}}^{\ge 0}$ and $x=\sum _{\nu \in {Q}^{+}}{x}_{\nu }$ where each ${x}_{\nu }$ is homogenous and $|{x}_{\nu }|=\nu$. Fix $\mu \in {Q}^{+}$. Then, the homogenity of the form $⟨,⟩$ gives $⟨ xμ , z ⟩= ⟨ xμ , zμ ⟩= ⟨ x , zμ ⟩=0$ for all $z\in {\stackrel{˜}{U}}^{\le 0}$. Thus ${x}_{\mu }\in {ℐ}^{+}$. Thus ${ℐ}^{+}$ is an homogenous ideal. (3)   Assume $x\in {\stackrel{˜}{U}}^{\ge 0}$. Then for every $y\in {\stackrel{˜}{U}}^{\ge 0}$ and every $z\in {\stackrel{˜}{U}}^{\le 0}$, $⟨ xy , z ⟩ = ⟨ y⊗x , Δz ⟩ = ⟨ y⊗x , ∑ z z(1) ⊗ K-|μ| z(2) ⟩ = ∑ z ⟨ y , z(1) ⟩ ⟨ x , K-|μ| z(2) ⟩ = 0.$ It follows that ${ℐ}^{+}$ is a right ideal of ${\stackrel{˜}{U}}^{\ge 0}$. The proof that ${ℐ}^{+}$ is a left ideal of ${\stackrel{˜}{U}}^{\ge 0}$?? (4)   First let us show that the left radical ${\left({ℐ}^{\otimes }\right)}^{+}$ of the form $⟨ , ⟩⊗: U ˜ ≥0 ⊗ U ˜ ≥0 × U ˜ ≤0 ⊗ U ˜ ≤0 → ℚq$ is ${ℐ}^{+}\otimes {\stackrel{˜}{U}}^{\ge 0}+{\stackrel{˜}{U}}^{\ge 0}\otimes {ℐ}^{+}$. Let $B$ be a basis of ${\stackrel{˜}{U}}^{\ge 0}$ consisting of homogenous elements. Let $∑ b,b′ cbb′b⊗b′ ∈ ℐ⊗+.$ Assume that ${b}^{\prime }$ is not in ${ℐ}^{+}$. Since each homogenous component of ${\stackrel{˜}{U}}^{\ge 0}$ is finite dimensional it follows that there is an element of ${\stackrel{˜}{U}}^{\ge 0}$ such that $⟨{z}_{b}^{\prime },b⟩={\delta }_{b}{b}^{\prime }$. Then $0= ⟨ ∑ b,b′ cbb′ b⊗b′ , z⊗zb′ ⟩ = ⟨ ∑ b cbb′b , z ⟩$ for all $z\in {\stackrel{˜}{U}}^{\le 0}$. It follows that $\sum _{b}{c}_{b{b}_{\prime }}b,z⟩\in {ℐ}^{+}$. This argument shows $ℐ⊗+⊆ ℐ+⊗ U ˜ ≥0 + U ˜ ≥0 ⊗ ℐ+.$ The othere inclusion is easy. The fact that $ℐ$ is a coideal now follows, since the equation $⟨ x , y1y2 ⟩= ⟨ Δx , y1⊗y2 ⟩$ implies that if $x\in {ℐ}^{+}$ then $\Delta \left(x\right)\in {\left({ℐ}^{+}\right)}^{\otimes }$
$\square$

3.2 The quantum Serre relations are in te ideal ${ℐ}^{+}$.

 Proof.
$\square$

## The double

4.1 The following relations are determined by the definition of the multiplication in $D\left({\stackrel{˜}{U}}^{\ge 0}\right)$: $EiFj- FjEi= Ki- K -i ∗ q-q-1 , Kλ , K μ ∗ = K μ ∗ Kλ, KμFi= q- ⟨ μ , i ⟩ FiKμ, K μ ∗ Ei= q ⟨ μ , i ⟩ Ei K μ ∗ .$

4.2 Let $\stackrel{˜}{U}=D\left({\stackrel{˜}{U}}^{\ge 0}\right)/J$ where $J$ is the ideal generated by the relations ${K}_{\mu }={K}_{\mu }^{\ast }$. Clearly this ideal is also a coideal. Thus $\stackrel{˜}{U}$ is a Hopf algebra with generators $Ei, Fi, Kμ,$ which satisfy the relations $K0=1, KλKμ= Kλ+μ, KμEi= q ⟨ μ , i′ ⟩ EiKμ, i∈I,μ∈Q, KμFi= q -⟨ μ , i′ ⟩ FiKμ, i∈I,μ∈Q, EiFj- FjEi= Ki- K -i ∗ q-q-1 ,$ and has coproduct given by $ΔKμ= Kμ⊗Kμ, ΔEi= Ei⊗1+ Ki⊗Ei, ΔFi= Fi⊗K-i+ 1 ⊗ Fi.$ The triangular decomposition shows that this is a presentation of $\stackrel{˜}{U}$.

4.3 The map $Φ: U ˜ ⊗ U0 → D U ˜ ≥0 Kμ⊗Kν ↦ Kμ+ν K -ν ∗ Ei⊗1 ↦ Ei Fi⊗1 ↦ Fi$ is a homomorphism of algebras.

## Bibliography

[DRV] Z. Daugherty, A. Ram, and R. Virk, Affine and graded BMW algebras, in preparation.