${\tilde{U}}^{\ge 0}$: The form, the quantum Serre relations, and the quantum double

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

Last updates: 12 April 2011

The Hopf algebras ${\tilde{U}}^{\ge 0}$ and ${\tilde{U}}^{\le 0}$

1.1 Let $I$ be an infinite set and let $Q={\sum }_{i\in I}\mathbb{Z}i$ 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 $\mathbb{Z}$ valued bilineare pairing between $Q$ and $Q^{\vee }$.

1.2 Let ${\tilde{U}}^{\ge 0}$ be the Hopf algebra over $\mathbb{Q}\left(q\right)$ with generators $$E_i,i\in I,K_{\mu },\mu \in Q,$$ and relations $$\begin{array}{c}{K}_0=1,K_{\lambda }{K}_{\mu }=K_{\lambda +\mu },\\ K_{\mu }{E}_i=q^{⟨\mu ,i^{\prime }⟩}{E}_i{K}_{\mu },i\in I,\mu \in Q.\end{array}$$ and with coproduct given by $$\begin{array}{c}\Delta \left(K_{\mu }\right)=K_{\mu }\otimes K_{\mu },\\ \Delta \left(E_i\right)=E_i\otimes 1+K_i\otimes E_i.\end{array}$$

1.3 ${\tilde{U}}^{\le 0}$ be the Hopf algebra over $\mathbb{Q}\left(q\right)$ with generators $$F_i,i\in I,K_{\mu }^{\ast },\mu \in Q,$$ and relations $$\begin{array}{c}{K}_0^{\ast }=1,K_{\lambda }^{\ast }{K}_{\mu }^{\ast }=K_{\lambda +\mu }^{\ast },\\ K_{\mu }^{\ast }{F}_i=q^{⟨-\mu ,i^{\prime }⟩}{F}_i{K}_{\mu }^{\ast },i\in I,\mu \in Q,\end{array}$$ and with coproduct given by $$\begin{array}{c}\Delta \left(K_{\mu }^{\ast }\right)=K_{\mu }^{\ast }\otimes K_{\mu }^{\ast },\\ \Delta \left(F_i\right)=F_i\otimes K_{-i}^{\ast }+1\otimes F_i.\end{array}$$

1.4 $$\begin{array}{c}\Delta \left(x\right)=\sum _x{x}_{\left(1\right)}{K}_{\mid x_{\left(2\right)}\mid }\otimes x_{\left(2\right)},\\ \Delta \left(y\right)=\sum _y{y}_{\left(1\right)}\otimes K_{-\mid y_{\left(1\right)}\mid }^{\ast }\otimes y_{\left(2\right)}.\end{array}$$

1.5 $$\begin{array}{c}\Delta \left(E^{\left(p\right)}\right)=\sum _{t+t^{\prime }=p}{q}^{t{t}^{\prime }}{E}_i^{\left(t\right)}{K}_{t{t}^{\prime }}\otimes E_i^{\left(t^{\prime }\right)},\\ \Delta \left(F^{\left(p\right)}\right)=\sum _{t+t^{\prime }=p}{q}^{t{t}^{\prime }}{F}_i^{\left(t\right)}{K}_{t^{\prime }i}\otimes F_i^{\left(t^{\prime }\right)}\end{array}$$

1.6 $$\begin{array}{cc}ϵ\left(K_{\mu }\right)=1,& ϵ\left(E_i\right)=0,\\ ϵ\left(K_{\mu }^{\ast }\right)=1,& ϵ\left(F_i\right)=0,\\ S\left(K_{\mu }\right)=K_{-\mu },& S\left(E_i\right)=?\\ S\left(K_{\mu }^{\ast }\right)=K_{-\mu }^{\ast },& S\left(F_i\right)=??\\ S^{-1}\left(K_{\mu }\right)=K_{-\mu },& S^{-1}\left(E_i\right)=-E_i{K}_{-i},\\ S\left(K_{\mu }^{\ast }\right)=K_{-\mu }^{\ast },& S\left(F_i\right)=-K_i^{\ast }{F}_i.\end{array}$$

The form

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

There is a unique bilinear pairing $$\tilde{U}\ge 0\hspace{0.17em}\times \hspace{0.17em}\tilde{U}\le 0\to \mathbb{Q}\left(q\right)$$ given by $$\begin{array}{c}⟨E_i,F_j⟩=\frac{-{\delta }_{ij}}{q-q^{-1}},\\ ⟨x,y_1{y}_2⟩=⟨\Delta \left(x\right),y_1\otimes y_2⟩,\\ ⟨x_1\otimes x_2,\Delta \left(y\right)⟩=⟨x_1{x}_2,y⟩.\end{array}$$

		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 $x\in {\tilde{U}}^{\ge 0}$ and $y\in {\tilde{U}}^{\ge 0}$ are homogenous and $\left|x\right|\ne \left|y\right|$ then $⟨x,y⟩=0$. (3)   If $x\in {\tilde{U}}^{\ge 0}$ and $y\in {\tilde{U}}^{\le 0}$ then $$⟨x{K}_{\mu },y⟩=⟨x,y{K}_{\mu }^{\ast }⟩=⟨x,y⟩.$$ (4)   If $x\in {\tilde{U}}^{\ge 0}$ and $y\in {\tilde{U}}^{\le 0}$ then $$\begin{array}{c}⟨K_{\mu }x,y⟩=⟨x,y⟩⟨K_{\mu },K_{-\left|x\right|}^{\ast }⟩,\\ ⟨x,K_{\mu }^{\ast }y⟩=⟨x,y⟩⟨K_{\left|x\right|},K_{\mu }^{\ast }⟩.\end{array}$$ (5)   $⟨K_{\mu },K_{\lambda }^{\ast }=q^{-⟨\lambda ,\mu ⟩}⟩$. (6)   If $x,z\in {\tilde{U}}^{\ge 0}$ and $y,w\in {\tilde{U}}^{\le 0}$ then $$⟨xz,yw⟩=\sum _{x,yz,w}⟨x_{\left(1\right)},y_{\left(2\right)}⟩⟨x_{\left(2\right)},w_{\left(2\right)}⟩⟨z_{\left(1\right)},y_{\left(1\right)}⟩⟨z_{\left(2\right)},w_{\left(1\right)}⟩q^{-⟨\left|x_{\left(1\right)}\right|,\left|z_{\left(1\right)}\right|⟩}{q}^{-⟨\left|x_{\left(2\right)}\right|,\left|z_{\left(2\right)}\right|⟩}{q}^{⟨\left|x_{\left(2\right)}\right|,\left|z_{\left(1\right)}\right|⟩}.$$

$\square$

The radical of the form $⟨,⟩$

3.1 Let ${ℐ}^{+}$ and ${ℐ}^{-}$ be the left and right radicals of the form $⟨,⟩$ respectively: $$\begin{array}{c}{ℐ}^{+}=\left\{x\in {\tilde{U}}^{\ge 0}\mid ⟨x,y⟩=0, \text{for all} y\in {\tilde{U}}^{\le 0}\right\},\\ {ℐ}^{-}=\left\{y\in {\tilde{U}}^{\le 0}\mid ⟨x,y⟩=0 \text{for all} x\in {\tilde{U}}^{\ge 0}\right\}.\end{array}$$ Then

(1)   ${ℐ}^{+}\cap {\tilde{U}}^0=0$.

(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 $x\in {\tilde{U}}^{\ge 0}$ and $x=\sum _{\nu \in Q^{+}}{x}_{\nu }$ where each $x_{\nu }$ is homogenous and $\left|x_{\nu }\right|=\nu$. Fix $\mu \in Q^{+}$. Then, the homogenity of the form $⟨,⟩$ gives $$⟨x_{\mu },z⟩=⟨x_{\mu },z_{\mu }⟩=⟨x,z_{\mu }⟩=0$$ for all $z\in {\tilde{U}}^{\le 0}$. Thus $x_{\mu }\in {ℐ}^{+}$. Thus ${ℐ}^{+}$ is an homogenous ideal. (3)   Assume $x\in {\tilde{U}}^{\ge 0}$. Then for every $y\in {\tilde{U}}^{\ge 0}$ and every $z\in {\tilde{U}}^{\le 0}$, $$\begin{array}{rcl}⟨xy,z⟩& =& ⟨y\otimes x,\Delta \left(z\right)⟩\\ & =& ⟨y\otimes x,\sum _z{z}_{\left(1\right)}\otimes K_{-\left|\mu \right|}{z}_{\left(2\right)}⟩\\ & =& \sum _z⟨y,z_{\left(1\right)}⟩⟨x,K_{-\left|\mu \right|}{z}_{\left(2\right)}⟩\\ & =& 0.\end{array}$$ It follows that ${ℐ}^{+}$ is a right ideal of ${\tilde{U}}^{\ge 0}$. The proof that ${ℐ}^{+}$ is a left ideal of ${\tilde{U}}^{\ge 0}$?? (4)   First let us show that the left radical ${\left({ℐ}^{\otimes }\right)}^{+}$ of the form $${⟨,⟩}^{\otimes }:\left({\tilde{U}}^{\ge 0}\otimes {\tilde{U}}^{\ge 0}\right)\hspace{0.17em}\times \hspace{0.17em}\left({\tilde{U}}^{\le 0}\otimes {\tilde{U}}^{\le 0}\right)\to \mathbb{Q}\left(q\right)$$ is ${ℐ}^{+}\otimes {\tilde{U}}^{\ge 0}+{\tilde{U}}^{\ge 0}\otimes {ℐ}^{+}$. Let $B$ be a basis of ${\tilde{U}}^{\ge 0}$ consisting of homogenous elements. Let $$\sum _{b,b^{\prime }}{c}_{bb\prime }b\otimes b^{\prime }\in {\left({ℐ}^{\otimes }\right)}^{+}.$$ Assume that $b^{\prime }$ is not in ${ℐ}^{+}$. Since each homogenous component of ${\tilde{U}}^{\ge 0}$ is finite dimensional it follows that there is an element of ${\tilde{U}}^{\ge 0}$ such that $⟨z_b^{\prime },b⟩={\delta }_b{b}^{\prime }$. Then $$0=⟨\sum _{b,b^{\prime }}{c}_{b{b}^{\prime }}b\otimes b^{\prime },z\otimes z_{b^{\prime }}⟩=⟨\sum _b{c}_{b{b}_{\prime }}b,z⟩$$ for all $z\in {\tilde{U}}^{\le 0}$. It follows that $\sum _b{c}_{b{b}_{\prime }}b,z⟩\in {ℐ}^{+}$. This argument shows $${\left({ℐ}^{\otimes }\right)}^{+}\subseteq {ℐ}^{+}\otimes {\tilde{U}}^{\ge 0}+{\tilde{U}}^{\ge 0}\otimes {ℐ}^{+}.$$ The othere inclusion is easy. The fact that $ℐ$ is a coideal now follows, since the equation $$⟨x,y_1{y}_2⟩=⟨\Delta \left(x\right),y_1\otimes y_2⟩$$ 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 $D\left({\tilde{U}}^{\ge 0}\right)$: $$\begin{array}{c}{E}_i{F}_j-F_j{E}_i=\frac{K_i-K_{-i}^{\ast }}{q-q^{-1}},\\ K_{\lambda },K_{\mu }^{\ast }=K_{\mu }^{\ast }{K}_{\lambda },\\ K_{\mu }{F}_i=q^{-⟨\mu ,i⟩}{F}_i{K}_{\mu },\\ K_{\mu }^{\ast }{E}_i=q^{⟨\mu ,i⟩}{E}_i{K}_{\mu }^{\ast }.\end{array}$$

4.2 Let $\tilde{U}=D\left({\tilde{U}}^{\ge 0}\right)/J$ where $J$ is the ideal generated by the relations $K_{\mu }=K_{\mu }^{\ast }$. Clearly this ideal is also a coideal. Thus $\tilde{U}$ is a Hopf algebra with generators $$E_i,F_i,K_{\mu },$$ which satisfy the relations $$\begin{array}{c}{K}_0=1,K_{\lambda }{K}_{\mu }=K_{\lambda +\mu },\\ K_{\mu }{E}_i=q^{⟨\mu ,i^{\prime }⟩}{E}_i{K}_{\mu },i\in I,\mu \in Q,\\ K_{\mu }{F}_i=q^{-⟨\mu ,i^{\prime }⟩}{F}_i{K}_{\mu },i\in I,\mu \in Q,\\ E_i{F}_j-F_j{E}_i=\frac{K_i-K_{-i}^{\ast }}{q-q^{-1}},\end{array}$$ and has coproduct given by $$\begin{array}{c}\Delta \left(K_{\mu }\right)=K_{\mu }\otimes K_{\mu },\\ \Delta \left(E_i\right)=E_i\otimes 1+K_i\otimes E_i,\\ \Delta \left(F_i\right)=F_i\otimes K_{-i}+1\hspace{0.17em}\otimes \hspace{0.17em}{F}_i.\end{array}$$ The triangular decomposition shows that this is a presentation of $\tilde{U}$.

4.3 The map $$\begin{array}{cccc}\Phi :& \tilde{U}\otimes U^0& \to & D\left({\tilde{U}}^{\ge 0}\right)\\ & K_{\mu }\otimes K_{\nu }& \mapsto & K_{\mu +\nu }{K}_{-\nu }^{\ast }\\ & E_i\otimes 1& \mapsto & E_i\\ & F_i\otimes 1& \mapsto & F_i\end{array}$$ is a homomorphism of algebras.

Notes and References

Bibliography

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

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