[Paper Review] Poisson Hopf algebras associated to quantized enveloping algebras

Toshiyuki Tanisaki|ArXiv.org|Feb 12, 2008

Algebraic structures and combinatorial models7 references3 citations

TL;DR

This paper establishes a direct construction of the Poisson Hopf algebra isomorphism $ Z_{ ho} \cong U_1 \cong \mathbb{C}[M] $, where $ Z_{ ho} $ is the Frobenius center of the specialized quantized enveloping algebra $ U_{ ho} $, $ U_1 $ is the classical limit, and $ \mathbb{C}[M] $ is the coordinate ring of a reductive algebraic group $ M $. The isomorphism is shown to preserve Poisson structures via a direct argument based on the Drinfel'd pairing, offering a more transparent alternative to dual-based proofs in prior work.

ABSTRACT

We study certain Poisson structures related to quantized enveloping algebras. In particular, we give a description of the Poisson structure of a certain manifold associated to the ring of differential operators.

Motivation & Objective

• To provide a direct, non-dual proof of the Poisson Hopf algebra isomorphism $ Z_{\zeta} \cong U_1 \cong \mathbb{C}[M] $, where $ Z_{\zeta} $ is the Frobenius center of the specialized quantized enveloping algebra $ U_{\zeta} $.
• To clarify the Poisson structure on the coordinate ring $ \mathbb{C}[M] $ of the algebraic group $ M $ associated to a finite-dimensional complex simple Lie algebra $ \mathfrak{g} $.
• To describe the Poisson algebra structure on the ring of differential operators associated to $ U_{\zeta} $, linking it to the Poisson geometry of $ G \times M $.

Proposed method

• Constructs the isomorphism $ U_1 \cong \mathbb{C}[M] $ directly using the Drinfel'd pairing, avoiding reliance on dual Hopf algebra arguments.
• Uses the comultiplication $ \Delta $, counit $ \varepsilon $, and antipode $ S $ of the quantized enveloping algebra $ U $, with Sweedler notation for iterated comultiplications.
• Applies the Drinfel'd pairing to define a Lie algebra isomorphism between $ \mathfrak{g} $ and $ \mathfrak{m} $, the Lie algebra of the group $ M \subset G \times G $.
• Derives the Poisson bracket on $ \mathbb{C}[G] \otimes \mathbb{C}[M] $ via the formula $ \{\varphi, \psi\} = -\sum_r (L_{\xi_r}\varphi)(R_{\eta_r}\psi) $, where $ \{\xi_r\}, \{\eta_r\} $ are dual bases under the modified Killing form $ \tilde{\kappa} $.
• Establishes the key identity $ \overline{(\sum_j x_j \otimes v_j - 1 \otimes 1)/(q - q^{-1})} = \sum_r \xi_r \otimes \eta_r $ in the completion of $ U(\mathfrak{g}) \otimes \mathbb{C}[M]^* $, proving the Poisson compatibility.
• Uses Lemma 8.4 to show that the map $ \Xi: \mathbb{C}[M] \to \mathfrak{g} $ satisfies $ \tilde{\kappa}(\theta(\Xi(\varphi)), \eta) = \langle \varphi, \eta \rangle $, confirming the Poisson structure via the Killing form.

Experimental results

Research questions

• RQ1 How can the Poisson Hopf algebra isomorphism $ Z_{\zeta} \cong U_1 \cong \mathbb{C}[M] $ be proven directly, without relying on duality arguments?
• RQ2 What is the explicit form of the Poisson bracket on the ring of differential operators associated to $ U_{\zeta} $, and how does it relate to the geometry of $ G \times M $?
• RQ3 How does the Drinfel'd pairing induce a Poisson structure on $ \mathbb{C}[M] $, and how is this compatible with the classical limit $ U_1 $?
• RQ4 Can the isomorphism between $ Z_{\zeta} $, $ U_1 $, and $ \mathbb{C}[M] $ be shown to preserve Poisson structures through a direct computation in the quantum setting?

Key findings

• The Poisson Hopf algebra isomorphism $ Z_{\zeta} \cong U_1 \cong \mathbb{C}[M] $ is established via a direct argument based on the Drinfel'd pairing, providing a more transparent alternative to dual-based proofs.
• The Poisson bracket on $ \mathbb{C}[G] \otimes \mathbb{C}[M] $ is explicitly given by $ \{\varphi, \psi\} = -\sum_r (L_{\xi_r}\varphi)(R_{\eta_r}\psi) $, where $ \{\xi_r\}, \{\eta_r\} $ are dual bases under $ \tilde{\kappa} $.
• The map $ \Xi: \mathbb{C}[M] \to \mathfrak{g} $ defined by $ x \mapsto (x - \varepsilon(x)1)/(q - q^{-1}) $ satisfies $ \tilde{\kappa}(\theta(\Xi(\varphi)), \eta) = \langle \varphi, \eta \rangle $, confirming the Poisson structure on $ \mathbb{C}[M] $.
• The Poisson tensor $ \delta $ on $ G \times M $ satisfies $ \delta_{(g,m)}((L^*_{\eta})_g, (R^*_{\xi})_m) = -\tilde{\kappa}(\xi, \eta) $, linking the Poisson geometry to the Killing form.
• The isomorphism $ U_1 \cong \mathbb{C}[M] $ is shown to be a Poisson Hopf algebra isomorphism, with the Poisson structure on $ \mathbb{C}[M] $ induced by the classical limit of the quantum group structure.
• The construction avoids dual Hopf algebra arguments, offering a more direct and geometrically transparent proof of the isomorphism compared to earlier approaches.