Timeline for Concrete examples of quantum duality principle
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 6, 2023 at 6:23 | comment | added | Adrien | A quantization is $D(G)[[\hbar]]$ where $D(G)$ is the algebra of differential operators on $G$, which you can identify with the Hopf algebraic Heisenberg double, which is $U(\mathfrak g) \ltimes O(G)$ but this time with respect to the action of $U(\mathfrak g)$ via, say, left invariant differential operators. | |
| Jul 6, 2023 at 6:22 | comment | added | Adrien | IN that case the Drinfeld double is (as a Lie bialgebra) the semidirect product $\mathfrak g \ltimes \mathfrak g^*$ where $\mathfrak g$ acts by the coadjoint action. As a group this is the semi-direct product $G \ltimes \mathfrak g^*$ with a certain Poisson structure. One quantization of this is $ O(G)\ltimes U(\mathfrak g)[[\hbar]]$ where the action is again the adjoint one, and the coproduct is the obvious one. The Heisenberg double in that case is just the cotangent bundle $T^*G$ with its symplectic structure. | |
| Jul 6, 2023 at 6:14 | comment | added | yohei ohta | Thank you for your reply. I was using $\mathcal{C}_h(G^*)$ in the sense of the quantization of $\mathcal{C}(G^*)$, but then $\mathcal{C}(G^*)=U_h(\mathfrak{g})$, so my question makes no sense! The example you gave looks very concrete and good. I would like to understand the relation between the Drinfeld double and the Heisenberg double of the Poisson Lie group and the Drinfeld double and the Heisenberg double of the Hopf algebra through concrete examples. I will try to write down these relations with your examples. | |
| Jul 5, 2023 at 15:47 | comment | added | Adrien | What is your definition of $\mathscr{C}_h(G^*)$ ? The only definitions I know are either very abstract or use the duality principle. IMHO the simplest example is: take $G$ with the zero Poisson structure. Its dual is then $\mathfrak g^*$, with Poisson structure given by the Lie bracket of $\mathfrak g$ and group structure given by the underlying vector space. Then the quantum formal series Hopf algebra associated with $U(\mathfrak g)[[\hbar]]$ is the subalgebra generated by $\hbar \mathfrak g$, which is indeed a quantization of the formal Poisson group associated with $\mathfrak g^*$. | |
| Jul 5, 2023 at 11:15 | history | edited | yohei ohta | CC BY-SA 4.0 |
added 9 characters in body
|
| Jul 5, 2023 at 11:07 | history | asked | yohei ohta | CC BY-SA 4.0 |