Abstract
We introduce a general notion of quantum universal enveloping algebroids (QUE algebroids), or quantum groupoids, as a unification of quantum groups and star-products. Some basic properties are studied including the twist construction and the classical limits. In particular, we show that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. Conversely, we formulate a conjecture on the existence of a quantization for any Lie bialgebroid, and prove this conjecture for the special case of regular triangular Lie bialgebroids. As an application of this theory, we study the dynamical quantum groupoid D⊗ℏUℏg, which gives an interpretation of the quantum dynamical Yang-Baxter equation in terms of Hopf algebroids.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 539-581 |
| Number of pages | 43 |
| Journal | Communications In Mathematical Physics |
| Volume | 216 |
| Issue number | 3 |
| DOIs | |
| State | Published - Feb 1 2001 |
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All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics
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Quantum groupoids. / Xu, Ping.
In: Communications In Mathematical Physics, Vol. 216, No. 3, 01.02.2001, p. 539-581.Research output: Contribution to journal › Article
TY - JOUR
T1 - Quantum groupoids
AU - Xu, Ping
PY - 2001/2/1
Y1 - 2001/2/1
N2 - We introduce a general notion of quantum universal enveloping algebroids (QUE algebroids), or quantum groupoids, as a unification of quantum groups and star-products. Some basic properties are studied including the twist construction and the classical limits. In particular, we show that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. Conversely, we formulate a conjecture on the existence of a quantization for any Lie bialgebroid, and prove this conjecture for the special case of regular triangular Lie bialgebroids. As an application of this theory, we study the dynamical quantum groupoid D⊗ℏUℏg, which gives an interpretation of the quantum dynamical Yang-Baxter equation in terms of Hopf algebroids.
AB - We introduce a general notion of quantum universal enveloping algebroids (QUE algebroids), or quantum groupoids, as a unification of quantum groups and star-products. Some basic properties are studied including the twist construction and the classical limits. In particular, we show that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. Conversely, we formulate a conjecture on the existence of a quantization for any Lie bialgebroid, and prove this conjecture for the special case of regular triangular Lie bialgebroids. As an application of this theory, we study the dynamical quantum groupoid D⊗ℏUℏg, which gives an interpretation of the quantum dynamical Yang-Baxter equation in terms of Hopf algebroids.
UR - http://www.scopus.com/inward/record.url?scp=0035529585&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0035529585&partnerID=8YFLogxK
U2 - 10.1007/s002200000334
DO - 10.1007/s002200000334
M3 - Article
AN - SCOPUS:0035529585
VL - 216
SP - 539
EP - 581
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
SN - 0010-3616
IS - 3
ER -