Quantum Group (Formal-Deformation Convention)
The term quantum group is used in several closely related ways. In this knowledge base, the primary meaning is a Hopf algebra equipped with explicit deformation data and a classical limit.
Definition
Let \(k\) be a field, let \(H_0\) be a Hopf algebra over \(k\), and write \(A=k[[\hbar]]\). A formal quantum group deforming \(H_0\) is a Hopf algebra \(H_\hbar\) over \(A\), together with an isomorphism of Hopf algebras
$$ H_\hbar/\hbar H_\hbar \;\cong\; H_0, $$
such that \(H_\hbar\) is complete and separated for the \(\hbar\)-adic topology and is topologically free as an \(A\)-module. Equivalently, as a topological \(A\)-module it has the form \(M[[\hbar]]\) for some \(k\)-vector space \(M\). Topological freeness is the flatness condition that prevents the deformation from acquiring \(\hbar\)-torsion.
The multiplication, unit, comultiplication, counit, and antipode of \(H_\hbar\) must all be continuous and must satisfy the Hopf algebra axioms. Reduction modulo \(\hbar\) recovers every structure map of \(H_0\).
Two standard forms
- A quantized universal enveloping algebra \(U_\hbar(\mathfrak g)\) deforms the cocommutative Hopf algebra \(U(\mathfrak g)\), usually starting from a Lie bialgebra \(\mathfrak g\).
- A quantized coordinate algebra \(\mathcal O_\hbar(G)\) deforms the commutative coordinate Hopf algebra \(\mathcal O(G)\), usually starting from a Poisson--Lie group \(G\).
These are dual viewpoints when suitable finiteness or topological duality hypotheses are available; they are not literally dual for arbitrary infinite-dimensional Hopf algebras.
Classical limit
The first-order failure of commutativity or cocommutativity records the classical infinitesimal structure. For example, if \(H_0\) is commutative, \(a,b\in H_\hbar\), and \(a_0,b_0\in H_0\) are their reductions, then
$$ \{a_0,b_0\} = \left(\frac{ab-ba}{\hbar}\right)\bmod \hbar $$
defines the associated Poisson bracket. Dually, when the undeformed coproduct is cocommutative, the first-order term of \((\Delta_\hbar-\Delta_\hbar^{\mathrm{op}})/\hbar\) gives the corresponding Lie cobracket.
The \(q\)-notation
Many examples are written as \(U_q(\mathfrak g)\) or \(\mathcal O_q(G)\). To retain a classical limit, a quantum group over \(k(q)\) must be accompanied by a suitable integral form over a ring such as \(k[q,q^{-1}]\), or by a completion near \(q=1\), whose specialization at \(q=1\) is the classical Hopf algebra. A Hopf algebra over \(k(q)\) without chosen specialization data does not by itself determine a deformation of a classical group.
Thus, in this project, "quantum group" does not mean every noncommutative Hopf algebra. The deformation parameter, flatness or topological-freeness condition, and classical specialization are part of the definition.
References
- V. G. Drinfeld, Quantum groups, Proceedings of the International Congress of Mathematicians, Berkeley (1986).
- M. Jimbo, A q-difference analogue of U(g) and the Yang--Baxter equation, Letters in Mathematical Physics 10 (1985).
- C. Kassel, Quantum Groups, Graduate Texts in Mathematics 155 (1995).
- V. Chari and A. Pressley, A Guide to Quantum Groups (1994).