Editing Universal enveloping algebra (section)

== Hopf algebras and quantum groups ==
The construction of the [[group ring|group algebra]] for a given [[group (mathematics)|group]] is in many ways analogous to constructing the universal enveloping algebra for a given Lie algebra. Both constructions are universal and translate representation theory into module theory. Furthermore, both group algebras and universal enveloping algebras carry natural [[coalgebra|comultiplications]] that turn them into [[Hopf algebra]]s. This is made precise in the article on the [[tensor algebra]]: the tensor algebra has a Hopf algebra structure on it, and because the Lie bracket is consistent with (obeys the consistency conditions for) that Hopf structure, it is inherited by the universal enveloping algebra.

Given a Lie group {{math|''G''}}, one can construct the vector space {{math|C(''G'')}} of continuous complex-valued functions on {{math|''G''}}, and turn it into a [[C*-algebra]]. This algebra has a natural Hopf algebra structure: given two functions
<math>\varphi, \psi\in C(G)</math>, one defines multiplication as
:<math>(\nabla(\varphi, \psi))(x)=\varphi(x)\psi(x)</math>
and comultiplication as
:<math>(\Delta(\varphi))(x\otimes y)=\varphi(xy),</math>
the counit as
:<math>\varepsilon(\varphi)=\varphi(e)</math>
and the antipode as
:<math>(S(\varphi))(x)=\varphi(x^{-1}).</math>
Now, the [[Gelfand–Naimark theorem]] essentially states that every commutative Hopf algebra is isomorphic to the Hopf algebra of continuous functions on some compact topological group {{math|''G''}}—the theory of compact topological groups and the theory of commutative Hopf algebras are the same. For Lie groups, this implies that {{math|C(''G'')}} is isomorphically dual to <math>U(\mathfrak{g})</math>; more precisely, it is isomorphic to a subspace of the dual space <math>U^*(\mathfrak{g}).</math>

These ideas can then be extended to the non-commutative case. One starts by defining the [[quasi-triangular Hopf algebra]]s, and then performing what is called a [[quantum deformation]] to obtain the '''quantum universal enveloping algebra''', or [[quantum group]], for short.