Editing Hopf algebra (section)

=== Quantum groups and non-commutative geometry ===
{{Main|quantum group}}

Most examples above are either commutative (i.e. the multiplication is [[commutative]]) or co-commutative (i.e.<ref name=Und57>{{harvnb|Underwood|2011|p=57}}</ref> Δ = ''T'' ∘ Δ where the ''twist map''<ref name=Und36>{{harvnb|Underwood|2011|p=36}}</ref> ''T'': ''H'' ⊗ ''H'' → ''H'' ⊗ ''H'' is defined by ''T''(''x'' ⊗ ''y'') = ''y'' ⊗ ''x''). Other interesting Hopf algebras are certain "deformations" or "[[quantization (physics)|quantization]]s" of those from example 3 which are neither commutative nor co-commutative. These Hopf algebras are often called ''[[quantum groups]]'', a term that is so far only loosely defined. They are important in [[noncommutative geometry]], the idea being the following: a standard algebraic group is well described by its standard Hopf algebra of regular functions; we can then think of the deformed version of this Hopf algebra as describing a certain "non-standard" or "quantized" algebraic group (which is not an algebraic group at all). While there does not seem to be a direct way to define or manipulate these non-standard objects, one can still work with their Hopf algebras, and indeed one ''identifies'' them with their Hopf algebras. Hence the name "quantum group".