**Quantum Groups and Non-commutative Geometry**

All examples above are either commutative (i.e. the multiplication is commutative) or co-commutative (i.e. Δ = *T* Δ where *T*: *H* ⊗ *H* → *H* ⊗ *H* is defined by *T*(*x* ⊗ *y*) = *y* ⊗ *x*). Other interesting Hopf algebras are certain "deformations" or "quantizations" 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".

Read more about this topic: Hopf Algebra

### Other articles related to "quantum groups, geometry, group, quantum group":

**Quantum Groups and Non-commutative Geometry**

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