# Hopf Algebras

Hopf Algebras

In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.

Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, in group theory (via the concept of a group ring), and in numerous other places, making them probably the most familiar type of bialgebra. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems on the other.

### Other articles related to "hopf algebras, hopf algebra, algebras, algebra":

Hopf Algebras - Analogy With Groups
... can be axiomatized by the same diagrams (equivalently, operations) as a Hopf algebra, where G is taken to be a set instead of a module ... in the group the antipode is the inverse In this philosophy, a group can be thought of as a Hopf algebra over the "field with one element" ...
Hopf Algebra - Quantum Groups and Non-commutative Geometry
... 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 ... 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 ...
Vladimir Drinfeld - Contributions To Mathematics
... Drinfeld coined the term "Quantum group" in reference to Hopf algebras, which are deformations of simple Lie algebras, and connected them to the study of the Yangâ€“Baxter equation, which is a ... He also generalized Hopf algebras to quasi-Hopf algebras and introduced the study of Drinfeld twists, which can be used to factorize the R-matrix ... also collaborated with Alexander Beilinson to rebuild the theory of vertex algebras in a coordinate-free form, which have become increasingly important to conformal field theory, string theory ...
Representation Theory - Branches and Topics - Associative Algebras - Hopf Algebras and Quantum Groups
... Hopf algebras provide a way to improve the representation theory of associative algebras, while retaining the representation theory of groups and Lie algebras as special cases ... The Hopf algebras associated to groups have a commutative algebra structure, and so general Hopf algebras are known as quantum groups, although this term is often restricted to ... representation theory of Lie groups and Lie algebras, for instance through the crystal basis of Kashiwara ...