# Hopf algebra

In

mathematics , a**Hopf algebra**, named afterHeinz Hopf , is a structure that is simultaneously a (unital associative) algebra, acoalgebra , and has anantiautomorphism , with these structures compatible.Hopf algebras occur naturally in

algebraic topology , where they originated and are related to theH-space concept, ingroup scheme theory, ingroup theory (via the concept of agroup ring ), and in numerous other places, making them probably the most familiar type ofbialgebra . 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.**Formal definition**Formally, a Hopf algebra is a

bialgebra "H" over a field "K" together with a "K"-linear map $Scolon\; H\; o\; H$ (called the**antipode**) such that the following diagram commutes:Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. In the sumlessSweedler notation , this property can also be expressed as:$S(c\_\{(1)\})c\_\{(2)\}=c\_\{(1)\}S(c\_\{(2)\})=epsilon(c)1qquadmbox\{\; for\; all\; \}cin\; H.$As for algebras, one can replace the underlying field "K" with a

commutative ring "R" in the above definition.The definition of Hopf algebra is self-dual (as reflected in the symmetry of the above diagram), so if one can define a dual of "H" (which is always possible if "H" is finite-dimensional), then it is automatically a Hopf algebra.

**Properties of the antipode**"S" is sometimes required to have a "K"-linear inverse, which is automatic in the finite-dimensional case, or if "H" is

commutative orcocommutative (or more generally quasitriangular).In general, "S" is an

antihomomorphism , so $S^2$ is ahomomorphism , which is therefore an automorphism if "S" was invertible (as may be required).If $S^2\; =\; Id$, then the Hopf algebra is said to be

**involutive**(and the underlying algebra with involution is a*-algebra ). If "H" is finite-dimensional, commutative, or cocommutative, then it is involutive.If a bialgebra "B" admits an antipode "S", then "S" is unique ("a bialgebra admits at most 1 Hopf algebra structure").

The antipode is an analog to the inversion map on a group that sends $g$ to $g^\{-1\}$. [

*[*]*http://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/QuantGrp/ln2_1.pdf Quantum groups lecture notes*]**Examples****Group algebra.**Suppose "G" is a group. Thegroup algebra "KG" is aunital associative algebra over "K". It turns into a Hopf algebra if we define

* Δ : "KG" → "KG" dirprod "KG" by Δ("g") = "g" dirprod "g" for all "g" in "G"

* ε : "KG" → "K" by ε("g") = 1 for all "g" in "G"

* "S" : "KG" → "KG" by "S"("g") = "g"^{ -1}for all "g" in "G".**Functions on a finite group.**Suppose now that "G" is a "finite" group. Then the set "K"^{"G"}of all functions from "G" to "K" with pointwise addition and multiplication is a unital associative algebra over "K", and "K"^{"G"}dirprod "K"^{"G"}is naturally isomorphic to "K"^{"G"x"G"}(for "G" infinite, "K"^{"G"}dirprod "K"^{"G"}is a proper subset of "K"^{"G"x"G"}). The set "K"^{"G"}becomes a Hopf algebra if we define

* Δ : "K"^{"G"}→ "K"^{"G"x"G"}by Δ("f")("x","y") = "f"("xy") for all "f" in "K"^{"G"}and all "x","y" in "G"

* ε : "K"^{"G"}→ "K" by ε("f") = "f"("e") for every "f" in "K"^{"G"}[here "e" is theidentity element of "G"]

* "S" : "K"^{"G"}→ "K"^{"G"}by "S"("f")("x") = "f"("x"^{-1}) for all "f" in "K"^{"G"}and all "x" in "G".**Regular functions on an algebraic group.**Generalizing the previous example, we can use the same formulas to show that for a givenalgebraic group "G" over "K", the set of allregular function s on "G" forms a Hopf algebra.**Universal enveloping algebra.**Suppose "g" is aLie algebra over the field "K" and "U" is itsuniversal enveloping algebra . "U" becomes a Hopf algebra if we define

* Δ : "U" → "U" dirprod "U" by Δ("x") = "x" dirprod 1 + 1 dirprod "x" for every "x" in "g" (this rule is compatible withcommutator s and can therefore be uniquely extended to all of "U").

* ε : "U" → "K" by ε("x") = 0 for all "x" in "g" (again, extended to "U")

* "S" : "U" → "U" by "S"("x") = -"x" for all "x" in "g".**Cohomology of Lie groups**The cohomology algebra of a Lie group is a Hopf algebra: the multiplication is provided by the cup-product, and the comultiplication :$H^*(G)\; ightarrow\; H^*(G\; imes\; G)\; cong\; H^*(G)otimes\; H^*(G)$by the group multiplication $G\; imes\; G\; ightarrow\; G$.This observation was actually a source of the notion of Hopf algebra. Using this structure, Hopf proved a structure theorem for the cohomology algebra of Lie groups.

**Theorem (Hopf)**[*Hopf, 1941.*] Let "A" be a finite-dimensional, graded commutative, gradedcocommutative Hopf algebra over a field of characteristic 0. Then "A" (as an algebra) is a free exterior algebra with generators of odd degree.**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" $circ$ Δ where "T": "H" dirprod "H" → "H" dirprod "H" is defined by "T"("x" dirprod "y") = "y" dirprod "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 innoncommutative 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".**Related concepts**Graded Hopf algebras are often used inalgebraic topology : they are the natural algebraic structure on the direct sum of all homology orcohomology groups of anH-space .Locally compact quantum group s generalize Hopf algebras and carry a topology. The algebra of allcontinuous function s on aLie group is a locally compact quantum group.Quasi-Hopf algebra s are also generalizations of Hopf algebras, where coassociativity only holds up to a twist.**Analogy with groups**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 this case:

* the field "K" is replaced by the 1-point set

* there is a natural counit (map to 1 point)

* there is a natural comultiplication (the diagonal map)

* the unit is the identity element of the group

* the multiplication is the multiplication in the group

* the antipode is the inverseIn this philosophy, a group can be thought of as a Hopf algebra over the "field with one element ". [*[*]*http://sbseminar.wordpress.com/2007/10/07/group-hopf-algebra/ Group = Hopf algebra « Secret Blogging Seminar*] , [*http://www.youtube.com/watch?v=p3kkm5dYH-w Group objects and Hopf algebras*] , video of Simon Willerton.**See also***

Quasitriangular Hopf algebra

*Algebra/set analogy

*Representation theory of Hopf algebras

*Ribbon Hopf algebra

*Superalgebra

* Supergroup

*Anyonic Lie algebra **Notes****References*** Pierre Cartier, [

*http://inc.web.ihes.fr/prepub/PREPRINTS/2006/M/M-06-40.pdf "A primer of Hopf algebras"*] , IHES preprint, September 2006, 81 pages

* Jurgen Fuchs, "Affine Lie Algebras and Quantum Groups", (1992), Cambridge University Press. ISBN 0-521-48412-X

* H. Hopf, Uber die Topologie der Gruppen-Mannigfaltigkeiten und ihrer Verallgemeinerungen, Ann. of Math. 42 (1941), 22-52. Reprinted in Selecta Heinz Hopf, pp. 119-151, Springer, Berlin (1964). MathSciNet | id = 4784

* Ross Moore, Sam Williams and Ross Talent: [*http://www-texdev.ics.mq.edu.au/Quantum/Quantum.ps Quantum Groups: an entrée to modern algebra*]

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