Editing Hopf algebra (section)

== Examples ==
{| class="wikitable"
|-
!  !! Depending on !! Comultiplication !! Counit !! Antipode !! Commutative !! Cocommutative !! Remarks
|-
| [[group ring|group algebra]] ''KG'' || [[group (mathematics)|group]] ''G'' || Δ(''g'') = ''g'' ⊗ ''g'' for all ''g'' in ''G'' || ''ε''(''g'') = 1 for all ''g'' in ''G'' || ''S''(''g'') = ''g''<sup>−1</sup> for all ''g'' in ''G'' || if and only if ''G'' is abelian || yes ||
|-
| functions ''f'' from a finite{{efn|The finiteness of ''G'' implies that ''K<sup>G</sup>'' ⊗ ''K<sup>G</sup>'' is naturally isomorphic to ''K''<sup>''G''x''G''</sup>. This is used in the above formula for the comultiplication. For infinite groups ''G'', ''K<sup>G</sup>'' ⊗ ''K<sup>G</sup>'' is a proper subset of ''K''<sup>''G''x''G''</sup>. In this case the space of functions with finite [[support (mathematics)|support]] can be endowed with a Hopf algebra structure.}} group to ''K'', ''K<sup>G</sup>'' (with pointwise addition and multiplication) || finite group ''G'' || Δ(''f'')(''x'',''y'') = ''f''(''xy'') || ''ε''(''f'') = ''f''(1<sub>''G''</sub>)|| ''S''(''f'')(''x'') = ''f''(''x''<sup>−1</sup>)  || yes || if and only if ''G'' is abelian ||
|- 
|[[Representative function]]s on a compact group||[[compact group]] ''G'' || Δ(''f'')(''x'',''y'') = ''f''(''xy'') || ''ε''(''f'') = ''f''(1<sub>''G''</sub>)|| ''S''(''f'')(''x'') = ''f''(''x''<sup>−1</sup>)  || yes || if and only if ''G'' is abelian || Conversely, every commutative involutive [[reduced algebra|reduced]] Hopf algebra over '''C''' with a finite Haar integral arises in this way, giving one formulation of [[Tannaka–Krein duality]].<ref>{{citation|last=Hochschild|first=G|title=Structure of Lie groups|year=1965|pages=14–32|publisher=Holden-Day}}</ref>
|-
| [[Regular function]]s on an [[algebraic group]] || || Δ(''f'')(''x'',''y'') = ''f''(''xy'') || ''ε''(''f'') = ''f''(1<sub>''G''</sub>)|| ''S''(''f'')(''x'') = ''f''(''x''<sup>−1</sup>)  || yes || if and only if ''G'' is abelian || Conversely, every commutative Hopf algebra over a field arises from a [[group scheme]] in this way, giving an [[equivalence (category theory)|antiequivalence]] of categories.<ref>{{Citation | last1=Jantzen | first1=Jens Carsten | author1-link=Jens Carsten Jantzen | title=Representations of algebraic groups | publisher=[[American Mathematical Society]] | location=Providence, R.I. | edition=2nd | series=Mathematical Surveys and Monographs | isbn=978-0-8218-3527-2 | year=2003 | volume=107}}, section 2.3</ref>
|-
| [[Tensor algebra]] T(''V'') || [[vector space]] ''V'' || Δ(''x'') = ''x'' ⊗ 1 + 1 ⊗ ''x'', ''x'' in ''V'',  Δ(1) = 1 ⊗ 1 || ''ε''(''x'') = 0  || ''S''(''x'') = −''x'' for all ''x'' in 'T<sup>1</sup>(''V'') (and extended to higher tensor powers) || If and only if dim(''V'')=0,1 || yes || [[symmetric algebra]] and [[exterior algebra]] (which are quotients of the tensor algebra) are also Hopf algebras with this definition of the comultiplication, counit and antipode
|-
| [[Universal enveloping algebra]] U(g) || [[Lie algebra]] ''g'' || Δ(''x'') = ''x'' ⊗ 1 + 1 ⊗ ''x'' for every ''x'' in ''g'' (this rule is compatible with [[commutator]]s and can therefore be uniquely extended to all of ''U'') || ''ε''(''x'') = 0 for all ''x'' in ''g'' (again, extended to ''U'') || ''S''(''x'') = −''x''  || if and only if ''g'' is abelian || yes ||
|-
| [[Sweedler's Hopf algebra]] ''H''=''K''[''c'', ''x'']/''c<sup>2</sup>'' = 1, ''x''<sup>2</sup> = 0 and ''xc'' = −''cx''.|| ''K'' is a field with [[Field characteristic|characteristic]] different from 2 || Δ(''c'') = ''c'' ⊗ ''c'',  Δ(''x'') = ''c'' ⊗ ''x'' + ''x'' ⊗ 1, Δ(1) = 1 ⊗ 1 || ''ε''(''c'') = 1 and ''ε''(''x'') = 0 || ''S''(''c'') = ''c''<sup>−1</sup> = ''c'' and ''S''(''x'') = −''cx'' || no || no || The underlying [[vector space]] is generated by {1, ''c'', ''x'', ''cx''} and thus has dimension 4. This is the smallest example of a Hopf algebra that is both non-commutative and non-cocommutative.
|-
| [[ring of symmetric functions]]<ref>See
{{cite journal |first=Michiel |last=Hazewinkel |title=Symmetric Functions, Noncommutative Symmetric Functions, and Quasisymmetric Functions |journal=Acta Applicandae Mathematicae |volume=75 |issue=1–3 |pages=55–83 |date=January 2003 |doi=10.1023/A:1022323609001 |url= |arxiv=math/0410468|s2cid=189899056 }}</ref> ||   
|| in terms of complete homogeneous symmetric functions ''h''<sub>''k''</sub> (''k'' &ge; 1):

Δ(''h<sub>k</sub>'') = 1 ⊗ ''h<sub>k</sub>'' + ''h''<sub>1</sub> ⊗ ''h''<sub>''k''−1</sub> + ... +  ''h''<sub>''k''−1</sub> ⊗ ''h''<sub>1</sub> + ''h<sub>k</sub>'' ⊗ 1.
|| ''ε''(''h<sub>k</sub>'') = 0
||  ''S''(''h<sub>k</sub>'') = (−1)<sup>''k''</sup> ''e<sub>k</sub>''
|| yes || yes ||
|}

Note that functions on a finite group can be identified with the group ring, though these are more naturally thought of as dual – the group ring consists of ''finite'' sums of elements, and thus pairs with functions on the group by evaluating the function on the summed elements.

=== Cohomology of Lie groups ===
The cohomology algebra (over a field <math>K</math>) of a Lie group <math>G</math> is a Hopf algebra: the multiplication is provided by the [[cup product]], and the comultiplication 
:<math>H^*(G,K) \rightarrow H^*(G\times G,K) \cong H^*(G,K)\otimes H^*(G,K)</math>
by the group multiplication <math>G\times G\to G</math>. 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)'''<ref name="Hopf, 1941">{{cite journal|last1=Hopf|first1=Heinz|title=Über die Topologie der Gruppen–Mannigfaltigkeiten und ihre Verallgemeinerungen|journal=Ann. of Math. |series= 2|date=1941|volume=42|issue=1|pages=22–52|doi=10.2307/1968985|language=de|jstor=1968985}}<!--|access-date=7 March 2016--></ref> Let <math>A</math> be a finite-dimensional, [[Graded-commutative|graded commutative]], graded cocommutative Hopf algebra over a field of characteristic 0. Then <math>A</math> (as an algebra) is a free exterior algebra with generators of odd degree.

=== 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".