Sweedler's Hopf Algebra

In mathematics, introduced an example of an infinite-dimensional Hopf algebra, and Sweedler's Hopf algebra ''H''₄ is a certain 4-dimensional quotient of it that is neither commutative nor cocommutative.

Definition

The following infinite dimensional Hopf algebra was introduced by . The Hopf algebra is generated as an algebra by three elements ''x'', ''g'' and ''g''^-1.

The coproduct Δ is given by
:Δ(g) = ''g'' ⊗''g'', Δ(''x'') = 1⊗''x'' + ''x'' ⊗''g''

The antipode ''S'' is given by
:''S''(''x'') = –''x'' ''g''⁻¹, ''S''(''g'') = ''g''⁻¹

The counit ε is given by
:ε(''x'')=0, ε(''g'') = 1

Sweedler's 4-dimensional Hopf algebra ''H''₄ is the quotient of this by the relations
:''x''² = 0, ''g''² = 1, ''gx'' = –''xg''
so it has a basis 1, ''x'', ''g'', ''xg'' . Note that Montgomery describes a slight variant of this Hopf algebra using the opposite coproduct, i.e. the coproduct described above composed with the tensor flip on ''H''₄⊗''H''₄. This Hopf algebra is isomorphic to the Hopf algebra described here by the Hopf algebra homomorphism $g\mapsto g$ and $x\mapsto gx$.

Sweedler's 4-dimensional Hopf algebra is a quotient of the Pareigis Hopf algebra, which is in turn a quotient of the infinite dimensional Hopf algebra.

References

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*{{Citation , last1=Van Oystaeyen , first1=Fred , last2=Zhang , first2=Yinhuo , title=The Brauer group of Sweedler's Hopf algebra H₄ , doi=10.1090/S0002-9939-00-05628-8 , mr=1706961 , year=2001 , journal=Proceedings of the American Mathematical Society , issn=0002-9939 , volume=129 , issue=2 , pages=371–380, doi-access=free , author1-link=Fred Van Oystaeyen , hdl=10067/378420151162165141 , hdl-access=free

Hopf algebras