Sweedler's Hopf algebra

{{short description|Example of a non-commutative and non-cocommutative Hopf algebra}}

In mathematics, {{harvs|txt|first=Moss E.|last= Sweedler|authorlink=Moss Sweedler|year=1969|loc=p. 89–90}} introduced an example of an infinite-dimensional Hopf algebra, and Sweedler's Hopf algebra H₄ is a certain 4-dimensional quotient of it that is neither commutative nor cocommutative.

Definition

The following infinite dimensional Hopf algebra was introduced by {{harvtxt|Sweedler|1969|loc=pages 89–90}}. 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 H₄ is the quotient of this by the relations

:x² = 0, g² = 1, gx = –xg

so it has a basis 1, x, g, xg {{harv|Montgomery|1993|loc=p.8}}. 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 H₄⊗H₄. 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

{{Citation | last1=Armour | first1=Aaron | last2=Chen | first2=Hui-Xiang | last3=Zhang | first3=Yinhuo | title=Structure theorems of H₄-Azumaya algebras | doi=10.1016/j.jalgebra.2005.10.020 | mr=2264134 | year=2006 | journal=Journal of Algebra | issn=0021-8693 | volume=305 | issue=1 | pages=360–393| doi-access=free }}
{{Citation | last1=Montgomery | first1=Susan | title=Hopf algebras and their actions on rings | url=https://books.google.com/books?id=U895BpubbMkC | publisher=Published for the Conference Board of the Mathematical Sciences, Washington, DC | series=CBMS Regional Conference Series in Mathematics | isbn=978-0-8218-0738-5 | mr=1243637 | year=1993 | volume=82}}
{{Citation | last1=Sweedler | first1=Moss E. | title=Hopf algebras | url=https://books.google.com/books?id=8FnvAAAAMAAJ | publisher=W. A. Benjamin, Inc., New York | series=Mathematics Lecture Note Series | mr=0252485 | year=1969| isbn=9780805392548 }}
{{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 }}

Category:Hopf algebras