View source for Bicrossed product of Hopf algebra

{{short description|Concept in Hopf algebra}}
{{Orphan|date=March 2024}}

In [[quantum group]] and [[Hopf algebra]], the '''bicrossed product''' is a process to create new Hopf algebras from the given ones. It's motivated by the [[Zappa–Szép product]] of groups. It was first discussed by M. Takeuchi in 1981,<ref>{{citation|last=Takeuchi|first=M.|title=Matched pairs of groups and bismash products of Hopf algebras|journal=Comm. Algebra|volume=9|issue=8|pages=841&ndash;882|year=1981|doi=10.1080/00927878108822621}}</ref> and now a general tool for construction of Drinfeld quantum double.<ref>{{Citation|last1=Kassel|first1=Christian|title=Quantum groups|url=https://archive.org/details/quantumgroups0000kass|volume=155|year=1995|series=Graduate Texts in Mathematics|location=Berlin, New York|publisher=Springer-Verlag|doi=10.1007/978-1-4612-0783-2|isbn=9780387943701|url-access=registration}}</ref><ref>{{Citation|last1=Majid|first1=Shahn|title=Foundations of quantum group theory|year=1995|publisher=Cambridge University Press|doi=10.1017/CBO9780511613104|isbn=9780511613104}}</ref>

== Bicrossed product ==

Consider two bialgebras <math>A</math> and <math>X</math>, if there exist linear maps <math>\alpha:A\otimes X \to X</math>  turning  <math>X</math> a [[module coalgebra]] over <math>A</math>, and <math>\beta: A\otimes X\to A</math>  turning <math>A</math> into a right [[module coalgebra]] over  <math>X</math>. We call them a pair of matched bialgebras, if we set <math>\alpha(a\otimes x)=a\cdot x</math> and <math>\beta(a\otimes x)=a^x</math>, the following conditions are satisfied

<math>a\cdot (xy)=\sum_{(a),(x)}(a_{(1)} \cdot x_{(1)}) (a_{(2)}^{x_{(2)}} \cdot y)</math>

<math>a\cdot 1_X=\varepsilon_A(a)1_X</math>

<math>(ab)^x=\sum_{(b),(x)}a^{b_{(1)} \cdot x_{(1)}} b_{(2)}^{x_{(2)}}</math>

<math>1_A^x=\varepsilon_X(x)1_A</math>

<math>\sum_{(a),(x)}a_{(1)}^{x_{(1)}} \otimes a_{(2)}\cdot x_{(2)}=\sum_{(a),(x)}a_{(2)}^{x_{(2)}}\otimes  a_{(1)}\cdot x_{(1)}</math>

for all <math>a,b\in A</math> and <math>x,y\in X</math>. Here the Sweedler's notation of [[Hopf algebra|coproduct of Hopf algebra]] is used.

For matched pair of Hopf algebras <math>A</math> and <math>X</math>, there exists a unique Hopf algebra over <math>X\otimes A</math>, the resulting Hopf algebra is called bicrossed product of <math>A</math> and <math>X</math> and denoted by <math>X \bowtie A</math>,

* The unit is given by <math>(1_X\otimes 1_A)</math>; 
* The multiplication is given by <math>(x\otimes a)(y\otimes b)=\sum_{(a),(y)}x(a_{(1)}\cdot y_{(1)}) \otimes a_{(2)}^{y_{(2)}} b</math>;
* The counit is <math>\varepsilon(x\otimes a)=\varepsilon_X(x)\varepsilon_A(a)</math>;
* The coproduct is <math>\Delta(x\otimes a)=\sum_{(x),(a)} (x_{(1)}\otimes a_{(1)}) \otimes (x_{(2)}\otimes a_{(2)})</math>;
* The antipode is <math>S(x\otimes a)=\sum_{(x),(a)}S(a_{(2)})\cdot S(x_{(2)}) \otimes S(a_{(1)})^{S(x_{(1)})}</math>.

== Drinfeld quantum double ==

For a given Hopf algebra <math>H</math>, its dual space <math>H^*</math> has a canonical Hopf algebra structure and <math>H</math> and <math>H^{*cop}</math> are matched pairs. In this case, the bicrossed product of them is called Drinfeld quantum double <math>D(H)=H^{*cop}\bowtie H</math>.

== References ==
{{reflist}}

{{DEFAULTSORT:bicrossed product}}
[[Category:Hopf algebras]]