Английская Википедия:Bicrossed product of Hopf algebra

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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, and now a general tool for construction of Drinfeld quantum double.

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 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>.