# Zoran Skoda Yetter-Drinfeld module algebra

Yetter-Drinfeld modules form a monoidal category. Yetter-Drinfeld algebra is a monoid in that category. It is an algebra and a comodule with the YD condition and the property that the coaction $\rho : M\to M\otimes H^{op}$ is an algebra map.

Let $H$ be a Hopf algebra. The compatibility of the left action $\blacktriangleright$ and the right coaction $\rho : X\mapsto X_{}\otimes X_{}$ for the left-right Yetter-Drinfeld $H$-module $M$ compatibility is

$(h_{(1)}\blacktriangleright X_{})\otimes h_{(2)} X_{} = (h_{(2)}\blacktriangleright X)_{} \otimes (h _{(2)}\blacktriangleright X)_{} h_{(1)},$

for all $h\in H$ and $X\in M$.

Proposition. If $\rho:M\to M\otimes H^{op}$ in other to check the YD module property it is sufficient to check it on the generators of algebras $H$ and $M$.

Proof. It is clear that the YD condition is linear. We therefore have to check the compatibility for products if the factors satisfy it. For the products $X Y$ in $M$ we compute

$\array{ (f_{(1)}\blacktriangleright (X Y)_{})\otimes f_{(2)} (X Y)_{} &=& (f_{(1)}\blacktriangleright (X_{} Y_{}))\otimes f_{(2)}Y_{} X_{} \\ &=&(f_{(1)}\blacktriangleright X_{}) (f_{(2)}\blacktriangleright Y_{})\otimes f_{(3)}Y_{} X_{} \\ &=&(f_{(1)}\blacktriangleright X_{})(f_{(3)}\blacktriangleright Y_{})_{}\otimes (f_{(3)}\blacktriangleright Y_{})_{} f_{(2)} X_{} \\ &=& (f_{(2)}\blacktriangleright X)_{}(f_{(3)}\blacktriangleright Y)_{}\otimes (f_{(3)}\blacktriangleright Y_{})_{})f_{(2)}\blacktriangleright X)_{} f_{(1)} \\ &=& ((f_{(2)}\blacktriangleright X)(f_{(3)}\blacktriangleright Y))_{}\otimes ((f_{(2)}\blacktriangleright X)(f_{(3)}\blacktriangleright Y))_{} f_{(1)} \\ &=& (f_{(2)}\blacktriangleright (X Y))_{}\otimes (f_{(2)}\blacktriangleright (X Y))_{} f_{(1)}, }$

and for the products $f g$ in $H$ we compute

$\array{ (f g)_{(1)} X_{}\otimes (f g)_{(2)} X_{} &=& (f_{(1)}\blacktriangleright (g_{(1)}\blacktriangleright X_{}))\otimes f_{(2)} (g_{(2)} X_{}) \\ &=& (f_{(2)}\blacktriangleright (g_{(2)}\blacktriangleright X)_{})\otimes (f_{(2)} \blacktriangleright (g_{(2)}\blacktriangleright X)_{}) g_{(1)} \\ &=& (f_{(2)}\blacktriangleright (g_{(2)}\blacktriangleright X))_{}\otimes (f_{(2)} \blacktriangleright (g_{(2)}\blacktriangleright X))_{} f_{(1)} g_{(1)} \\ &=& ((f g)_{(2)}\blacktriangleright X)_{}\otimes ((f g)_{(2)}\blacktriangleright X)_{} (f g)_{(1)} }$

Braided commutativity:

$X_{} (X_{}\blacktriangleright Y) = Y X$

Products on the left

$X_{}X'_{}(X'_{}\blacktriangleright(X_{}\blacktriangleright Y)) = X_{}(X_{}\blacktriangleright Y) X' = Y X X'$

Products on the right

$X_{} (X_{}\blacktriangleright (Y Y')) = X_{} (X_{}\blacktriangleright Y)(X_{}\blacktriangleright Y') = Y X_{} (X_{}\blacktriangleright Y') =Y Y' X$

Last revised on June 22, 2015 at 17:45:59. See the history of this page for a list of all contributions to it.