Zoran Skoda
1(x)1-Delta(1)

For any element hh in a weak bialgebra HH, denote

I(h)=ϵ(1 (2)h)1 (1)11ϵ(1 (1)h)1 (2)=Π¯ L(h)11Π L(h). I(h) = \epsilon(1_{(2)}h)1_{(1)}\otimes 1 - 1\otimes \epsilon(1_{(1)}h)1_{(2)} = \bar\Pi^L(h)\otimes 1 - 1\otimes\Pi^L(h).

We claim that Δ(1)I(h)=0\Delta(1)\cdot I(h) = 0. Indeed,

1 (1)ϵ(1 (2)h)1 (1)1 (2) = ϵ((1 (1)1 (1)) (2))(1 (1)1 (1)) (1)ϵ(1 (2)h)1 (1)1 (2) = ϵ(1 (2)1 (2))ϵ(1 (3)h)1 (1)1 (1)1 (3) = ϵ(1 (2)1 (2)h)1 (1)1 (1)1 (3) = ϵ(1 (2)h)1 (1)1 (3) = 1 (1)ϵ(1 (2)h)1 (3) = 1 (1)ϵ(1 (2)1 (1)h)1 (3)1 (2) = 1 (1)ϵ(1 (2)1 (2))ϵ(1 (1)h)1 (3)1 (3) = 1 (1)1 (2)ϵ(1 (1)h)1 (2)\array{ 1_{(1)}\epsilon(1_{(2')}h)1_{(1')}\otimes 1_{(2)} &=& \epsilon((1_{(1)} 1_{(1')})_{(2)}) (1_{(1)} 1_{(1')})_{(1)}\epsilon(1_{(2')}h)1_{(1')}\otimes 1_{(2)}\\ &=& \epsilon(1_{(2)}1_{(2')})\epsilon(1_{(3')}h)1_{(1)}1_{(1')}\otimes 1_{(3)}\\ &=& \epsilon(1_{(2)}1_{(2')}h)1_{(1)}1_{(1)'}\otimes 1_{(3)}\\ &=& \epsilon(1_{(2)} h) 1_{(1)} \otimes 1_{(3)}\\ &=& 1_{(1)}\otimes\epsilon(1_{(2)}h)1_{(3)}\\ &=& 1_{(1)}\otimes\epsilon(1_{(2)}1_{(1')}h) 1_{(3)}1_{(2')}\\ &=& 1_{(1)}\otimes\epsilon(1_{(2)}1_{(2')})\epsilon(1_{(1')}h)1_{(3)}1_{(3')}\\ &=& 1_{(1)}\otimes 1_{(2)}\epsilon(1_{(1')}h)1_{(2')} }

It follows that (11Δ(1))I(h)=I(h)0=I(h)(1\otimes 1 - \Delta(1)) I(h) = I(h) - 0 = I(h). In particular, for every hh, the element I(h)I(h) belongs to the right principal ideal generated by 11Δ(1)1\otimes 1 - \Delta(1).

Last revised on May 24, 2019 at 12:36:23. See the history of this page for a list of all contributions to it.