The Homogeneous And Graded Coideal Subalgebras Of U_χ

The main object studied in this article is the homogeneous coideal subalgebras of the Drinfeld double of Nichols algebras of diagonal type Ux where the χ is a symmetrical bicharacter over a free abelian group with finite rank, and the χ corresponds to a finite Weyl groupoid. This article establishes the condition when the multiplication of a homogeneous coideal subalgebra of the nonnegative part of Uχ and that of the nonpositive part of Uχ is still a homogeneous coideal subalgebra [see Theorem.6.6]. And this result implies that the homogeneous coideal subalgebras of Uχ can be found out through checking the morphisms of Weyl groupoid.