Computing of the number of right coideal subalgebras of [formula omitted

Abstract

In this paper we complete the classification of right coideal subalgebras containing all grouplike elements for the multiparameter version of the quantum group U q ( so 2 n + 1 ) , q t ≠ 1 . It is known that every such subalgebra has a triangular decomposition U = U − H U + , where U − and U + are right coideal subalgebras of negative and positive quantum Borel subalgebras. We found a necessary and sufficient condition for the above triangular composition to be a right coideal subalgebra of U q ( so 2 n + 1 ) in terms of the PBW-generators of the components. Furthermore, an algorithm is given that allows one to find an explicit form of the generators. Using a computer realization of that algorithm, we determined the number r n of different right coideal subalgebras that contain all grouplike elements for n ⩽ 7 . If q has a finite multiplicative order t > 4 , the classification remains valid for homogeneous right coideal subalgebras of the multiparameter version of the Lusztig quantum group u q ( so 2 n + 1 ) (the Frobenius–Lusztig kernel of type B n ) in which case the total number of homogeneous right coideal subalgebras and the particular generators are the same.