The Weakening Of The Corresponding Quantum Group. Twisted Affine Kac-moody Algebra G Promotion

Suppose G is an untwisted affinc Kac-Moody algcbras, the quantum group of (?) is the associative algebra U_q((?)) with generators E_i, F_i, K_i^±1 and D^±1 subject to some relations.In this paper, in order to weaken the invertibility to regularity,in which instead {K_i,K_i^-1,D_<sub>-1]} of {K_i, K_i, D, D},we also introduce projector L,such that K_tK|-_i = K|-_i = L^n-1Lⁿ = L.We define the bialgebra B_q^d((?)),which contains a subalgebra m_q^d((?)),is a weak Hopf algebra.This weak Hopf algebra m_q^d((?)) isomorphic to the weak Hopf algebra defined by Shilinyang.Bialgebra B_q^d((?)) is a weak Hopf algebra when n=3.Since m_q^d((?))/(l - L^n-1) (?) U_q((?)),we use Verma-type modules M_J^q(λ) over U_q(G) to define Verma-type modules ^ωM_J^q(λ) over m_q^d((?)) .Moreover,we generalize the A — form of M_J^q(λ) to the weak A— form of ^ωM_J^q(λ),we also give some properties of ^ωM_J^q(λ).