Exchange The Hopf Road Coalgebra Groups Structural Classification

Usually, there are Sweedler's 4-dimensional Hopf algebra, group algebra and universal enveloping algebra of Lie algebras as examples of Hopf algebras. Let k be a field and q∈k^×. Let g be a finite dimensional semisimple Lie algebra with Cartan matrix A = ( a_ij)_n×n. If A is diagonalizable, the quantum universal enveloping algebra U_q( g )of g is a Hopf algebra over k . The structure of U_q( g ) varies as q varies in k^×. In general, it is not easy to establish a Hopf algebra.Recently, many mathematicians have devoted to studying the structure of algebra with the help of quiver. Quivers have played an important role in the representation theory of algebras. In 2002, C.Cibils and M.Rosso obtained a equivalent condition for a path coalgebra of a quiver to admit a graded Hopf algebra structure. The representation theory of (graded) Hopf algebras has applications in many branches of physics and mathematics such as the construction of solutions to the quantum Yang-Baxter equation , which was completed by S.Majid in 1990. Quivers have also been used in gauge and string theory. Recently, S.Zhang, Y.Zhang and H.X.Chen gave the classification of Hopf path coalgebra kQ^c with pointed arrows, Hopf path algebras kQ^s and type one Hopf path coalgebras kG[ kQ₁ ] and Hopf path algebra kQ^a with pointed arrows (when Q is finite) with pointed module structures.Let G be a group and kG be the group algebra of G over a field k . It is well known that the kG -Hopf bimodule category ^kG_kGM^kG_kG is equivalent to the direct product category∏_C∈K(G) M_kZu(C), where K( G) is the set of conjugate classes in G , u : K (G )�'G is a map such that u (C )∈C for any C∈K (G ), Z_u(C) ={ g∈G | gu (C ) = u (C )g} and M_kZu(C) denotes the category of right kZ_u(C) modules. In this paper, we discuss the isomorphic classication of Hopf path coalgebras kQ^c and the structures of Hopf subalgebras of kG[kQ₁] of kQ^c in case G = D₁ and D₂ , respectively. Some conclusions are described as following: Theorem 4.3 Suppose G = D₁= {1 , g}is a cyclic group of order 2. Let k be a fixed field with char ( k )≠2,r be a ramification data of G with r₁ = mand r_g = q, and...