RESEARCH ARTICLE

Automorphism group of Green ring of Sweedler Hopf algebra

  • Tingting JIA ,
  • Ruju ZHAO ,
  • Libin LI
Expand
  • School of Mathematical Science, Yangzhou University, Yangzhou 225002, China

Received date: 25 Sep 2015

Accepted date: 05 Dec 2015

Published date: 30 Aug 2016

Copyright

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

Let H2 be Sweedler’s 4-dimensional Hopf algebra and r(H2) be the corresponding Green ring of H2. In this paper, we investigate the automorphism groups of Green ring r(H2) and Green algebra F(H2) = r(H2) ⊗FZ, where F is a field, whose characteristics is not equal to 2. We prove that the automorphism group of r(H2) is isomorphic to K4, where K4 is the Klein group, and the automorphism group of F(H2) is the semidirect product of Z2 and G, where G= F \ {1/2} with multiplication given by a · b= 1− ab+ 2ab.

Cite this article

Tingting JIA , Ruju ZHAO , Libin LI . Automorphism group of Green ring of Sweedler Hopf algebra[J]. Frontiers of Mathematics in China, 2016 , 11(4) : 921 -932 . DOI: 10.1007/s11464-016-0565-4

1
Alperin R C. Homology of the group of automorphisms of k[x, y]. J Pure Appl Algebra, 1979, 15: 109–115

DOI

2
Chen H X, Huang H L, Ye Y, Zhang P. Monomial Hopf algebra. J Algebra, 2004, 275:212–232

DOI

3
Chen H X, Oystaeyen F V, Zhang Y H. The Green rings of Taft algebras. Proc Amer Math Soc, 2014, 142: 765–775

DOI

4
Chen H X, Wang W J. The coalgebra automorphisms of a Hopf algebras. http://www.paper.edu.cn (in Chinese)

5
Dicks W. Automorphisms of the polynomial ring in two variables. Publ Sec Mat Univ Autonoma Barcelona, 1983, 27: 155–162

DOI

6
Han J Z, Su Y C. Automorphism groups of Witt algebras. arXiv: 1502.01441v1

7
Huang H L, Chen H X, Zhang P. Generalized Taft Hopf algebras. Algebra Colloq, 2004, 11(3): 313–320

8
Li L B, Zhang Y H. The Green rings of the generalized Taft Hopf algebras. Contemp Math, 2013, 585: 275–288

DOI

9
Radford D E. On the coradical of a finite-dimensional Hopf algebra. Proc Amer Math Soc, 1975, 53: 9–15

DOI

10
Taft E J. The order of the antipode of a finite-dimensional Hopf algebra. Proc Natl Acad Sci USA, 1971, 68: 2631–2633

DOI

11
van der Kulk W. On polynomial rings in two variables. Nieuw Arch Wiskd (3), 1953, I: 33–41

12
Vesselin D, Yu J T. Automophisms of polynomial algebras and Dirichlet series. J Algebra, 2009, 321: 292–302

DOI

13
Yu J T. Recognizing automophisms of polynomial algebras. Contemp Math, 1998, 14:215–225

14
Zhao K M. Automorphisms of the binary polynomial algebras on integer rings. Chinese Ann Math Ser A, 1995, (4): 448–494 (in Chinese)

Outlines

/