Universal Central Extensions of the Matrix Leibniz Superalgebras 𝔰𝔩(m, n, 𝒜)

Abstract

The universal central extensions and their extension kernels of the smatrix Lie superalgebra 𝔰𝔩(m, n, 𝒜), the Steinberg Lie superalgebra 𝔰𝔩(m, n, 𝒜) in category SLeib of Leibniz superalgebras are determined under a weak assumption (compared with Mikhalev and Pinchuk, 2000) using the first Hochschild homology and the first cyclic homology group.

Key Words:

• Cyclic homology
• Kernel of central extension
• Leibniz or Steinberg superalgebras

Mathematics Subject Classification:

• Primary 17A32, 17A70
• Secondary 19D55

ACKNOWLEDGMENTS

D. Liu is supported by the NNSF (Grant 10671027, 10271076).

N. Hu is supported in part by the NNSF (Grant 10431040, 10271047), the TRAPOYT, and the FUDP from the MOE of China, the SRSTP from the STCSM, the Shanghai Priority Academic Discipline from the SMEC.

N. Hu thanks N. Bergeron, Y. Gao for financial support during a one-year visit in the Department of Mathematics and Statistics of York University. Part of this work was done during D. Liu's Ph.D. study in the Department of Mathematics at University of Bielefeld under the support of the “Asia–Euro Link Programme” and his post-doctor, study in the Department of Mathematics at Shanghai Jiaotong University. He is also deeply indebted to Profs. C. M. Ringel and Cuipo Jiang for their kind hospitality and continuous encouragement and instruction. Authors give their special thanks to Gao for the crucial comments. D. Liu is also supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 06KJB110003), and the China Postdoctoral Science Foundation.

Notes

Communicated by M. Rosso.

Dedicated to Professor Guangyu Shen on his 75th birthday.