Color cyclic homology and Steinberg Lie color algebras

Abstract

The present paper contains two interrelated developments. First, the basic properties of the construction theory over the Steinberg Lie color algebras are developed in analogy with Steinberg Lie algebra case. This is done on the example of the central closed of the Steinberg Lie color algebras. The second development is that we define the first ϵ-cyclic homology group HC₁(R, ϵ) of the Γ-graded associative algebra R (which could be seemed as the generalization of cyclic homology group and the \mathbb{Z}/\mathrm{2}\mathbb{Z}-graded version of cyclic homology that was introduced by Kassel) to calculate the universal central extension of Steinberg Lie color algebras.

Keywords

Steinberg Lie color algebra / ϵ-cyclic homology group / universal central extension

References

[1] Blaga P A. On the Morita invariance of the Hochschild homology of superalgebras. Stud Univ Babeş-Bolyai Math, 2006, 51(1): 41–48

[2] Chen H, Guay N. Central extensions of matrix Lie superalgebras over ℤ/2ℤ-graded algebras. Algebr Represent Theory, 2013, 16(2): 591–604

[3] Gao Y. Steinberg unitary Lie algebras and skew-dihedral homology. J Algebra, 1996, 17: 261–304

[4] Gao Y, Shang S. Universal coverings of Steinberg Lie algebras of small characteristic. J Algebra, 2007, 311: 216–230

[5] Garland H. The arithmetic theory of loop groups. Publ Math Publ Math Inst Hautes Études Sci, 1980, 52: 5–136

[6] Kassel C. Algèbres enveloppantes et homologie cyclique. C R Acad Sci Paris Sér I Math, 1986, 303(16): 779–782

[7] Kassel C, Loday J-L. Extensions centrales d’algèbres de Lie. Ann Inst Fourier, 1982, 32(4): 119–142

[8] Loday J-L. Cyclic Homology. Grundlehren Math Wiss, Vol 301. Berlin: Springer, 1992

[9] Neher N. An introduction to universal central extensions of Lie superalgebras. In: Groups, Rings, Lie and Hopf Algebras (St. John’s, NF, 2001). Math Appl, 555. Dordrecht: Kluwer Acad Publ, 2003, 141–166

[10] Rittenberg V, Wyler D. Sequences of ℤ2⊕ℤ2 graded Lie algebras and superalgebras. J Math Phys, 1978, 19: 2193–2200

[11] Scheunert M, Zhang R B. Cohomology of Lie superalgebras and of their generations. J Math Phys, 1998, 39: 5024–5061

[12] Shang S, Chen H, Gao Y. Central extensions of Steinberg Lie superalgebras of small rank. Comm Algebra, 2007, 35(12): 4225–4244

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Higher Education Press and Springer-Verlag Berlin Heidelberg