Restricted cohomology of modular Witt algebras

Evans, Tyler; Fialowski, Alice; Penkava, Michael

doi:10.1090/proc/12863

Restricted cohomology of modular Witt algebras
HTML articles powered by AMS MathViewer

by Tyler J. Evans, Alice Fialowski and Michael Penkava PDF

Proc. Amer. Math. Soc. 144 (2016), 1877-1886 Request permission

Abstract:

In this paper we compute the restricted $0$, $1$-, and $2$-dimensional cohomology groups with trivial coefficients of the modular Witt algebras $W$ for any prime $p>3$, and give explicit description of all one-dimensional restricted central extensions of $W$.

References

Richard E. Block, On the Mills-Seligman axioms for Lie algebras of classical type, Trans. Amer. Math. Soc. 121 (1966), 378–392. MR 188356, DOI 10.1090/S0002-9947-1966-0188356-3
D. B. Fuks and T. Dzh. Èvans, On the structure of the restricted Lie algebra for the Witt algebra in a finite characteristic, Funktsional. Anal. i Prilozhen. 36 (2002), no. 2, 69–74, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 36 (2002), no. 2, 140–144. MR 1922020, DOI 10.1023/A:1015622607840
Tyler J. Evans and Dmitry Fuchs, A complex for the cohomology of restricted Lie algebras, J. Fixed Point Theory Appl. 3 (2008), no. 1, 159–179. MR 2402915, DOI 10.1007/s11784-008-0060-y
D. B. Fuks, Cohomology of infinite-dimensional Lie algebras, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986. Translated from the Russian by A. B. Sosinskiĭ. MR 874337
D. B. Fuchs and I. M. Gelfand, Cohomology of the Lie algebra of vector fields on a circle, Functional Analysis and Its Applications, 2:342–343, 1968.
J. E. Humphreys, Existence of Levi factors in certain algebraic groups, Pacific J. Math. 23 (1967), 543–546. MR 218365
J. E. Humphreys, Restricted Lie algebras (and beyond), Algebraists’ homage: papers in ring theory and related topics (New Haven, Conn., 1981) Contemp. Math., vol. 13, Amer. Math. Soc., Providence, R.I., 1982, pp. 91–98. MR 685939
Nathan Jacobson, Abstract derivation and Lie algebras, Trans. Amer. Math. Soc. 42 (1937), no. 2, 206–224. MR 1501922, DOI 10.1090/S0002-9947-1937-1501922-7
N. Jacobson, Restricted Lie algebras of characteristic $p$, Trans. Amer. Math. Soc. 50 (1941), 15–25. MR 5118, DOI 10.1090/S0002-9947-1941-0005118-0
Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
George J. McNinch, Levi decompositions of a linear algebraic group, Transform. Groups 15 (2010), no. 4, 937–964. MR 2753264, DOI 10.1007/s00031-010-9111-8
M. Schottenloher, A mathematical introduction to conformal field theory, 2nd ed., Lecture Notes in Physics, vol. 759, Springer-Verlag, Berlin, 2008. MR 2492295
F. V. Weinstein, Filtering bases: a tool to compute cohomologies of abstract subalgebras of the Witt algebra, Unconventional Lie algebras, Adv. Soviet Math., vol. 17, Amer. Math. Soc., Providence, RI, 1993, pp. 155–216. MR 1254731
N. N. Yakovlev, Cohomologies of Witt algebras, Functional Analysis and Its Applications, 9(3):269–271, July-September 1975.