Lie algebra bundle

In Mathematics, a weak Lie algebra bundle

:$xi=(xi,\; p,\; X,\; heta),$

is a vector bundle $xi,$ over a base space "X" together with a morphism

:$heta\; :\; xi\; oplus\; xi\; ightarrow\; xi$

which induces a Lie algebra structure on each fibre $xi\_x,$.

A Lie algebra bundle $xi=(xi,\; p,\; X),$ is a vector bundle in whicheach fibre is a Lie algebra and for every "x" in "X", there is an open set $U$ containing "x", a Lie algebra "L" and a homeomorphism

:$phi:U\; imes\; L\; o\; p^\{-1\}(U),$

such that

:$phi\_x:x\; imes\; L\; ightarrow\; p^\{-1\}(x),$

is a Lie algebra isomorphism.

Any Lie algebra bundle is a weak Lie algebra bundle but the converse need not be true in general.

References

*A.Douady et M.Lazard, Espaces fibres en algebre de Lie et en groups, Invent. math., Vol. 1, 1966, pp.133-151
*B.S.Kiranagi, Lie Algebra bundles, Bull. Sci. Math., 2^e serie, 102(1978), 57-62.
*B.S.Kiranagi, Semi simple Lie algebra bundles, Bull. Math de la Sci. Math de la R.S.de Roumaine, 27 (75), 1983, 253-257.
*B.S.Kiranagi and G.Prema, On complete reducibility of Module Bundles, Bull. Austral. Math Soc., 28 (1983), 401-409.
*B.S.Kiranagi and G.Prema, Cohomology of Lie algebra bundles and its applications, Ind. J. Pure and Appli. Math. 16(7): 1985, 731/735.
*B.S.Kiranagi and G.Prema, Lie algebra bundles defined by Jordan algebra bundles, Bull. Math. Soc.Sci.Math.Rep.Soc. Roum., Noun. Ser. 33 (81), 1989, 255-264.
*B.S.Kiranagi and G.Prema, On complete reducibility of Bimodule bundles, Bull. Math. Soc. Sci.Math. Repose; Roum, Nouv.Ser. 33 (81), 1989, 249-255.
*B.S.Kiranagi and G.Prema, A decomposition theorem of Lie algebra Bundles, Communications in Algebra 18 (6), 1990, 1869-1877 .
*B.S.Kiranagi, G.Prema and C.Chidambara, Rigidity theorem for Lie algebra Bundles, Communications in Algebra 20 (6), 1992, pp. 1549 - 1556.

ee also

*Algebra bundle
*Adjoint bundle