Metric lie algebra pdf

We show that any metric lie algebra without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary lie. However, it is known that lie groups, which are not a direct product of compact and abelian groups, have no biinvariant metric. In this paper, we give a systematic study on isotropic ideals of metric n lie algebras. Let v be a vector space, and let b be a non degenerate sym metric bilinear form on v. Contents preface pagexi 1 introduction 1 2 liegroups. A metric lie algebra is a lie algebra equipped with an invariant nondegenerate symmetric bilinear form. Let g be a complex connected lie group, with lie algebra g lieg, and let k. We define the notion of a double extension of a metric lie nalgebra by another lie nalgebra and prove that all metric lie nalgebras are obtained from the simple and onedimensional ones by.

Among all the riemannian metrics on a lie groups, those for which the left. The present paper contains a systematic study of the structure of metric lie algebras, i. Consider representing the rotation axis u u1,u2,u3 of unit magnitude and the rotation angle. Aspects of uduality in blg models with lorentzian metric 3algebras. We show that any metric lie algebra without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary lie algebra l by an orthogonal lmodule a in a.

Metric 2step nilpotent lie algebras associated with graphs meera mainkar central michigan university vi workshop on differential geometry joint work with rachelle decoste and lisa demeyer. For symmetric spaces this lie group is called the transvection group 24. It is called indecomposable if it is not the direct sum of two metric lie algebras. Metric lie algebras and quadratic extensions springerlink. Lie n algebra and prove that all metric lie na lgebras are obtained from the simple and onedimensional ones b y itera ting the oper a tions of orthogonal direct sum and do uble extension. Chapter 18 metrics, connections, and curvature on lie groups. Lie derivative of a metric determinant physics forums. The paradigm of lie algebras is the vector space of matrices with the commutator of. Computing biinvariant pseudometrics on lie groups for. This lie 3algebra is isomorphic to a 4dimensional simple lie 3algebra a 4 see up to signs, and of course, has a metric lie 3algebra structure. Dg0312243 we developed a general classification scheme for metric lie algebras, i. Why the induced metric from the lie algebra of lie group. A metric lie algebra g,h,i is a lie algebra gwith an nondegenerate symmetric bilinear form h,i on g, which is adinvariant, i. We show that any metric lie algebra g without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary lie algebra l by an orthogonal lmodule a in a canonical way.

Notes on lie algebras cornell department of mathematics. In the literature these lie algebras are also called quadratic lie algebras. Introduction to lie groups and lie algebras stony brook math. Pdf metric lie algebras with maximal isotropic centre. In 12 we developed a general classification scheme for metric lie algebras, i. Im hoping to clear up some confusion i have over what the lie derivative of a metric determinant is. We also show a third characterisation for six extreme cases of 3leibniz algebras as graded lie superalgebras. Pdf aspects of uduality in blg models with lorentzian.