Lie Groups and Lie Algebras I pp 6-29 | Cite as

# Basic Notions

## Abstract

We will assume familiarity with the basic concepts of manifold theory. However in order to avoid misunderstandings some of them will be defined in the text. The basic field, by which we mean either the field ℝ of real numbers or the field ℂ of complex numbers, will be denoted by *K.* Unless stated otherwise, differentiability of functions will be understood in the following sense: in every case there exist as many derivatives as are needed. Differentiability of manifolds and maps is understood in the same sense. The Jacobian matrix of a system of differentiable functions *f* _{1},…, *f* _{m} of variables *x* _{1},…, *x* _{n} will be denoted by \(\frac{{\partial \left( {{{f}_{1}}, \ldots ,{{f}_{m}}} \right)}}{{\partial \left( {{{x}_{1}}, \ldots ,{{x}_{n}}} \right)}}
\).For *m = n* its determinant will be denoted by \(\frac{{D\left( {{{f}_{1}}, \ldots ,{{f}_{n}}} \right)}}{{D\left( {{{x}_{1}}, \ldots ,{{x}_{n}}} \right)}}
\)

## Keywords

Semidirect Product Differentiable Manifold Differentiable Structure Tangent Algebra Trivial Fibre Bundle## Preview

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