Lie theory

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Lie theory (/ˈl/ LEE) is an area of mathematics, developed initially by Sophus Lie. Early expressions of Lie theory are found in books composed by Lie with Friedrich Engel and Georg Scheffers from 1888 to 1896.

In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations.

The hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions.

In the longer term, it has not been the direct application of continuous symmetry to geometric questions that has made Lie theory a central chapter of contemporary mathematics. The fact that there is a good structure theory for Lie groups and their representations has made them integral to large parts of abstract algebra. Some major areas of application have been found, for example in automorphic representations and in mathematical physics, and the subject has become a busy crossroads.

Elementary Lie theory[edit]

The one-parameter groups are the first instance of Lie theory. The compact case arises through Euler's formula in the complex plane. Other one-parameter groups occur in the split-complex number plane as the unit hyperbola

\lbrace \exp(j t) = \cosh(t) + j \sinh(t) : t \in R \rbrace,

and in the dual number plane as the line \lbrace \exp(\epsilon t) = 1 + \epsilon t :  t \in R \rbrace. In these cases the Lie algebra parameters have names: angle, hyperbolic angle, and slope. Using the appropriate "angle", and a radial vector, any one of these planes can be given a polar decomposition. Any one of these decompositions, or Lie algebra renderings, may be necessary for rendering the Lie subalgebra of a 2 × 2 real matrix.

There is a classical 3-parameter Lie group and algebra pair: the quaternions of unit length which can be identified with the 3-sphere. Its Lie algebra is the subspace of quaternion vectors. Since the commutator ij − ji = 2k, the Lie bracket in this algebra is twice the cross product of ordinary vector analysis.

Another elementary 3-parameter example is given by the Heisenberg group and its Lie algebra. Standard treatments of Lie theory often begin with the Classical groups.

Aspects of Lie theory[edit]

The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups. Structure is captured in Root systems and root data. Lie theory is frequently built upon a study of the classical linear algebraic groups. Special branches include Weyl groups, Coxeter groups, and buildings. The classical subject has been extended to Groups of Lie type.

In 1900 David Hilbert challenged Lie theorists with his Fifth Problem presented at the International Congress of Mathematicians in Paris.

See also[edit]

Further reading[edit]