Lie theory

In mathematics, the researcher Sophus Lie (/ˈliː/ LEE) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is Lie sphere geometry. This article addresses his approach to transformation groups, which is one of the areas of mathematics, and was worked out by Wilhelm Killing and Élie Cartan. The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and root data. Lie theory has been particularly useful in mathematical physics since it describes the standard transformation groups: the Galilean group, the Lorentz group, the Poincaré group and the conformal group of spacetime. 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 and in the dual number plane as the line { exp ⁡ ( ε t ) = 1 + ε t : t ∈ R } . {displaystyle lbrace exp(varepsilon t)=1+varepsilon t:tin R brace .} 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. Early expressions of Lie theory are found in books composed by Sophus Lie with Friedrich Engel and Georg Scheffers from 1888 to 1896.