16 Facts About Lie group

In mathematics, a Lie group is a group that is a differentiable manifold.

Lie group stated that all of the principal results were obtained by 1884.

The term groupes de Lie first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse.

Lie group's ideas did not stand in isolation from the rest of mathematics.

The initial application that Lie group had in mind was to the theory of differential equations.

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Real Lie group is a group that is a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps.

Complex Lie group is defined in the same way using complex manifolds rather than real ones, and holomorphic maps.

Similarly, using an alternate metric completion of, one can define a p-adic Lie group over the p-adic numbers, a topological group which is an analytic p-adic manifold, such that the group operations are analytic.

Lie group can be defined as a topological group that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds.

Meanwhile, for every finite-dimensional matrix Lie algebra, there is a linear group with this algebra as its Lie algebra.

Lie group is said to be simply connected if every loop in can be shrunk continuously to a point in.

An example of a simply connected Lie group is the special unitary Lie group SU, which as a manifold is the 3-sphere.

Lie subgroup of a Lie group is a Lie group that is a subset of and such that the inclusion map from to is an injective immersion and group homomorphism.

Lie algebra of any compact Lie group can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones.

The structure of an abelian Lie group algebra is mathematically uninteresting ; the interest is in the simple summands.

Identity component of any Lie group is an open normal subgroup, and the quotient group is a discrete group.