An extension of Galois theory to the case of continuous symmetry groups lie groups lie algebras and some of their applications pdf one of Lie’s principal motivations. Lie group with complex multiplication. Lie group of “local” symmetries of a manifold. Lie group typically playing the role of a symmetry of a physical system.
Lie groups play an important role, via their connections with Galois representations in number theory. Thus both multiplication and inversion are differentiable maps. Lie subgroup” of a Lie group that is not closed. See the discussion below of Lie subgroups in the section on basic concepts. Restricting attention to matrix Lie groups simplifies the definition of the Lie algebra and the exponential map. The following are standard examples of matrix Lie groups.
Lie groups occur in abundance throughout mathematics and physics. In two dimensions, if we restrict attention to simply connected groups, then they are classified by their Lie algebras. Lie algebras of dimension two. 14, 52, 78, 133, and 248. The product of two Lie groups is a Lie group. Lie group is a Lie group.