and in the
dual number plane as the line
In these cases the Lie algebra parameters have names:
angle,
hyperbolic angle, and
slope.[2] These species of angle are useful for providing
polar decompositions which describe sub-algebras of 2 x 2 real matrices.[3]
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.
According to historian Thomas W. Hawkins, it was
Élie Cartan that made Lie theory what it is:
While Lie had many fertile ideas, Cartan was primarily responsible for the extensions and applications of his theory that have made it a basic component of modern mathematics. It was he who, with some help from
Weyl, developed the seminal, essentially algebraic ideas of
Killing into the theory of the structure and representation of
semisimple Lie algebras that plays such a fundamental role in present-day Lie theory. And although Lie envisioned applications of his theory to geometry, it was Cartan who actually created them, for example through his theories of symmetric and generalized spaces, including all the attendant apparatus (
moving frames, exterior
differential forms, etc.)[4]
Lie's three theorems
In his work on transformation groups, Sophus Lie proved three theorems relating the groups and algebras that bear his name. The first theorem exhibited the basis of an algebra through
infinitesimal transformations.[5]: 96 The second theorem exhibited
structure constants of the algebra as the result of
commutator products in the algebra.[5]: 100 The
third theorem showed these constants are anti-symmetric and satisfy the
Jacobi identity.[5]: 106 As Robert Gilmore wrote:
Lie's three theorems provide a mechanism for constructing the Lie algebra associated with any Lie group. They also characterize the properties of a Lie algebra. ¶ The converses of Lie’s three theorems do the opposite: they supply a mechanism for associating a Lie group with any finite dimensional Lie algebra ...
Taylor's theorem allows for the construction of a canonical analytic structure function φ(β,α) from the Lie algebra. ¶ These seven theorems – the three theorems of Lie and their converses, and Taylor's theorem – provide an essential equivalence between Lie groups and algebras.[5]
^"Lie’s lasting achievements are the great theories he brought into existence. However, these theories – transformation groups, integration of differential equations, the geometry of contact – did not arise in a vacuum. They were preceded by particular results of a more limited scope, which pointed the way to more general theories that followed. The line-sphere correspondence is surely an example of this phenomenon: It so clearly sets the stage for Lie’s subsequent work on contact transformations and symmetry groups." R. Milson (2000) "An Overview of Lie’s line-sphere correspondence", pp 1–10 of The Geometric Study of Differential Equations, J.A. Leslie & T.P. Robart editors,
American Mathematical SocietyISBN0-8218-2964-5 , quotation pp 8,9
M.A. Akivis & B.A. Rosenfeld (1993) Élie Cartan (1869–1951), translated from Russian original by V.V. Goldberg, chapter 2: Lie groups and Lie algebras,
American Mathematical SocietyISBN0-8218-4587-X .
P. M. Cohn (1957) Lie Groups, Cambridge Tracts in Mathematical Physics.
J. L. Coolidge (1940) A History of Geometrical Methods, pp 304–17, Oxford University Press (Dover Publications 2003).
Robert Gilmore (2008) Lie groups, physics, and geometry: an introduction for physicists, engineers and chemists, Cambridge University Press
ISBN9780521884006 .
F. Reese Harvey (1990) Spinors and calibrations, Academic Press,
ISBN0-12-329650-1 .
Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
ISBN978-3319134666.
Sattinger, David H.; Weaver, O. L. (1986). Lie groups and algebras with applications to physics, geometry, and mechanics. Springer-Verlag.
ISBN3-540-96240-9.