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Lie algebras were introduced to study the concept of infinitesimal transformations by Sophus Lie in the 1870s, [1] and independently discovered by Wilhelm Killing [2] in the 1880s. The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group was used.
The Lie algebra can be thought of as the infinitesimal vectors generating the group, at least locally, by means of the exponential map, but the Lie algebra does not form a generating set in the strict sense. [2] In stochastic analysis, an Itō diffusion or more general Itō process has an infinitesimal generator.
Chevalley used these bases to construct analogues of Lie groups over finite fields, called Chevalley groups. The Chevalley basis is the Cartan-Weyl basis, but with a different normalization. The generators of a Lie group are split into the generators H and E indexed by simple roots and their negatives . The Cartan-Weyl basis may be written as
Lie's fundamental theorems underline that Lie groups can be characterized by elements known as infinitesimal generators. These mathematical objects form a Lie algebra of infinitesimal generators. Deduced "infinitesimal symmetry conditions" (defining equations of the symmetry group) can be explicitly solved in order to find the closed form of ...
The Lie algebra bracket captures the essence of the Lie group product in a sense made precise by the Baker–Campbell–Hausdorff formula. The elements of s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} are the "infinitesimal generators" of rotations, i.e., they are the elements of the tangent space of the manifold SO(3) at the identity element.
The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones. The structure of an abelian Lie algebra is mathematically uninteresting (since the Lie bracket is identically zero); the interest is in the simple ...
The universal enveloping algebra of a free Lie algebra on a set X is the free associative algebra generated by X.By the Poincaré–Birkhoff–Witt theorem it is the "same size" as the symmetric algebra of the free Lie algebra (meaning that if both sides are graded by giving elements of X degree 1 then they are isomorphic as graded vector spaces).
The Lie algebra of SU(n), denoted by (), can be identified with the set of traceless anti‑Hermitian n × n complex matrices, with the regular commutator as a Lie bracket. Particle physicists often use a different, equivalent representation: The set of traceless Hermitian n × n complex matrices with Lie bracket given by − i times the ...