In mathematics, particularly in algebra and analysis, a Lie algebra (honoring the work and memory of Sophus Lie ) is a mathematical system that involves geometric objects linking Lie groups and differentiable varieties.
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- 1 Definitive framework
- 2 Alternative definition
- 1 Example
- 3 Verbigracia
- 4 Algebraic morphisms and subsystems
- 5 Lie algebra classes
- 6 References
- 7 Sources
A Lie algebra g is a vector K-space (where K is the body of real or complex numbers) along with a binary operation [·, ·]: g × g → g , called the Lie bracket, which satisfies the following conditions:
- It is bilinear, that is, [ ax+ by , z ] = a [ x , z ] + b [ y , z ] and [ z , ax + by ] = a [ z , x ] + b [ z , y ] for all a , b in K and all x , y , z in g .
- Satisfies Jacobi’s identity, that is, [[ x , y ], z ] + [[ z , x ], y ] + [[ y , z ], x ] = 0 for all x , y , z in g . That is, the sum of the associated products of 3 elements of g , considered in counterclockwise cycle is 0 .
- For all xin g [ x , x ] = 0.
Let us note that from the first and third joint properties, we conclude [ x , y ] = – [ y , x ] for all x , and in g it is anticommutative . Also keep in mind that the multiplication determined by the Lie bracket is not generally associative, that is, [[ x , y ], z ] is not necessarily equal to [ x , [ y , z ]].
Let E be an n-dimensional vector space over a field K- in what follows the field of real or complex numbers- in which a vector composition operation (sic) has been defined : each pair of vectors corresponds to a vector v = [u, w], called commutator of vectors u and w. This switching operation verifies the following conditions:
- [au + bw, v] = a [u, v] + b [w, v] where a and b are scalars
- [u, w] + [w, v] = 0
- [u, [v, w]] + [v, [w, u]] + [w, [u, w]] = 0
- All vector K-space with the commutation operation of the previous form is called Lie algebra on the K field. If K = R is the field of real numbers it will be called Lie’s real algebra ; if K = C, the field of complex numbers will be named Lie complex algebra 
- The Lie algebra of a Lie group, according to a proposition, is always real, we will call it plainly Lie algebra.
- If [v, w] = 0 for any two vectors v and w of E, the algebra E is called commutative 
- Lie’s real algebra R 3is obtained from the three-dimensional space R 3 with the vector product v × w as the commutator of the vectors v and w .
- Each vector space becomes a trivial abelianLie algebra if we define the Lie bracket as identically zero.
- The Euclidean space R 3becomes a Lie algebra with the Lie bracket given by the cross product.
- If an associative algebra A is givenwith multiplication *, a Lie algebra can be given by defining [ x , y ] = x * y – y * x . This expression is called the switch of x and y .
- Conversely, it can be shown that each Lie algebra can be immersed in another that arises from an associative algebrain that way.
- Other important examples of Lie algebras come from differential topology: vector fields in a differentiable variety form an infinite dimensional Lie algebra; for two vector fields X and Y , the Lie bracket [ X , Y ] is defined as:
[ X , Y ] f = (XY – YX) f
for each function f in the variety (here we consider the vector fields as differential operators that act on functions in a variety ). (The proper generalization of variety theory should determine this as the Lie algebra of the infinite dimensional Lie group of the diffeomorphisms of the variety .)
The vector space of the left-invariant vector fields in a Lie group is closed under this operation and is therefore a finite dimensional Lie algebra. One can alternatively think of the underlying space of the Lie algebra of a Lie group as the tangent space in the identity element of the group. Multiplication is the differential of the group switch, ( a , b ) | -> aba −1 b −1 , on the identity element.
As a concrete example, consider the Lie group SL (n, R ) of all n-by-n matrices at real values and determinant 1. The tangent space in the identity matrix can be identified with the space of all real matrices n -by-n with trace 0 and the Lie algebra structure that comes from the Lie group matches the one that arises from the matrix multiplication switch.
Algebraic morphisms and subsystems
A homomorphism φ: g -> h between the Lie algebra g and h on the same base body F is a linear F- function such that [φ ( x ), φ ( y )] = φ ([ x , y ] ) for all x and y in g . that the composition of such homomorphisms is again a homomorphism, and the Lie algebras on the body F , together with these morphisms , form a category . If such a homomorphism is bijective , it is called aisomorphism , and the two Lie algebras g and h are called isomorphs . For all practical purposes, isomorphic Lie algebras are identical.
A subalgebra of Lie algebra g is a linear sub space h of g such that [ x , y ] ∈ h for all x , y ∈ h . ie [ h , h ] ⊆ h . The subalgebra is then a Lie algebra.
An ideal of Lie algebra g is a linear subspace h of g such that [a, y] ∈ h for all a ∈ g and y ∈ h . ie [ g , h ] ⊆ h . All ideals are subalgebras. If h is an ideal of g , then the quotient space g / h becomes a Lie algebra defining [ x + h , y + h ] = [ x , y] + h for all x , y ∈ g . Ideals are precisely the nuclei of homomorphisms , and the fundamental theorem about homomorphisms is valid for Lie algebras .
Lie algebra classes
The Lie algebras real and complex can be classified to a certain degree, and this classification is an important step toward the classification of Lie groups. Each finite-dimensional real or complex Lie algebra is presented as the Lie algebra of a single Lie group simply connected to real or complex (Ado’s theorem), but there may be more than one group, even more than a related group, giving place to the same algebra. For example, the groups SO (3) (orthogonal matrices 3 × 3 of determinant 1) and SU (2) (unit matrices 2 × 2 of determinant 1), both give rise to the same Lie algebra, namely R 3 with the cross product. A Lie algebra is abelian if the Lie bracket is overridden, ie [ x, y ] = 0 for all x and y . More generally, a Lie g algebra is nilpotent if the descending central series
g ⊇ [ g , g ] ⊇ [[ g , g ], g ] ⊇ [[[ g , g ], g ], g ] ⊇ …
eventually becomes (English earlier or later) to zero. By Engel’s theorem, a Lie algebra is nilpotent if and only if for each u in g , the function ad ( u ): g -> g defined by
ad ( u ) ( v ) = [ u , v ]
it is nilpotent . More generally still, a Lie g algebra is soluble if the derived series
g ⊇ [ g , g ] ⊇ [[ g , g ], [ g , g ]] ⊇ [[[ g , g ], [ g , g ]], [[ g , g ], [ g , g ]] ] ⊇ …
eventually becomes (English earlier or later) to zero. A maximum soluble subalgebra is called a Borel subalgebra .
A Lie g algebra is called semi-simple if the only soluble ideal of g is trivial. Equivalent, g is semi-simple if and only if the Killing form K ( u , v ) = tr (ad ( u ) ad ( v )) is non-degenerate; here tr denotes the trace operator. When the body F is of zero characteristic, g is semi-simple if and only if each representation is totally reducible, that is, for each invariant subspace of the representation there is an invariant complement (Weyl’s theorem). A Lie algebra is simpleif you have no nontrivial ideal. In particular, a simple Lie algebra is semi-simple , and more generally, semi-simple Lie algebras are direct sum of simple . The Lie algebras complex semi-simple are classified through their root systems