A Lie algebra, named after Sophus Lie, is an algebraic structure, which is mainly used for the study of geometric objects such as Lie groups and differentiable manifolds.
- 4.1 Definition
- 4.2 Ideal
- 4.3 set of Ado
- 5.1 Abelian Lie algebra
- 5.2 Nilpotent Lie algebra 5.2.1 definition
- 5.2.2 set of angel
- 5.5.1 set of Weyl
- 5.5.2 decomposition
- 5.5.3 classification
A Lie algebra is a vector space over a field together with an inner join
Which is called the Lie bracket and satisfying the following conditions:
- It is bilinear, that is linear in both arguments. Consequently, it applies and for all and all.
- It satisfies the Jacobi identity. The Jacobi identity is: applies to all.
- It applies to everyone.
Involve the first and third characteristic are taken together for all the anti-symmetry. When the body does not have characteristic 2, as can be seen from the anti-symmetry alone again the third property derived (you choose ). Lie bracket is not associative in general: does not have to be the same. Instead of a body and of a vector space can be defined more generally for a commutative unitary ring a Lie algebra.
- The vector space forms a Lie algebra, if one defines the Lie bracket as the cross product.
- The general linear Lie algebra of a vector space is the Lie algebra of endomorphisms of the commutator as Lie bracket. If, in particular, writes instead of one.
- An Ideal is made of endomorphisms with trace. It's called " special linear Lie algebra " and is denoted by or. Example, describes the Lie group of all matrices with real elements and determinant 1, then the tangent space to the identity matrix with the space of all real matrices can be identified with trace 0, and the matrix multiplication of the Lie group supplies over the commutator, the Lie bracket the Lie algebra.
- Generally, one can make any associative algebra to a Lie algebra by using as Lie bracket the commutator
In physics are the Lie groups or important as they describe rotations of the real or complex space in n dimensions. As I said, not only reminds the notation for the Lieprodukt to the cross product of vectors in three dimensions. The group property here means specifically, for example, that the product of two turns around a respective axis and rotation about a third axis must be displayed so that it is passed to the exponential function. In fact, the groups in the form of complex numbers can be represented, with the self-adjoint operators corresponding to the elements of the Lie algebra. In total, unitary operators are obtained in a Hilbert space. For details, see quantum mechanics, gauge theory and quantum chromodynamics.
Smooth vector fields
The smooth vector fields on a differentiable manifold form an infinite-dimensional Lie algebra. The vector fields operate as a Lie derivative on the ring of smooth functions. Be two smooth vector fields and a smooth function. We define the Lie bracket by
Lie algebra of a Lie group
The vector space of left-invariant vector fields on a Lie group is closed under this Kommutatoroperation and forms a finite-dimensional Lie algebra.
Smooth functions with the Poisson bracket
The smooth functions on a symplectic manifold form a Lie algebra with the Poisson bracket. Comparisons Poisson manifold.
Let and be two Lie algebras. A linear mapping is called a Lie algebra homomorphism if and only if for all.
In the category of Lie algebras, the Lie algebras the objects and the Lie algebra homomorphisms are the arrows.
A subalgebra of a Lie algebra is a vector subspace which is closed under the Lie bracket. That is, applies to all. A subalgebra of a Lie algebra is itself a Lie algebra.
A subalgebra is called an ideal if for all and true.
The ideals are precisely the kernels of homomorphisms Lie algebra.
On the quotient space is defined by a Lie algebra, the quotient algebra. Here were.
Set of Ado
The set of Ado (after the Russian mathematician Igor Dmitrievich Ado ) states that every finite-dimensional complex Lie algebra isomorphic to a subalgebra of is for a sufficiently large. That is, it can represent any finite-dimensional complex Lie algebra as a Lie algebra of matrices.
Types of Lie algebras
Abelian Lie algebra
A Lie algebra is abelian if the Lie bracket is identically zero.
Each vector space is an abelian Lie algebra, if one defines each Lie bracket than zero.
Nilpotent Lie algebra
Let be a Lie algebra. A descending central series is
Defined. Occasionally it is also written.
A Lie algebra is called nilpotent if its lower central series descending is zero, that is, for an index is valid.
Set of angel
Let be a finite-dimensional complex Lie algebra, then the following two statements are equivalent:
This set is named after the mathematician Friedrich Engel.
Solvable Lie algebra
Let be a Lie algebra. We define the derived (or -derived ) by series:
The derived series is occasionally also similar written.
A Lie algebra is called solvable if its derived series eventually becomes zero, ie for large.
A maximal solvable subalgebra is called Borel subalgebra.
Simple Lie algebra
A Lie algebra is called simple if it has no non-trivial ideals and is not abelian.
The Lie algebras simplicity is used differently. This can lead to confusion. If one conceives a Lie algebra as algebraic structure, as is the requirement that it must not be abelian, unnatural.
Semi- Simple Lie algebra
A Lie algebra is called semisimple if it is the direct sum of simple Lie algebras.
For a finite-dimensional Lie algebra the following statements are equivalent:
Set of Weyl
Let be a semisimple, finite dimensional complex Lie algebra, then any finite dimensional representation of completely reducible, ie as a direct sum of irreducible representations dismantled. The set is named after Hermann Weyl.
Semi- Simple Lie algebras have a decomposition
In a Cartan subalgebra and root spaces, see root system # Lie algebras.
Simple half complex Lie algebras can be classified by their root systems; this classification was completed in 1900 by Élie Cartan.
Reductive Lie algebra
A Lie algebra is called reductive if
With the center of the Lie algebra
Applies. A Lie algebra is precisely then reductive if every finite- dimensional representation is completely reducible. In particular, half- Simple Lie algebras are reductively by the theorem of Weyl.
Real Lie algebras
A selection of real Lie algebras