Lie algebra

A Lie algebra , named after Sophus Lie is an algebraic structure provided with a clamp Lie is provided, d. H. there is an antisymmetric connection that fulfills the Jacobi identity . Lie algebras are mainly used to study geometric objects such as Lie groups and differentiable manifolds .

definition

A Lie algebra is a vector space over a body together with an internal link ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle K}$

${\displaystyle [\cdot ,\cdot ]\colon {\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}},\quad (x,y)\mapsto [x,y],}$

which is called a Lie bracket and satisfies the following conditions:

• It is bilinear, that is, linear in both arguments. Thus it is and for all and all . ${\displaystyle [ax+by,z]=a[x,z]+b[y,z]}$ ${\displaystyle [z,ax+by]=a[z,x]+b[z,y]}$ ${\displaystyle a,b\in K}$ ${\displaystyle x,y,z\in {\mathfrak {g}}}$
• It suffices for the Jacobi identity . The Jacobi identity is: applies to everyone . ${\displaystyle [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0}$ ${\displaystyle x,y,z\in {\mathfrak {g}}}$
• It applies to everyone . ${\displaystyle [x,x]=0}$ ${\displaystyle x\in {\mathfrak {g}}}$

The first and third properties taken together imply antisymmetry for all . If the body does not have characteristic 2, one can derive the third characteristic again from the antisymmetry alone (one choose ). ${\displaystyle [x,y]=-[y,x]}$ ${\displaystyle x,y\in {\mathfrak {g}}}$ ${\displaystyle K}$ ${\displaystyle y=x}$

Lie brackets are generally not associative: they don't have to be the same . However, the flexibility law always applies to Lie brackets . ${\displaystyle [[x,y],z]}$ ${\displaystyle [x,[y,z]]}$ ${\displaystyle [[x,y],x]=[x,[y,x]]}$

Instead of a field and a vector space, a Lie algebra can be defined more generally for a commutative unitary ring.

Examples

From algebra

• The vector space forms a Lie algebra if the Lie bracket is defined as the cross product . ${\displaystyle \mathbb {R} ^{3}}$
• The general linear Lie algebra for a vector space is the Lie algebra of the endomorphisms of with the commutator ${\displaystyle {\mathfrak {gl}}(V)}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle V}$

${\displaystyle [A,B]=AB-BA}$

as a lie bracket. Is special , you write or instead . ${\displaystyle V=K^{n}}$ ${\displaystyle {\mathfrak {gl}}_{n}(K)}$ ${\displaystyle {\mathfrak {gl}}(n,K)}$ ${\displaystyle {\mathfrak {gl}}(V)}$

• The endomorphisms with trace in also form a Lie algebra. It is called "special linear Lie algebra" and is denoted by or . This notation is derived from the Lie group of all matrices with real elements and determinant 1, because the tangent space of the identity matrix can be identified with the space of all real matrices with trace 0, and the matrix multiplication of the Lie group yields via the commutator the Lie bracket of Lie algebra. ${\displaystyle 0}$ ${\displaystyle {\mathfrak {gl}}(V)}$ ${\displaystyle {\mathfrak {sl}}(V)}$ ${\displaystyle {\mathfrak {sl}}_{n}(K)}$ ${\displaystyle {\rm {SL}}(n,\mathbb {R} )}$ ${\displaystyle (n\times n)}$ ${\displaystyle (n\times n)}$
• In general, any associative algebra can be made a Lie algebra by using the commutator as a Lie bracket ${\displaystyle A}$

${\displaystyle [x,y]=x\cdot y-y\cdot x}$

elects. Conversely, one can show that every Lie algebra can be understood as embedded in an associative algebra with a commutator, the so-called universal enveloping algebra .

• The derivatives on a (not necessarily associative) algebra become a Lie algebra with the commutator bracket.

From physics

In physics, the Lie groups or are important because they describe rotations of real or complex space in dimensions. For example, the commutator relation of the particular orthogonal group on which it is based is Lie algebra ${\displaystyle \mathrm {SO} (n)}$ ${\displaystyle \mathrm {SU} (n)}$ ${\displaystyle n}$ ${\displaystyle \mathrm {SO} (3)}$ ${\displaystyle {\mathfrak {so}}(3)}$

${\displaystyle [L_{i},L_{j}]=-\sum _{k=1}^{3}\varepsilon _{ijk}L_{k}}$

in the base of the three matrices ${\displaystyle 3\times 3}$

${\displaystyle (L_{i})_{jk}=\varepsilon _{ijk}}$

where denotes the Levi-Civita symbol . The three coordinate transformations for rotations around the coordinate axes are obtained by applying the matrix exponential to the generators ${\displaystyle \varepsilon }$

${\displaystyle R_{i}=\exp \left(\theta L_{i}\right)}$.

In general, every element of the Lie group and thus every rotation in three-dimensional real space can be determined by the exponential of a linear combination of basis vectors of Lie algebra ${\displaystyle \mathrm {SO} (3)}$ ${\displaystyle {\mathfrak {so}}(3)}$

${\displaystyle R=\exp \left(\sum _{i=1}^{3}\theta _{i}L_{i}\right)}$

represent.

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. Let be two smooth vector fields and one smooth function. We define the Lie bracket through ${\displaystyle X,Y}$ ${\displaystyle f}$

${\displaystyle \ [X,Y]f:=(XY-YX)f}$.

Lie algebra of a Lie group

The vector space of the left-invariant vector fields on a Lie group is closed under this commutator operation and forms a finite-dimensional Lie algebra.

Smooth functions with Poisson brackets

The smooth functions on a symplectic manifold form a Lie algebra with the Poisson bracket . Compare Poisson manifolds .

Constructions

From given Lie algebras one can construct new ones, see

• Affine Lie algebra
• Semi-direct sum

Homomorphism

Let and be two Lie algebras. A linear mapping is called a Lie algebra homomorphism if holds for all . ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {h}}}$ ${\displaystyle \varphi \colon {\mathfrak {g}}\longrightarrow {\mathfrak {h}}}$ ${\displaystyle [\varphi (x),\varphi (y)]=\varphi ([x,y])}$ ${\displaystyle x,y\in {\mathfrak {g}}}$

In the Lie algebras category, the Lie algebras are the objects and the Lie algebra homomorphisms are the arrows.

Sub algebra

definition

A subalgebra of a Lie algebra is a subspace that is closed under the Lie bracket. That is, for all true . A sub-algebra of a Lie algebra is itself a Lie algebra. ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {h}}\subseteq {\mathfrak {g}}}$ ${\displaystyle x,y\in {\mathfrak {h}}}$ ${\displaystyle [x,y]\in {\mathfrak {h}}}$

ideal

A sub-algebra is called an ideal if it applies to all and . ${\displaystyle {\mathfrak {i}}\subseteq {\mathfrak {g}}}$ ${\displaystyle [x,y]\in {\mathfrak {i}}}$ ${\displaystyle x\in {\mathfrak {g}}}$ ${\displaystyle y\in {\mathfrak {i}}}$

The ideals are exactly the kernels of the Lie algebra homomorphisms.

The quotient space is defined by a Lie algebra, the quotient algebra. Were there . ${\displaystyle {\mathfrak {g}}/{\mathfrak {i}}}$ ${\displaystyle [x+{\mathfrak {i}},y+{\mathfrak {i}}]:=[x,y]+{\mathfrak {i}}}$ ${\displaystyle x,y\in {\mathfrak {g}}}$

Ado theorem

Ado's theorem (after the Russian mathematician Igor Dmitrijewitsch Ado ) states that every finite-dimensional complex Lie algebra is isomorphic to a subalgebra that is large enough . This means that any finite-dimensional complex Lie algebra can be represented as a Lie algebra of matrices. ${\displaystyle {\mathfrak {gl}}_{n}(\mathbb {C} )}$ ${\displaystyle n}$

Types of Lie algebras

Abelian Lie algebra

A Lie algebra is Abelian if the Lie bracket is identical to zero.

Every vector space forms an Abelian Lie algebra if one defines every Lie bracket as zero.

Nilpotent Lie algebra

definition

Be a Lie algebra. A descending central row is through ${\displaystyle {\mathfrak {g}}}$

${\displaystyle {\mathcal {C}}^{0}{\mathfrak {g}}={\mathfrak {g}},\;\;\;{\mathcal {C}}^{1}{\mathfrak {g}}=[{\mathfrak {g}},{\mathfrak {g}}],\;\;\;{\mathcal {C}}^{2}{\mathfrak {g}}=[{\mathfrak {g}},{\mathcal {C}}^{1}{\mathfrak {g}}],}$

general

${\displaystyle {\mathcal {C}}^{n+1}{\mathfrak {g}}=[{\mathfrak {g}},{\mathcal {C}}^{n}{\mathfrak {g}}]}$

Are defined. Occasionally it is also written. ${\displaystyle {\mathfrak {g}}^{n}}$

A Lie algebra is called nilpotent if its descending central sequence becomes zero, that is, if it holds for an index . ${\displaystyle {\mathcal {C}}^{N}{\mathfrak {g}}=\{0\}}$ ${\displaystyle N}$

Set of angels

Let be a finite-dimensional complex Lie algebra, then the following two statements are equivalent: ${\displaystyle {\mathfrak {g}}}$

1. Lie algebra is nilpotent. ${\displaystyle {\mathfrak {g}}}$
2. For each is a nilpotent linear map. ${\displaystyle x\in {\mathfrak {g}}}$ ${\displaystyle {\rm {ad}}(x)\colon {\mathfrak {g}}\rightarrow {\mathfrak {g}},\ y\mapsto [x,y]}$

This sentence is named after the mathematician Friedrich Engel .

Solvable Lie algebra

Be a Lie algebra. We define the derived (or derived ) series by: ${\displaystyle {\mathfrak {g}}}$

${\displaystyle {\mathcal {D}}^{0}{\mathfrak {g}}={\mathfrak {g}},\;\;\;{\mathcal {D}}{\mathfrak {g}}=[{\mathfrak {g}},{\mathfrak {g}}],\;\;\;{\mathcal {D}}^{2}{\mathfrak {g}}=[{\mathcal {D}}{\mathfrak {g}},{\mathcal {D}}{\mathfrak {g}}]}$, general . ${\displaystyle {\mathcal {D}}^{n+1}{\mathfrak {g}}=[{\mathcal {D}}^{n}{\mathfrak {g}},{\mathcal {D}}^{n}{\mathfrak {g}}]}$

The derived series is occasionally written or similar. ${\displaystyle {\mathfrak {g}}^{(n)}}$

A Lie algebra is said to be solvable if its derivative series finally becomes zero, i.e. H. for big . The Cartan criterion is an equivalent condition for the case of characteristic 0 of the basic body. From the set of Lie to give properties of finite, resolvable complex Lie algebras. ${\displaystyle {\mathcal {D}}^{N}{\mathfrak {g}}=\{0\}}$ ${\displaystyle N}$

A maximally resolvable subalgebra is called a Borel subalgebra .

The greatest solvable ideal in a finite-dimensional Lie algebra is the sum of all solvable ideals and is called the radical of the Lie algebra.

Simple Lie algebra

A Lie algebra is called simple if it has no non-trivial ideal and is not Abelian.

The Lie algebras use simplicity differently. This can lead to confusion. If you understand a Lie algebra as an algebraic structure, the requirement that it must not be Abelian is unnatural.

Semi-simple Lie algebra

→ Main article : Semi-simple Lie algebra

A Lie algebra is called semi-simple if it is the direct sum of simple Lie algebras. ${\displaystyle {\mathfrak {g}}}$

For a finite-dimensional Lie algebra , the following statements are equivalent: ${\displaystyle {\mathfrak {g}}}$

1. ${\displaystyle {\mathfrak {g}}}$ is semi-easy.
2. The radical of disappears, i.e. H. there are no nontrivial solvable ideals. ${\displaystyle {\mathfrak {g}}}$
3. Cartan criterion : The killing form : is not degenerate ( denotes the trace of endomorphisms). ${\displaystyle \ k(u,v)={\rm {tr}}({\rm {ad}}(u)\circ {\rm {ad}}(v))}$ ${\displaystyle {\rm {tr}}}$

Weyl's theorem

→ Main article : Weyl's theorem (Lie algebra)

Let be a semi-simple, finite-dimensional, complex Lie algebra, then every finite-dimensional representation of is completely reducible, i.e. can be decomposed as a direct sum of irreducible representations. The set is named after Hermann Weyl . ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {g}}}$

Disassembly

Semi-simple Lie algebras have a decomposition

${\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus \bigoplus _{\alpha }{\mathfrak {g}}_{\alpha }}$

into a Cartan subalgebra and root spaces , see root system # Lie algebras . ${\displaystyle {\mathfrak {h}}}$ ${\displaystyle {\mathfrak {g}}_{\alpha }}$

classification

Semi-simple complex Lie algebras can be classified based on their root systems ; this classification was completed in 1900 by Élie Cartan .

Reductive Lie algebra

A Lie algebra is called reductive if ${\displaystyle {\mathfrak {g}}}$

${\displaystyle {\mathfrak {g}}={\mathfrak {z}}({\mathfrak {g}})\oplus \left[{\mathfrak {g}},{\mathfrak {g}}\right]}$

with the center of Lie algebra

${\displaystyle {\mathfrak {z}}({\mathfrak {g}})=\left\{X\in {\mathfrak {g}}:\left[X,Y\right]=0\ \forall \ Y\in {\mathfrak {g}}\right\}}$

applies. In this case is a semi-simple Lie algebra . ${\displaystyle \left[{\mathfrak {g}},{\mathfrak {g}}\right]}$

A Lie algebra is reductive if and only if every finite-dimensional representation is completely reducible. In particular, semisimple Lie algebras are reductive according to Weyl's theorem.

Real Lie algebras

A selection of real Lie algebras

1. one-dimensional: with ${\displaystyle \mathbb {R} }$ ${\displaystyle [.\,,.]\equiv 0}$
2. There are exactly two isomorphism classes of two-dimensional real Lie algebras with and . ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle [.\,,.]\equiv 0}$ ${\displaystyle [.\,,.]\not \equiv 0}$
3. three-dimensional:
   1. ${\displaystyle \mathbb {R} ^{3}}$
   2. Heisenberg algebra
   3. ${\displaystyle {\mathfrak {su}}(2)\cong {\mathfrak {so}}(3,\mathbb {R} )}$
   4. ${\displaystyle {\mathfrak {sl}}(2,\mathbb {R} )}$
4. six-dimensional: ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle \cong {\mathfrak {so}}(3,1)}$

literature

• Igor Frenkel , James Lepowsky , Arne Meurman : Vertex Operator Algebras and the Monster , Academic Press, New York (1989) ISBN 0-12-267065-5
• Anthony W. Knapp: Lie Groups Beyond an Introduction , Birkhäuser (2002) ISBN 0-8176-4259-5
• Jean-Pierre Serre : Complex Semisimple Lie Algebras , Springer, Berlin, 2001. ISBN 3-5406-7827-1

Web links

Wikibooks: Full Proof of Classification  - Learning and Teaching Materials

• Leistner: The classical Lie algebras and their root systems.