Lie bracket

The Lie bracket is an object from mathematics , in particular from the field of algebra and differential geometry . The Lie bracket is the multiplicative connection in a Lie algebra , i.e. a kind of multiplication on a set with a special algebraic structure . Examples of such a connection are the trivial Lie bracket , the matrix commutator, the cross product or the Poisson bracket . The Lie bracket and Lie algebra are named after the mathematician Sophus Lie .

definition

Let be a vector space over the body . An inner connection ${\displaystyle V}$ ${\displaystyle K}$

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

is called a Lie bracket if it has the following three properties:

• It is bilinear , that is, linear in both arguments. So it applies

${\displaystyle [ax+by,z]=a[x,z]+b[y,z]}$

and

${\displaystyle [z,ax+by]=a[z,x]+b[z,y]}$

for everyone and everyone . ${\displaystyle a,b\in K}$ ${\displaystyle x,y,z\in V}$

• It applies to everyone . ${\displaystyle [x,x]=0}$ ${\displaystyle x\in V}$
• It satisfies the Jacobi identity , that is, it applies

${\displaystyle [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0}$

for everyone . ${\displaystyle x,y,z\in V}$

A vector space together with a Lie bracket is called a Lie algebra .

properties

Antisymmetry

The antisymmetry of the Lie bracket follows from the first and the second property of the definition, i.e. for all . If the body does not have the characteristic , one can derive the property again from the antisymmetry alone . By exposing . ${\displaystyle [x,y]=-[y,x]}$ ${\displaystyle x,y\in V}$ ${\displaystyle K}$ ${\displaystyle 2}$ ${\displaystyle [x,x]=0}$ ${\displaystyle y=x}$

flexibility

Lie brackets are generally not associative , which means that the term does not have to be the same as the term . However, the Lie bracket satisfies the flexibility law , so it applies to all elements . ${\displaystyle [[x,y],z]}$ ${\displaystyle [x,[y,z]]}$ ${\displaystyle [[x,y],x]=[x,[y,x]]}$ ${\displaystyle x,y\in V}$

Examples

Trivial lie bracket

Is any vector space and are and are two elements of the space, then can be through ${\displaystyle V}$ ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle [a,b]:=0}$

a Lie bracket can always be defined. Vector spaces with a trivial Lie bracket are also called Abelian Lie algebras .

Matrix commutator

Let , and three matrices with entries in a field (for example the field of real or the field of complex numbers ). The commutator for square matrices is defined by ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle n\times n}$ ${\displaystyle K}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle [\cdot ,\cdot ]}$

${\displaystyle [A,B]:=A\cdot B-B\cdot A}$,

where is called the matrix multiplication . The calculation rules apply to the commutator ${\displaystyle \cdot }$ ${\displaystyle \lambda ,\mu \in K}$

${\displaystyle {\begin{aligned}\left[\lambda A+\mu B,C\right]&=(\lambda A+\mu B)\cdot C-C\cdot (\lambda A+\mu B)\\&=\lambda (A\cdot C-C\cdot A)+\mu (B\cdot C-C\cdot B)\\&=\lambda [A,C]+\mu [B,C]\,,\end{aligned}}}$

${\displaystyle [A,A]=A\cdot A-A\cdot A=0}$ and

${\displaystyle {\begin{aligned}\left[A,[B,C]\right]+\left[B,[C,A]\right]+\left[C,[A,B]\right]=&[A,B\cdot C-C\cdot B]+[B,C\cdot A-A\cdot C]+[C,A\cdot B-B\cdot A]\\=&A\cdot (B\cdot C-C\cdot B)-(B\cdot C-C\cdot B)\cdot A+B\cdot (C\cdot A-A\cdot C)\\&-(C\cdot A-A\cdot C)\cdot B+C\cdot (A\cdot B-B\cdot A)-(A\cdot B-B\cdot A)\cdot C\\=&0\,.\end{aligned}}}$

Therefore the commutator in the space of the matrix is ​​a Lie bracket. ${\displaystyle n\times n}$

The Pauli matrices are now used as a concrete example

${\displaystyle \sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{2}={\begin{pmatrix}0&-\mathrm {i} \\\mathrm {i} &0\end{pmatrix}},\quad \sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}$

viewed over the body of complex numbers. If one forms the commutator of and , then applies ${\displaystyle \mathbb {C} }$ ${\displaystyle \sigma _{1}}$ ${\displaystyle \sigma _{3}}$

${\displaystyle {\begin{aligned}\left[\sigma _{1},\sigma _{3}\right]&=\sigma _{1}\cdot \sigma _{3}-\sigma _{3}\cdot \sigma _{1}\\&={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\cdot {\begin{pmatrix}1&0\\0&-1\end{pmatrix}}-{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\cdot {\begin{pmatrix}0&1\\1&0\end{pmatrix}}\\&={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}-{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}\\&=-2\mathrm {i} {\begin{pmatrix}0&-\mathrm {i} \\\mathrm {i} &0\end{pmatrix}}\\&=-2\mathrm {i} \,\sigma _{2}\,.\end{aligned}}}$

Cross product

→ Main article : Cross product

For is the cross product ${\displaystyle a,b\in \mathbb {R} ^{3}}$

${\displaystyle a\times b={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}\times {\begin{pmatrix}b_{1}\\b_{2}\\b_{3}\end{pmatrix}}:={\begin{pmatrix}a_{2}b_{3}-a_{3}b_{2}\\a_{3}b_{1}-a_{1}b_{3}\\a_{1}b_{2}-a_{2}b_{1}\end{pmatrix}}}$

a lie bracket. Compared to the examples above, this multiplication is usually not written in brackets. The bilinearity and the identity can be read directly from the definition. In order to recognize the Jacobi identity , the term ${\displaystyle a\times a=0}$

${\displaystyle a\times \left(b\times c\right)+b\times \left(c\times a\right)+c\times \left(a\times b\right)}$

can be calculated component by component.

Lie bracket of vector fields

→ Main article : Lie derivative

Let and be two vector fields on the -dimensional smooth manifold . The Lie derivative is then defined by ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle n}$ ${\displaystyle M}$

${\displaystyle ({\mathcal {L}}_{X}Y)f=X(Y(f))-Y(X(f))}$.

This operator fulfills the defining properties of a Lie bracket. Therefore one also writes . ${\displaystyle (X,Y)\mapsto {\mathcal {L}}_{X}Y}$ ${\displaystyle [X,Y]:={\mathcal {L}}_{X}Y}$

Jacobi bracket

Let be a commutative ring , a commutative algebra over, and two derivatives of . Then it's through ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle \delta _{1},\delta _{2}\in \operatorname {Der} (B)}$ ${\displaystyle B}$

${\displaystyle [\delta _{1},\delta _{2}]:=\delta _{1}\delta _{2}-\delta _{2}\delta _{1}}$

defined operation a Lie bracket on the space of the derivatives. It is called the Jacobi bracket . Since the vector fields from the previous example are special derivatives and their Lie bracket is defined accordingly, this Lie bracket is a concrete example of a Jacobi bracket.

Poisson bracket

→ Main article : Poisson bracket

The Poisson bracket is a two-digit operation that operates on the algebra of smooth functions . It fulfills the defining properties of a Lie bracket and also the product rule ${\displaystyle \{\cdot ,\cdot \}}$

${\displaystyle \{fg,h\}=f\{g,h\}+\{f,h\}g}$

for all smooth functions , and . Poisson brackets are often used on functions that map from a smooth manifold into the real numbers. Such manifolds with a fixed Poisson bracket are called Poisson manifolds . For example, each symplectic manifold can naturally be enclosed in Poisson brackets. In local coordinates , the Poisson bracket has the representation ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle (q_{1},\ldots ,q_{n},p_{1},\ldots ,p_{n})}$

${\displaystyle \{f,g\}=\sum _{i=1}^{n}\left({\frac {\partial f}{\partial q_{i}}}{\frac {\partial g}{\partial p_{i}}}-{\frac {\partial f}{\partial p_{i}}}{\frac {\partial g}{\partial q_{i}}}\right)}$.

Individual evidence

1. ↑ ^{a b} James E. Humphreys: Introduction to Lie algebras and representation theory . Springer, New York 1997, ISBN 3-540-90053-5 , pp. 4 .
2. ^ R. Abraham, Jerrold E. Marsden , T. Ratiu: Manifolds, tensor analysis, and applications (= Applied mathematical sciences 75). 2nd Edition. Springer, New York NY et al. 1988, ISBN 0-387-96790-7 , pp. 278-279.
3. ^ Günter Scheja, Uwe Storch: Textbook of Algebra [Electronic Resource] . Vieweg + Teubner Verlag, Wiesbaden 1988, ISBN 978-3-322-80092-3 , p. 105-106 .