sl (2, C)

This article deals with Lie algebra , for the group see Special Linear Group . ${\displaystyle {\mathfrak {sl}}(2,C)}$ ${\displaystyle SL(2,C)}$

In mathematics , the Lie algebra is the prototype of a complex simple Lie algebra . This is a three-dimensional, complex, simple Lie algebra. These properties already clearly identify it as a Lie algebra. ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$

That is the three-dimensional Lie algebra of the special linear group . It is defined using the complex number field and has two real forms , the Lie algebra and the Lie algebra . ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle SL(2,\mathbb {C} )}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle {\mathfrak {su}}(2)}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {R} )}$

The group plays a role particularly in the special theory of relativity , since it is the simply connected superposition of the actual orthochronous Lorentz transformations . ${\displaystyle SL(2,\mathbb {C} )}$ ${\displaystyle SO_{0}(3,1)}$

Commutator relations

We consider the vector space spanned by the base x, y, h . This is then determined by the following commutator relations: ${\displaystyle g=\langle \{x,y,h\}\rangle _{\mathbb {C} }}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$

${\displaystyle [x,y]=h,\quad [h,x]=2x,\quad [h,y]=-2y}$

A frequently used implementation takes place using the following non-marking 2 × 2 matrices:

${\displaystyle x={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad y={\begin{pmatrix}0&0\\1&0\end{pmatrix}},\quad h={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$

Alternative realization through the cross product

By defining the cross product in and the following vectors ${\displaystyle \mathbb {C} ^{3}}$

${\displaystyle x=(1,\mathrm {i} ,0),\quad y=(-1,\mathrm {i} ,0),\quad h=(0,0,2\mathrm {i} )}$

the result is the same algebra:

${\displaystyle x\times y=h,\quad h\times x=2x,\quad h\times y=-2y}$

properties

${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$is a simple (especially semi-simple ) Lie algebra.

Proof: Be a nontrivial ideal in and be with . If , then , so and so . So we can assume or o. B. d. A . From then follows and with it , so again . ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle ax+bh+cy\in {\mathfrak {a}}\setminus 0}$ ${\displaystyle a,b,c\in \mathbb {C} }$ ${\displaystyle a=c=0}$ ${\displaystyle h\in {\mathfrak {a}}}$ ${\displaystyle 2x=\left[h,x\right]\in {\mathfrak {a}}}$ ${\displaystyle 2y=\left[h,y\right]\in {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {a}}={\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle a\not =0}$ ${\displaystyle c\not =0}$ ${\displaystyle a\not =0}$ ${\displaystyle \left[y,\left[y,ax+bh+cy\right]\right]=\left[y,-ah+2by\right]=-2ay}$ ${\displaystyle y\in {\mathfrak {a}}}$ ${\displaystyle h=\left[x,y\right]\in {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {a}}={\mathfrak {sl}}(2,\mathbb {C} )}$

Structure of the Lie algebra sl (2, C)

Killing form

The killing form of can be made explicit by the formula ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$

${\displaystyle B(v,w)=4\,\operatorname {Spur} (vw)}$

calculate so it is

${\displaystyle B(x,x)=B(y,y)=0,\ B(h,h)=8}$

${\displaystyle B(x,y)=4,\ B(x,h)=B(y,h)=0.}$

Cartan Involution

A maximally compact subgroup of the Lie group is , its Lie algebra is spanned by and . ${\displaystyle SL(2,\mathbb {C} )}$ ${\displaystyle K=SU(2)}$ ${\displaystyle {\mathfrak {k}}={\mathfrak {su}}(2)}$ ${\displaystyle i(x+y),\ x-y}$ ${\displaystyle ih}$

A Cartan involution of is given by ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$

${\displaystyle \theta (A)=-{\overline {A}}^{T}}$.

${\displaystyle {\mathfrak {k}}={\mathfrak {su}}(2)}$is their own space to eigenvalue . The Cartan decomposition is obtained ${\displaystyle 1}$

${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )={\mathfrak {k}}\oplus {\mathfrak {p}}}$,

where is the eigenspace to the eigenvalue . ${\displaystyle {\mathfrak {p}}=\left\{A\in {\mathfrak {sl}}(2,\mathbb {C} ):A={\overline {A}}^{T}\right\}}$ ${\displaystyle -1}$

Iwasawa decomposition

An Iwasawa decomposition of is ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$

${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )={\mathfrak {k}}\oplus {\mathfrak {a}}\oplus {\mathfrak {n}}}$

with . ${\displaystyle {\mathfrak {k}}={\mathfrak {su}}(2),\ {\mathfrak {a}}=\left\{{\begin{pmatrix}\lambda &0\\0&-\lambda \end{pmatrix}}:\lambda \in \mathbb {R} \right\},\ {\mathfrak {n}}=\left\{{\begin{pmatrix}0&n\\0&0\end{pmatrix}}:n\in \mathbb {C} \right\}}$

Real forms

It has two real forms : its compact real form is , its split real form is . ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {su}}(2)}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {R} )}$

Cartan subalgebras

Is a maximal abelian subalgebra

${\displaystyle {\mathfrak {h}}_{0}=\left\{{\begin{pmatrix}\lambda &0\\0&-\lambda \end{pmatrix}}:\lambda \in \mathbb {C} \right\}}$.

${\displaystyle {\mathfrak {h}}_{0}}$is a Cartan sub-algebra .

Every Cartan sub-algebra is to be conjugated; i.e., it is of the form ${\displaystyle {\mathfrak {h}}\subset {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {h}}_{0}}$

${\displaystyle {\mathfrak {h}}=g{\mathfrak {h}}_{0}g^{-1}:=\left\{ghg^{-1}:h\in {\mathfrak {h}}_{0}\right\}}$

for a . ${\displaystyle g\in SL(2,\mathbb {C} )}$

Root system

The root system to be ${\displaystyle {\mathfrak {h}}_{0}}$

${\displaystyle R=\left\{\alpha _{12}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}},\ \alpha _{21}={\begin{pmatrix}-1&0\\0&1\end{pmatrix}}\right\}}$.

The dual roots are

${\displaystyle \alpha _{12}^{*}{\begin{pmatrix}\lambda &0\\0&-\lambda \end{pmatrix}}=2\lambda ,\ \alpha _{12}^{*}{\begin{pmatrix}\lambda &0\\0&-\lambda \end{pmatrix}}=-2\lambda }$.

The associated root spaces are

${\displaystyle {\mathfrak {g}}_{\alpha _{12}}=\mathbb {C} {\begin{pmatrix}0&1\\0&0\end{pmatrix}},\ {\mathfrak {g}}_{\alpha _{21}}=\mathbb {C} {\begin{pmatrix}0&0\\1&0\end{pmatrix}}}$.

The Weyl group is the symmetrical group . ${\displaystyle S_{2}}$

See also

• Representation theory of sl (2, C)

Web links

• Nicolas Perrin: The Lie Algebra ${\displaystyle {\mathfrak {sl}}_{2}}$ PDF
• Abhinav Shrestha: Representations of semisimple Lie algebras PDF