Structural constant

In mathematics, structure constants contain the entire information of a (finite-dimensional) Lie algebra and thus, in particular, all local information of each Lie group assigned to it .

definition

Let be a finite-dimensional Lie algebra with the Lie bracket and be a vector space basis of this Lie algebra. Since every element in vector spaces can be represented as a linear combination with respect to a basis, the decomposition exists for all of them ${\displaystyle V}$ ${\displaystyle [\cdot ,\cdot ]}$ ${\displaystyle \{x_{1},\ldots ,x_{n}\}}$ ${\displaystyle i,j\in {1,\ldots n}}$

${\displaystyle [x_{i},x_{j}]=\sum _{k=1}^{n}c_{ij}^{k}x_{k}}$

the Lie bracket of Lie algebra. The constants (i.e. from the set of complex numbers) are called structural constants of Lie algebra. ${\displaystyle n^{3}}$ ${\displaystyle c_{ij}^{k}\in \mathbb {C} }$

properties

• Antisymmetry

The structural constants are antisymmetric in the lower indices due to the antisymmetry of the Lie bracket;

${\displaystyle c_{ij}^{k}=-c_{ji}^{k}}$

From this it follows for structure constants with identical lower indices . ${\displaystyle c_{ii}^{k}=0}$

• Jacobi identity

Due to the Jacobi identity for the Lie bracket, a Jacobi identity follows for the structural constants:

${\displaystyle \sum _{l=1}^{n}\left(c_{il}^{m}c_{jk}^{l}+c_{jl}^{m}c_{ki}^{l}+c_{kl}^{m}c_{ij}^{l}\right)=0}$

• Tensor structure

The structural constants are - tensors . This means that the following applies to a change of base : ${\displaystyle (2,1)}$ ${\displaystyle \textstyle x_{i}\to x_{i}'=\sum _{j=1}^{n}a_{i}^{j}x_{j}}$

${\displaystyle \sum _{k=1}^{n}{c'}_{ij}^{k}a_{k}^{l}=\sum _{r,s=1}^{n}c_{rs}^{l}a_{i}^{r}a_{j}^{s}}$

example

As an example of structure constants, the Lie algebra , which is important in physics, is given in the basis of the Pauli matrices . The Lie bracket in this illustration is the commutator and it applies ${\displaystyle {\mathfrak {su}}(2)}$ ${\displaystyle \sigma _{i},i=1,2,3}$

${\displaystyle [\sigma _{i},\sigma _{j}]=\sum _{k=1}^{3}2\mathrm {i} \varepsilon _{ij}^{k}\sigma _{k}}$

with the totally antisymmetric Levi Civita symbol . ${\displaystyle \varepsilon _{ij}^{k}}$

literature

• Manfred Böhm: Lie groups and Lie algebras in physics . Springer, Berlin 2011, ISBN 978-3-642-20378-7 , pp. 179-180 .
• Hans Samelson: Notes on Lie Algebras . Springer, New York 1990, ISBN 978-0-387-97264-0 , pp. 5 .