2022-02-20

31: Lie Algebra

<The previous article in this series | The table of contents of this series | The next article in this series>
definition of Lie algebra

Topics


About: Lie algebra

The table of contents of this article

Starting Context

Target Context


  • The reader will have a definition of Lie algebra.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

Main Body

1: Structured Description


Here is the rules of Structured Description.

Entities:
F: ∈{ the fields }
V: ∈{ the vectors spaces over F}, with [∙,∙]:V×V→V
//

Conditions:
∀v1,v2,v3∈V,∀r1,r2∈F
(
1) [r1v1+r2v2,v3]=r1[v1,v3]+r2[v2,v3] ∧ [v3,r1v1+r2v2]=r1[v3,v1]+r2[v3,v2]
∧
2) [v2,v1]=−[v1,v2]
∧
3) ∑cyclic[v1,[v2,v3]]=0
)
//

2: Natural Language Description


Any vectors space, V, over any field, F, with any bracket, [∙,∙]:V×V→V, such that for any v1,v2,v3∈V and any r1,r2∈F, 1) [r1v1+r2v2,v3]=r1[v1,v3]+r2[v2,v3] and [v3,r1v1+r2v2]=r1[v3,v1]+r2[v3,v2]; 2) [v2,v1]=−[v1,v2] 3) ∑cyclic[v1,[v2,v3]]=0

3: Note


Inevitably, for each v∈V, [v,0]=[0,v]=0: [v,0]=[v,0v+0v]=0[v,v]+0[v,v]=0+0=0; [0,v]=[0v+0v,v]=0[v,v]+0[v,v]=0+0=0.

Lie algebra is a not-necessarily-associative algebra: [r1v1+r2v2,r1′v1′+r2′v2′]=r1[v1,r1′v1′+r2′v2′]+r2[v2,r1′v1′+r2′v2′]=r1(r1′[v1,v1′]+r2′[v1,v2′])+r2(r1′[v2,v1′]+r2′[v2,v2′])=(r1r1′)[v1,v1′]+(r1r2′)[v1,v2′])+(r2r1′)[v2,v1′]+(r2r2′)[v2,v2′]), but the associativity, [[v1,v2],v3]=[v1,[v2,v3]], is not guaranteed to hold.

References
<The previous article in this series | The table of contents of this series | The next article in this series>
Posted by at
Labels: