What is the simplicity of the Special Linear Lie Algebra?

Click For Summary
SUMMARY

The Special Linear Lie Algebra, denoted as sl_n(ℂ), is defined as the set of n x n matrices with a trace of zero. It is classified as simple because it contains no non-trivial ideals other than the zero ideal and itself. The defining property of a simple Lie algebra is that for any ideal I, if x is in L and y is in I, then the Lie bracket [x,y] must also be in I. This establishes the simplicity of sl_n(ℂ) as it adheres to the criteria set forth in the definition.

PREREQUISITES
  • Understanding of Lie algebras and their properties
  • Familiarity with matrix theory, specifically n x n matrices
  • Knowledge of the concept of ideals in algebra
  • Basic comprehension of the trace of a matrix
NEXT STEPS
  • Study the structure and properties of Lie algebras in detail
  • Learn about the classification of simple Lie algebras
  • Explore the relationship between Lie algebras and Lie groups
  • Investigate examples of other simple Lie algebras beyond sl_n(ℂ)
USEFUL FOR

Mathematicians, particularly those specializing in algebra and representation theory, as well as students studying advanced linear algebra and Lie theory.

heras1985
Messages
7
Reaction score
0
Hi,
Show that the Special linear Lie algebra is simple.
I tried it with induction but without result.
 
Physics news on Phys.org
Welcome to PF.

What exactly have tried? A good point to start is citing the definition of "Special linear Lie algebra" and "simple".
 
Definition of simple:
L is called simple if it has no ideals except {0} and L.
I is an ideal of L if x\in L, y\in I \Rightarrow [x,y]\in I
The matrices whose trace is 0 form the special linear lie algebra sl_n(\mathbb{C}). The special linear lie algebra is the lie algebra of the special linear group (this group is form by the matrices whose determinant is 1).
 
Last edited:

Thread 'Confusion about the Moyal-Weyl twist'

I am studying the mathematical formalism behind non-commutative geometry approach to quantum gravity. I was reading about Hopf algebras and their Drinfeld twist with a specific example of the Moyal-Weyl twist defined as F=exp(-iλ/2θ^(μν)∂_μ⊗∂_ν) where λ is a constant parametar and θ antisymmetric constant tensor. {∂_μ} is the basis of the tangent vector space over the underlying spacetime Now, from my understanding the enveloping algebra which appears in the definition of the Hopf algebra...
View full post »

Similar threads

Undergrad What Are Classical Lie Algebras A, B, C, and D?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad About derivations of lie algebra
  • · Replies 5 ·
Replies
5
Views
2K
Graduate What is the definition of quotient Lie algebra?
  • · Replies 15 ·
Replies
15
Views
3K
About semidirect product of Lie algebra
  • · Replies 19 ·
Replies
19
Views
3K
Undergrad The Lie algebra of ##\frak{so}(3)## without complexification
  • · Replies 2 ·
Replies
2
Views
3K
Graduate About Universal enveloping algebra
  • · Replies 4 ·
Replies
4
Views
2K
Graduate How do we prove that a nonzero nilpotent Lie algebra has a nontrivial center?
  • · Replies 9 ·
Replies
9
Views
2K
Graduate Topology on a space of Lie algebras
  • · Replies 2 ·
Replies
2
Views
3K
High School What unique contributions to math did linear algebra make?
  • · Replies 19 ·
Replies
19
Views
4K
Graduate Questions about solvable Lie algebras
  • · Replies 7 ·
Replies
7
Views
2K