Lie Algebra of the reals with addition

In summary, the Lie Algebra of the Lie Group (R,+) is the upper triangular 2x2 matrices with diagonal elements equal to zero. The exponential map is the normal exponential map, and the adjoint representation is the trivial representation since the group is abelian. This is to be expected as the conjugation action of an abelian group is trivial.
  • #1
holy_toaster
32
0
The following question may be trivial, but I just can't get it figuered out:

Consider the real numbers R with the addition operation + as a Lie Group (R,+). What is the Lie Algebra of this Lie Group? Is it again (R,+), this time considered as a vector space? If so, what is the exponential map and the adjoint representation of the Lie group on the Lie Algebra?
 
Physics news on Phys.org
  • #2
Now, the set (R,+) is isomorphic to SUT2(R) (= the set of all 2x2 upper triangular matrices with diagonal entries =1). An isomorphism is given by

[tex]\mathbb{R}\rightarrow SUT_2(\mathbb{R}):c\rightarrow \left(\begin{array}{cc} 1 & c\\ 0 & 1\end{array}\right).[/tex]

So the Lie Algebra of (R,+) will be the Lie-algebra of SUT2(R). But this Lie algebra is well known to be the upper triangular 2x2-upper triangular matrices with diagonal elements zero. The exponential map being the normal exponential map. Of course, this Lie-algebra can also be written as (R,+) with Lie bracket [a,b]=0. The exponential is then

[tex]exp(a)\cong exp\left(\begin{array}{cc}0 & a\\ 0 & 0\end{array}\right)=I+\left(\begin{array}{cc} 0 & a\\ 0 & 0\end{array}\right)\cong a[/tex]

So the exponential map is simply the identity map.
 
  • #3
Aha. Thanks that helps. But if I calculate now the adjoint representation of the group over the Lie Algebra, I get that it is the trivial representation, i.e. every group element acts as the identity. Can that be true?
 
  • #4
holy_toaster said:
Aha. Thanks that helps. But if I calculate now the adjoint representation of the group over the Lie Algebra, I get that it is the trivial representation, i.e. every group element acts as the identity. Can that be true?

Yes, that is to be expected. The adjoint representation of any abelian Lie group is trivial (according to wiki). The thing is that the conjugation action G->Aut(G) is trivial since the group is abelian. Thus it is not surprising that adjoint representation is also trivial.
 
  • #5
Yes, I see. Thank you.
 

1. What is meant by "Lie Algebra of the reals with addition"?

The Lie Algebra of the reals with addition refers to the mathematical structure that describes the behavior of vector addition in the real numbers. It is a special type of algebraic structure known as a Lie algebra, which studies the properties of vector spaces and their associated operations.

2. How is the Lie Algebra of the reals with addition different from other Lie algebras?

The Lie Algebra of the reals with addition is unique in that it only deals with the operation of addition, whereas other Lie algebras may involve multiple operations such as multiplication, division, or exponentiation. This makes it a simpler and more specialized type of Lie algebra.

3. What are some applications of the Lie Algebra of the reals with addition?

The Lie Algebra of the reals with addition has many applications in physics, particularly in the study of symmetries and conservation laws in classical mechanics. It is also used in mathematical finance to model the behavior of financial instruments and in computer graphics to create smooth animations.

4. How is the Lie Algebra of the reals with addition related to Lie groups?

Lie groups and Lie algebras are closely related, with Lie groups being the continuous versions of Lie algebras. In the case of the Lie Algebra of the reals with addition, the corresponding Lie group is the group of real numbers under addition. The Lie algebra provides a way to study and understand the group's underlying structure and properties.

5. Are there any practical uses for the Lie Algebra of the reals with addition?

Yes, the Lie Algebra of the reals with addition has many practical applications in various fields such as physics, engineering, and computer science. It is a powerful tool for analyzing and solving problems involving vector addition and can provide insights into the structure of mathematical systems and their behavior.

Similar threads

  • Differential Geometry
Replies
20
Views
2K
  • Differential Geometry
Replies
1
Views
1K
  • Differential Geometry
Replies
3
Views
2K
  • Differential Geometry
Replies
17
Views
3K
  • High Energy, Nuclear, Particle Physics
Replies
27
Views
3K
Replies
9
Views
1K
Replies
7
Views
2K
Replies
7
Views
2K
Replies
6
Views
876
Replies
14
Views
2K
Back
Top