Question about Haar measures on lie groups

In summary, the conversation discusses integrating over a Lie group in the fundamental representation and whether the Haar measure would change when passing to the adjoint representation. The conclusion is that the measure should not change due to its invariance under left and right translations. The conversation also includes a specific example of the Itzykson-Zuber integral and a resource that may provide further information.
  • #1
Luck0
22
1
I'm not sure if this question belongs to here, but here it goes

Suppose you have to integrate over a lie group in the fundamental representation. If you pass to the adjoint representation of that group, does the Haar measure have to change? I think that it should not change because it is invariant under left and right translations, is it correct?
 
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  • #3
dextercioby said:
Integrate what?

Some function f: G -> ℝ, where G is the Lie group. For example, the Itzykson-Zuber integral

∫dUexp(-tr(XUYU†)), where X, Y are n x n hermitean matrices and U ∈ U(n)
 

Related to Question about Haar measures on lie groups

What is a Haar measure on a Lie group?

A Haar measure on a Lie group is a mathematical concept used to measure the size or volume of subsets of a Lie group. It is a generalization of the notion of length, area, or volume in Euclidean space. It allows for the integration of functions on a Lie group, which is important in many areas of mathematics and physics.

Why is a Haar measure important in Lie group theory?

A Haar measure is important in Lie group theory because it allows for the definition of various important concepts such as integration and averaging on a Lie group. It also enables the study of measure-preserving actions of Lie groups, which has applications in probability theory and ergodic theory.

How is a Haar measure constructed on a Lie group?

A Haar measure is constructed on a Lie group by first defining a translation-invariant measure on a subgroup of the group, and then extending it to the entire group using a partition of unity. This construction ensures that the Haar measure is invariant under left and right translations, which is a necessary property for a measure on a Lie group.

What are some properties of a Haar measure on a Lie group?

A Haar measure on a Lie group has several important properties, including translation invariance, non-triviality, and uniqueness up to a scalar multiple. It is also a Borel measure, meaning it assigns a measure to all Borel subsets of the group. Additionally, the Haar measure is always positive and finite on compact subsets of the Lie group.

How is a Haar measure used in applications?

A Haar measure has many applications in mathematics and physics. It is used in harmonic analysis, representation theory, and differential geometry, among others. In physics, the Haar measure is essential in the formulation of quantum mechanics and in the study of symmetries in physical systems. Additionally, it has applications in signal processing, data analysis, and machine learning.

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