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Chapter 29: Problem 11

Find the invariant Haar measure of the general linear group in two dimensions.

Short Answer

Expert verified
The invariant Haar measure of the general linear group in two dimensions \(GL(2, \mathbb{R})\) is given by the absolute value of the determinant function, \( \mu(A) = |\det(A)| \), which is right- and left-invariant.

Step by step solution

01

Definition of the General Linear Group

The general linear group \(GL(2, \mathbb{R})\) consists of all 2x2 matrices with real numbers and non-zero determinant. It can be denoted by \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), with \( a, b, c, d \in \mathbb{R} \) and \( ad - bc \neq 0 \).
02

Haar Measure Function Construction

The Haar measure of \(A\) in \(GL(2, \mathbb{R})\) should be a multiple of the determinant of \(A\). This is because it's well known that the determinant function is an invariant measure on \(GL(2, \mathbb{R})\). So, we can define the Haar measure \( \mu \) of \( GL(2, \mathbb{R}) \) as an absolute value of the determinant function: \( \mu(A) = |\det(A)| \).
03

Show the Invariance of Haar Measure

To show that the Haar measure \( \mu \) is right-invariant, we need to prove that for any two matrices \( A, B \in GL(2, \mathbb{R}) \), \( \mu(AB) = \mu(A) \). Since the determinant of a product of two matrices equals the product of their determinants, \( |\det(AB)| = |\det(A)||\det(B)| \). And because \( |\det(B)| = 1 \), it follows that \( \mu(AB) = \mu(A) \). The same proof applies for the left-invariance of \( \mu \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

General Linear Group
The general linear group, often abbreviated as \(GL(n, \mathbb{R})\), is an essential concept in mathematics dealing with matrices. It consists of all \(n \times n\) matrices with real numbers as entries for which the determinant is non-zero. For instance, in the case of two dimensions, it is denoted by \(GL(2, \mathbb{R})\).

If we consider a 2x2 matrix \(A\), it can be represented as follows:
  • \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\),
  • with the condition that \(ad - bc eq 0\).
This non-zero determinant condition ensures matrices in \(GL(2, \mathbb{R})\) are invertible, meaning they have an inverse that is also a 2x2 matrix with real numbers. Thus, these groups play a significant role in linear transformations and geometric transformations in space by including only those matrices that preserve the structure of space, given their invertibility.
Matrix Determinant
The determinant of a matrix, indicated as \(\det(A)\) for a matrix \(A\), is a special number derived from a square matrix. It plays a crucial role in various fields of mathematics like linear algebra, calculus, and more.

For the 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), its determinant is calculated as:
  • \(\det(A) = ad - bc\)
This calculation is based on the differences between the products of the diagonal elements. The determinant can
  • help identify invertibility of the matrix鈥� a non-zero determinant signifies that the matrix is invertible,
  • be used to find the area scaling factor for geometric transformation,
  • help solve linear equations through methods like Cramer's Rule.
A determinant that's zero indicates the matrix does not have a unique solution or is singular.
Invariant Measure
In the mathematical context, a measure is a system to assign a size or magnitude to sets in a consistent way. An invariant measure is particularly special because its value remains unchanged under transformations.

In the realm of the general linear group \(GL(2, \mathbb{R})\), this concept is exemplified by the Haar measure. It is an invariant measure that assigns to each matrix the absolute value of its determinant, denoted as \(\mu(A) = |\det(A)|\). This assignment makes the Haar measure unique because it stays constant under both right and left multiplication by other matrices in the group.Here鈥檚 why that is significant:
  • When even transforming the matrices, this measure retains its value, ensuring its invariance.
  • It helps in analyzing properties like volume and density within the context of non-Euclidean spaces.
This invariant nature is intrinsic in studying symmetrical properties and understanding algebraic structures in higher mathematics.

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Most popular questions from this chapter

Show that the invariant Haar measure for a compact group satisfies \(d \boldsymbol{\mu}_{g}=d \boldsymbol{\mu}_{g^{-1}}\). Hint: Define a measure \(\boldsymbol{v}\) by \(d \boldsymbol{v}_{g} \equiv d \boldsymbol{\mu}_{g^{-1}}\) and show that \(\boldsymbol{v}\) is left-invariant. Now use the uniqueness of the left-invariant Haar measure for compact groups.

Let \(G\) be a Lie group. Let \(S\) be a subgroup of \(G\) that is also a submanifold of \(G\). Show that \(S\) is a Lie group.

Let \(v\) be a Lie algebra. Verify that \(\operatorname{ad}_{\mathbf{x}}\) is a derivation of \(\mathfrak{v}\) for any \(\mathbf{X} \in \mathfrak{v}\), and that \(\operatorname{ad}_{[\mathbf{X}, \mathbf{Y}]}=\left[\mathrm{ad} \mathbf{x}, \mathrm{ad}_{\mathbf{Y}}\right]\).

Show that if \(\psi\) is an automorphism of \(\mathfrak{v}\), then $$ \operatorname{ad}_{\psi(\mathbf{X})}=\psi \circ \mathrm{ad}_{\mathbf{X}} \circ \psi^{-1} \quad \forall \mathbf{X} \in \mathfrak{v} $$ Hint: Apply both sides to an arbitrary element of \(\mathfrak{v}\).

Let \(\mathbf{D}_{1}\) and \(\mathbf{D}_{2}\) be derivations of a Lie algebra \(\mathfrak{v}\). Show that \(\mathbf{D}_{1} \mathbf{D}_{2} \equiv\) \(\mathbf{D}_{1} \circ \mathbf{D}_{2}\) is not a derivation, but \(\left[\mathbf{D}_{1}, \mathbf{D}_{2}\right]\) is.
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