Question

Show that the groups \(S L(n, \mathbb{R})\) and \(S O(n)\) are closed subgroups of \(G L \cdot(N, \mathbb{R})\), and that \(U(n)\) and \(S U(n)\) are closed subgroups of \(G L(n, C)\). Show furthcrmore that \(S O(n)\) and \(U(n)\) are compact Lie subgroups.

Short answer

The groups \(SL(n, \mathbb{R})\), \(SO(n)\), \(U(n)\) and \(SU(n)\) are closed subgroups of the relevant larger groups. In addition, \(SO(n)\) and \(U(n)\) are compact. This is because these groups are both closed and bounded.

Step-by-step solution

Define the groups

Let's start by defining each of the groups. The special linear group \(SL(n, \mathbb{R})\) comprises of all \(n \times n\) matrices with real entries and determinant equal to 1. Orthogonal group \(SO(n)\) consists of all \(n \times n\) real orthogonal matrices with determinant 1. Unitary group \(U(n)\) is the group of \(n \times n\) unitary matrices, and the special unitary group \(SU(n)\) consists of \(n \times n\) unitary matrices with determinant 1.

Show that these are closed subgroups

A subgroup of \(GL(n, \mathbb{R})\) or \(GL(n, C)\) is closed if it is closed under the operations of the group, which in this case are multiplication and inverse. To prove that \(SL(n, \mathbb{R})\), \(SO(n)\), \(U(n)\) and \(SU(n)\) are closed subgroups we show that multiplying any two matrices in these groups or taking the inverse of a matrix in these groups results in another matrix in the same group. Let's take \(SL(n, \mathbb{R})\) for example. If two matrices from \(SL(n, \mathbb{R})\) are multiplied, the determinant of the product is equal to the product of the determinants (which are both 1), thus the determinant of the product is still 1 and hence belongs to \(SL(n, \mathbb{R})\). The same argument could be made for \(SO(n)\), \(U(n)\), and \(SU(n)\).

Show that \(SO(n)\) and \(U(n)\) are compact subgroups

A Lie group is compact if it is both closed and bounded. We have already shown that \(SO(n)\) and \(U(n)\) are closed. It remains to show that they are bounded. To check for boundedness of \(SO(n)\), we see that each entry of these matrices is a real number between -1 and 1, thus it is bounded. For \(U(n)\), each entry is a complex number of modulus 1, which is also bounded.Therefore, \(SO(n)\) and \(U(n)\) are both closed and bounded, making them compact Lie subgroups.

Key concepts

Special Linear Group

The Special Linear Group, denoted as \(SL(n, \mathbb{R})\), is a fascinating concept in the world of Lie groups. It consists of all \(n \times n\) matrices that have real number entries and a determinant of exactly 1. This condition of the determinant being 1 ensures that each matrix in \(SL(n, \mathbb{R})\) has an inverse, maintaining the structure of a group.

• The determinant is a product of the matrix's eigenvalues, so having a determinant of 1 indicates a balanced scaling effect by the matrix.
• Matrix multiplication and taking inverses are the operations that need to adhere to the determinant condition.

By being a subset of \(GL(n, \mathbb{R})\), the General Linear Group of all invertible matrices, \(SL(n, \mathbb{R})\) is more restricted in nature, which highlights its 'special' status. Exploring how these matrices interact under multiplication is crucial to understanding their role as a closed subgroup.

Orthogonal Group

The Orthogonal Group, represented as \(SO(n)\), consists of \(n \times n\) matrices that are both orthogonal and have a determinant of 1. Orthogonal matrices possess the property that their transpose is also their inverse, meaning \(A^TA = I\), where \(I\) is the identity matrix.

• The defining properties imply that orthogonal matrices maintain vector lengths and angles, making them vital in geometry.
• As a subset of the real matrices, \(SO(n)\) showcases symmetry and structure.

By maintaining real number entries and specific eigenvalue properties, orthogonal matrices ensure that operations like multiplication and inversion remain within the group. This makes \(SO(n)\) a closed subgroup of \(GL(n, \mathbb{R})\). Furthermore, the constraints on the entries (between -1 and 1) support its status as a compact Lie group.

Unitary Group

Unitary matrices form the Unitary Group \(U(n)\), characterized by \(n \times n\) matrices that, like orthogonal matrices, have their conjugate transpose serving as their inverse. Specifically, \(UU^* = I\), where \(U^*\) represents the conjugate transpose.

• Entries of a unitary matrix are complex numbers, adding a layer of complexity compared to real orthogonal matrices.
• The conjugate transpose is crucial in maintaining the unitary condition, providing insights into quantum mechanics and other fields.

The unitarity condition ensures preservation of norm, making \(U(n)\) particularly significant. These matrices are instrumental in defining symmetries in complex vector spaces, and as closed subgroups, they help form the structure of \(GL(n, \mathbb{C})\). Further, with moudulus 1 entries being bounded, \(U(n)\) is compact.

Compact Lie Group

A compact Lie group is one that is both closed and bounded; these attributes make these groups quite interesting in mathematical and physical contexts.

• Closedness ensures that all points of accumulation for sequences of group elements remain within the group.
• Boundedness, on the other hand, implies that there is a finite limit to the extent of any sequence of group elements.

The groups \(SO(n)\) and \(U(n)\) are prime examples of compact Lie groups. We've proven them to be closed subgroups, and their entries (real numbers for \(SO(n)\) and complex numbers of unit modulus for \(U(n)\)) confirm their boundedness. This compact nature implies significant implications, like finite measure under the Haar measure, which is an essential concept in harmonic analysis and the theory of integrations over groups.

Matrix Determinants

Determining whether matrices belong to certain groups often hinges on their determinants.

• The determinant of a matrix gives a scalar value that is a fundamental property reflecting certain attributes of the matrix, such as scaling factor, volume change factor in transformations, and singularity (when zero).
• In groups like \(SL(n, \mathbb{R})\) and \(SU(n)\), the condition of having a determinant of 1 is vital for defining the group.

Understanding how operations affect the determinant is crucial. When matrices are multiplied, their determinants multiply too. Similarly, the determinant of an inverse matrix is the reciprocal of the determinant of the original matrix. These properties confirm the structure and stability of special types of matrix groups. Grasping determinants helps in appreciating transformations and inherent symmetries involved in these groups.