Transcribed TextTranscribed Text

Problem 4. Let V denotes the vector space Rntm with basis e1 Consider the following bi-linear form on V + - - - Let O(m,n n) denote the set of all matrices A in GLm+n which preserve the form B, i.e., for any vectors u. e E V and SO(m.n) be the set of matrices A in O(m,n which also satisfy the condition det(A) = 1. a) Show that O(m. n) and SO(m, n) are subgroups of Glatai b) Find a condition which is satisfied by all matrices in O(m,n); c) Explain why the Lie algebras of O(m. n) and SO(m,n) are the same; d) Describe the Lie algebra so(m,n) of O(m,n); Problem 2. Let G denotes the group GL,(R) for n > 2. Let H be a normal subgroup of G. a) Find all possibilities for H assuming it is discrete; b) Find all possibilities for H assuming it is closed subgroup, i.e., if lim h, E H and h, - g for some g E G, then g € H; c) Give an example of non closed normal subgroup; d) Describe all normal subgroups of GL"(R). e) Find all values of m and n such that the Lie algebra so(m, n) is simple; f) Give (infinitely many) examples of pairs (m,n) such that the group SO(m,1 n) is not connected. Hint: For parts b) and d) you need to find conditions like AAT = Id and A + AT = 0. Part f) is harder than the rest.

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

Matrix Groups Problems
    $25.00 for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Mathematics - Other Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats