Hey! Sign in to get help with your study questionsNew here? Join for free to post

Mappings

Announcements Posted on
How proud of your uni are you? Take our survey for the chance to win a holiday! 09-02-2016
  1. Offline

    ReputationRep:
    If V is a 3-dimensional Lie algebra with basis vectors E,F,G with Lie bracket relations [E,F]=G, [E,G]=0, [F,G]=0 and V' is the Lie algebra consisting of all 3x3 strictly upper triangular matrices with complex entries then would you say the following 2 mappings (isomorphisms) are different? I had to give an example of 2 different isomorphisms between these vector spaces.

    \varphi : V \to V' given by

    \varphi(aE+bF+cG)=\left( \begin{array}{ccc} 0 & a & c\\ 0 & 0 & b\\ 0 & 0 & 0 \end{array} \right)\;,\;\;\;\;\;a,b,c \in \mathbb{C}

    and \bar{\varphi} : V \to V' given by

    \bar{\varphi}(E)=\left( \begin{array}{ccc} 0 & 1 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array} \right)

    \bar{\varphi}(F)=\left( \begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{array} \right)

    \bar{\varphi}(G)=\left( \begin{array}{ccc} 0 & 0 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array} \right)
  2. Offline

    ReputationRep:
    Maybe this post is a little late given that the thread is a week old now, but recall that Lie algebra isomorphisms are also linear transformations, so are determined by their values on the basis elements E,F and G. So it seems your two maps are the same. However, the map you've given is an isomorphism and from this, it doesn't look too difficult to find a second?

Reply

Submit reply

Register

Thanks for posting! You just need to create an account in order to submit the post
  1. this can't be left blank
    that username has been taken, please choose another Forgotten your password?
  2. this can't be left blank
    this email is already registered. Forgotten your password?
  3. this can't be left blank

    6 characters or longer with both numbers and letters is safer

  4. this can't be left empty
    your full birthday is required
  1. By joining you agree to our Ts and Cs, privacy policy and site rules

  2. Slide to join now Processing…

Updated: June 16, 2012
TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

Today on TSR

Student Money Week continues...

Find out which Q&As are happening today

Poll
How should university tuition be funded?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read here first

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups
Study resources
Quick reply
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.