Let <mi class="MJX-tex-caligraphic" mathvariant="script">g be a Lie algebra and let a ,

cricafh 2022-05-24 Answered
Let g be a Lie algebra and let a , b , c g be such that a b = b a and [ a , b ] = c 0. Let h = s p a n   { a , b , c }. How to prove that h is isomorphic to the strictly upper triangular algebra n ( 3 , F )?
Problem: If h n ( 3 , F ) then a , b , c n ( 3 , F ) with a b = b a and [ a , b ] = c as in h But then c must equal 0 whereas c h is not 0?
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Answers (1)

Terrance Phillips
Answered 2022-05-25 Author has 10 answers
The Lie algebra with [ a , b ] = c is the 3-dimensional Heisenberg Lie algebra h 1 . It has a faithful linear representation given by
a = ( 0 1 0 0 0 0 0 0 0 ) , b = ( 0 0 0 0 0 1 0 0 0 ) , c = ( 0 0 1 0 0 0 0 0 0 ) ,
Obviously this matrix Lie algebra is given by n 3 , so that n 3 h 1 .
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