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# lecture notes9 - MATH 223A NOTES 2011 LIE ALGEBRAS 31 8.4...

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MATH 223A NOTES 2011 LIE ALGEBRAS 31 8.4. Root strings. This subsection is based on Erdmann and Wildon “Introduction to Lie Algebras” an undergraduate textbook which is the place to look if you don’t understand something. We first review what we have so far using an example. 8.4.1. example: sl (3 , F ) . Let L = sl (3 , F ). This is 8 dimensional with H being the 2- dimensional subalgebra of diagonal matrices with trace zero. H = h 1 0 0 0 h 2 0 0 0 h 3 : h 1 + h 2 + h 3 = 0 The o ff -diagonal entries have an obvious basis given by x ij , the matrix with 1 in the ij position and 0 elsewhere: x 12 = 0 1 0 0 0 0 0 0 0 , x 23 = 0 0 0 0 0 1 0 0 0 , x 13 = 0 0 1 0 0 0 0 0 0 and x 21 , x 32 , x 31 . Note that each x ij is an eigenvector: For example, [ hx 12 ] = hx 12 x 12 h = ( h 1 h 2 ) x 12 So, (1) x 12 L α where α ( h ) = h 1 h 2 , (2) x 23 L β where β ( h ) = h 2 h 3 and (3) x 13 L α + β with ( α + β )( h ) = h 1 h 3 . We also have (4) x 21 L α (5) x 32 L β (6) x 31 L α β . This gives the root space decomposition: L = H L α L β L α + β L α L β L α β What is h α ? [ x 12 x 21 ] = 1 0 0 0 1 0 0 0 0 = h α This matrix is h α since α ( h α ) = h 1 h 2 = 2. x a = x 12 , y α = x 21 and S α = span ( x 12 , x 21 , h α ) Find the decomposition of L into irreducible S α -modules.

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32 MATH 223A NOTES 2011 LIE ALGEBRAS One component is V (2) = L α Fh α L α . To find the others we draw the root spaces in the following pattern: L α + β L β = V (1) L α H L α = V (2) V (0) L β L α β = V (1) Claim : L α + β V β = V (1).
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