Advanced TopicsLIE ALGEBRAS(Date Due: 30thJuly)ASSIGNMENT 2Question 1.LetLbe the real vector...

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Advanced TopicsLIE ALGEBRAS(Date Due: 30thJuly)ASSIGNMENT 2Question 1.LetLbe the real vector spaceR3. Givenx,y?L, define[x,y] :=x×y,where×denotes the usualcross productof vectors.Sow thatLis a Lie algebra and determine its structure constants relative to the standardbasis forR3.Question 2.Letdbe a derivation of the Lie algebraL. Show that ifdcommutes with every innerderivation, thend(L)?C(L),whereC(L) denotes thecentreofL.Question 3.Letx?gl(n,F) havendistinct eigenvalues?1,?2,···,?ninF. Prove that the eigenvaluesof adxare then2scalars?i-?j,(1=i, j=n).(Note that onlyn2-n+ 1 scalars are paiwise distinct from each other since?i-?i= 0for alli.)1
	
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Advanced Topics LIE ALGEBRAS (Date Due: 30 July)
ASSIGNMENT 2
Question 1.
3
Let L be the real vector spaceR . Given x;y2L, dene
[x;y] :=xy;
where denotes the usual cross product of vectors.
Sow that L is a Lie algebra and determine its structure constants relative to the standard
3
basis forR .
Question 2.
Let  be a derivation of the Lie algebra L. Show that if  commutes with every inner
derivation, then
(L)C(L);
whereC(L) denotes the centre of L.
Question 3.
Letx2gl(n;F) haven distinct eigenvalues ; ; ; inF. Prove that the eigenvalues
1 2 n
2
of ad are the n scalars
x
   ; (1i; jn):
i j
2
(Note that only n  n + 1 scalars are paiwise distinct from each other since    = 0
i i
for all i.)
1

David · Accepted Answer

Exercise-1:
We have
[x, y] = x× y
For being Lie algebra, this binary operation should satisfy the following condi-
tion:
Bilinear For a, b ∈ R and x, y, z ∈ R3, we have
[ax+ by, z] = (ax+ by)× z = ax× z + by × z
= a(x× z) + b(y × z) = a[x, z] + b[y, z]
1. We have [x, x] = x× x = 0
2. We have
[x, [yz]] + [z[xy]] + [y[zx]]
= x× (y × z) + z × (x× y) + y × (z × z)
= (x.z)y − (x.y)z + (z.y)x− (z.x)y + (y.x)z − (y.z)x
= 0
Hence Jacobi identity satisfies.
This proves that R3 with [x, y] binary operation is Lie algebra.
Now we have e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1) be the basis element.
Now if
[ei,

Advanced TopicsLIE ALGEBRAS(Date Due: 30thJuly)ASSIGNMENT 2Question 1.LetLbe the real vector spaceR3. Givenx,y?L, define[x,y] :=x×y,where×denotes the usualcross productof vectors.Sow thatLis a Lie...

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