I want to Calculate the Killing form for the two-dimensional non-abelian Lie algebra.
The killing form is defined as:
K: L × L —> F
K(x,y) = tr (ad_x • ad_y) for x , y in L (lie-algebra).
The two-dim non-abelian lie-algebra is
span {x, y} , [x,y] = y.
So, I would calculate the matrix of this lie-algebra.
Can I take the basis of it to be {x,y} ?
So the entries of the matrix are
K(x,x), K(x,y) = K(y,x), K(y,y) (it is a symmetrix matrix).
So, I need to compute the matrices: ad_x and ad_y.
Is
ad_x =
0 0
0 y
And
ad_y=
0 0
-y 0
?
It is just a liitle weird having this basis..
The killing form is defined as:
K: L × L —> F
K(x,y) = tr (ad_x • ad_y) for x , y in L (lie-algebra).
The two-dim non-abelian lie-algebra is
span {x, y} , [x,y] = y.
So, I would calculate the matrix of this lie-algebra.
Can I take the basis of it to be {x,y} ?
So the entries of the matrix are
K(x,x), K(x,y) = K(y,x), K(y,y) (it is a symmetrix matrix).
So, I need to compute the matrices: ad_x and ad_y.
Is
ad_x =
0 0
0 y
And
ad_y=
0 0
-y 0
?
It is just a liitle weird having this basis..