Question

Lie Groups and Lie Algebras Kirillov. Group Theory

3.15. Let G be a co Is,-algebra g = Lie(G), and lette simply omplex connected simply-connected Lie grou p, with Lie e the R-linear map θ :. g → g by θ(x + y) = x-y, x, y et. at θ is an automorphism of g (considered as a real Lie alge- (1)/ Define bra), of the ndthat it can be uniquely lifted to an automorphism θ: G-G group G (considered as a real Lie group). =C", show that K is a real Lie group with Lie algebra e. GO (2) Let K t Sn(n) be the unitary quaternionic groun

Answer 1

Given that G is the complex complex connected simply a connected lie group,With lie algebra g=Lie(G) and let us take other variable r be a real form of G.

In the group theory of lie groups,The exponential map is a map from the lie algebra of a lie group to the group which allows one to recapture the local group structure from the lie algebra.The existence of the exponential map is one of the primary reasons that a lie algebras are a useful tool for studying lie groups.

An example of a lie group representation is the adjoint representation of a lie-Group G; each element g in a lie group G defines an automorphism of G by conjugation means if any function gives complex conjugate of itself,then it forms automorphism.The differential of that function also an automorphism of the lie-algebra.

(1).R is a linear map that means the elements in input are same for output after transformation and theta:g-->g by given relation theta(x+it)=x-iy. Here,In this transformation, theta transformed the equation (x+iy) into complex conjugate of this that is (x-iy).So,From this transformation, theta is an automorphism of g from above given lie algebra.And this also lifted to an automorphism of theta:G-->G of the group G and it is real lie group because already given that x,y are real values belongs to r region.

So,Finally lie group G also forms the real linear map.

(2).Let k=G^theta  is also a real group with real algebra r which belongs to real values of x,y because From the above explanation,we know that G forms the real group and theta is transformation variable for forming automorphism of G.

Any power of the real group forms the real group only.So,here G has a power theta and this value becomes real only.

From the above data,theta=(x-iy)/(x+iy).

Multiply with (x+iy) in both numerator and denominator

We will get theta=(x²-y²)/(x+iy)².

Normalized value of (x+iy)=√(x²+y²).

Finally,theta=(x²-y²)/(x²+y²).This is also real and G also real group.

So,Finally,G^theta also a real lie group with lie algebra r From above explanation.