Find Lorentzian Scalar Product on 4-D Lie Algebra G

[kroni]

Hi everybody,

Let G a four dimmensionnal Lie group with g as lie algebra. Let T1 ... T4 the four generator. I would like to find à lorentzian scalar product (1-3 Signature) on it and left invariant. A classical algebra take tr (AB^t) as scalar product but I don't find à lorentzian équivalent. Did it contraint the generator or the structure constant to respect some criterion ?

Thanks for your answer

Clement

 

[fresh_42]

Hi everybody,

Let G a four dimmensionnal Lie group with g as lie algebra. Let T1 ... T4 the four generator. I would like to find à lorentzian scalar product (1-3 Signature) on it and left invariant. A classical algebra take tr (AB^t) as scalar product but I don't find à lorentzian équivalent. Did it contraint the generator or the structure constant to respect some criterion ?

Thanks for your answer

Clement

I've read your post now several times. I'd like to help you but I don't understand it. The $T_i$ are generators of $g$ or of $G$? What do you mean by a left invariant scalar product? And Lorentzian in this context means exactly what? Since a four dimensional Lie Algebra isn't semisimple I have difficulties to understand which bilinear form you're looking for.