SL$(2, mathbb{C})$ double cover of SO${}^{+}$(1, 3)

In this note we present explicit and elementary formulas for the correspondence between the group of special Lorentz transformation SO^+(3,1), on the one hand, ...

The Special Linear Group in Two Complex Dimensions S L ( 2 , C ) is the double cover of the Restricted Lorentz Group in 4 Dimensions S O ( 3 , 1 ) o .

In this note we present explicit and elementary formulas for the correspondence between the group of special Lorentz transformation SO+(3, 1), on the one hand, ...

The special linear group SL(2, C) is a double covering of the restricted Lorentz group. · The symplectic group Sp(2, C) is isomorphic to SL(2, C); it is used to ...

(a) Show that the Lie algebra so(3,C) is isomorphic to sl(2,C). (b) Construct a Lie group homomorphism SL(2,C) → SO(3,C) which realizes the iso- morphism ...

1.1 Introduction. The spin homomorphism. SL2(C) → SO1,3(R) is a homomorphism of classical matrix Lie groups. The lefthand group con- sists of 2 × 2 complex ...

Regarding surjectivity, SL(2,C) and SO+(1,3) have the same dimension and are both connected, is that sufficient to conclude the homomorphism is ...

This only holds for finite dimensional dims. Unitary infinite dimensional reps of SO(3,1) are not equivalent to SU(2)xSU(2).

sl(2,C) can be viewed as either a real 6-dimensional Lie algebra or a complex 3-dimensional. Lie algebra. However, that's not the whole story. As a complex ...

Since SL(2,C) is connected, the image of f is exactly the connected component of the identity: SO+ (3,1). The kernel of f are the matrices that ...