On the extensions of infinite-dimensional representations of Lie semigroups.

Mirotin, Adolf R.

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[For the entire collection see Zbl 0742.00067.]The author formulates several theorems about invariant orders in Lie groups (without proofs). The main theorem: a simply connected Lie group $G$ admits a continuous invariant order if and only if its Lie algebra $L (G)$ contains a pointed invariant cone. V. M. Gichev has proved this theorem for solvable simply connected Lie groups (1989). If $G$ is solvable and simply connected then all pointed invariant cones $W$ in $L (G)$ are global in $G$ (a Lie wedge $W \subset L (G)$ ...

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It is shown, using techniques inspired by the method of orbits, that each non-zero mass, positive energy representation of the Poincaré group $𝒫^{1, 1} = S O (1, 1) \otimes_{s} ℝ^{2}$ can be obtained via contraction from the discrete series of representations of $S U (1, 1)$ .