Abstract:
The concept of a discontinuity group for a locally bounded homomorphism of a locally compact group is introduced and its properties are studied.We prove an analogue of Lie's theorem on the triangular form of finite-dimensional locally bounded not necessarily continuous representations of solvable Lie groups and the continuity of a locally bounded representation of a block triangular form under the condition of continuity of diagonal blocks. From this, we deduce the continuity of locally bounded finite-dimensional representations of connected solvable Lie groups on the commutator subgroup. The fact of automatic continuity of all finite-dimensional representations of semisimple compact Lie groups is used and the fact of automatic continuity of all finite-dimensional representations of semisimple non-compact Lie groups is additionally established in order to reduce the proof of the main assertion to the semisimple case. The assertion of automatic continuity of a mapping on the commutator subgroup is extended to the case of locally bounded homomorphisms of connected Lie groups. Thus, all locally bounded homomorphisms of a connected Lie group into a connected Lie group are continuous if and only if the group is perfect, i.e., coincides with its group-theoretical commutator subgroup (without applying the closure operation).
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