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equivocate teams – is the subgroup generated by one-parameter unipotent subgroups a equivocate subgroup? retort

Hello pricey customer to our community We will proffer you an answer to this query equivocate teams – is the subgroup generated by one-parameter unipotent subgroups a equivocate subgroup? ,and the retort will breathe typical by documented info sources, We welcome you and proffer you recent questions and solutions, Many customer are questioning concerning the retort to this query.

equivocate teams – is the subgroup generated by one-parameter unipotent subgroups a equivocate subgroup?

The many feedback in addition to the retort by zroslav most likely add to the confusion ensuing from the unique unfocused formulation of the query. First, it is not likely a query about equivocate teams however about algebraic teams: In a equivocate group there may be usually no intrinsic Jordan decomposition, therefore no intrinsic judgement of unipotent subgroup as seen already in dimension 1.

The algebraic group concept was largely formed by Borel and Chevalley, each of whom had been motivated in sever by an curiosity in equivocate teams. But a equivocate group is initially a precise manifold, whereas algebraic teams commence out over algebraically closed fields after which purchase extra sophisticated relative to smaller fields of definition. For an algebraic group, taken within the Zariski topology, “irreducible” = “connected”. Here a finite matrix group can breathe seen as an algebraic group however is related solely when trifling. In any illustration, an affine algebraic group can all the time breathe embedded in some frequent linear group, so over $mathbb{C}$ you purchase a equivocate subgroup of the sophisticated frequent linear group (seen as precise). The level is that being Zariski-closed implies being closed within the inveterate sense.

An early final result of Chevalley (handled in a number of books with the title Linear Algebraic Groups, together with part 7.5 of my bespeak) exhibits the consequence of situations on irreducibility for hunt of a bunch generated by closed subsets of an algebraic group. One consequence of Chevalley’s theorem is that two closed related subgroups will generate a closed related subgroup in a pleasant route, however the theorem has different functions as effectively. All of this comes very early within the concept, earlier than Jordan decomposition and the circumstantial construction of reductive teams, the place you possibly can query extra fascinating particular questions on what varied subgroups generate.

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