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of a dense subset in the general linear group. It can be considered as a special case of the
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subgroups generalises the way a square real matrix can be written as a product of an
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of an element in algebraic group as a product of semisimple and unipotent elements
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of a Lie group as the product of semisimple, abelian, and nilpotent subgroups.
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analogues, making it harder to summarise the facts into a unified theory.
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writes a semisimple real Lie algebra as the sum of eigenspaces of a
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and associated objects, by showing how they are built up out of
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The same ideas are often applied to Lie groups, Lie algebras,
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can be regarded as a generalization of the principle of
305:writes a finite dimensional Lie algebra as a
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153:. They are essential technical tools in the
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161:; they can also be used to study the
145:are used to analyse the structure of
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59:adding citations to reliable sources
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278:Gram–Schmidt orthogonalization
191:Jordan–Chevalley decomposition
165:of such groups and associated
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294:writes a parabolic subgroup
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209:semisimple algebraic group
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285:Langlands decomposition
274:upper triangular matrix
333:Birkhoff decomposition
256:of a semisimple group
184:List of decompositions
247:Iwasawa decomposition
155:representation theory
337:Bruhat decomposition
326:Bruhat decomposition
236:Cartan decomposition
198:Bruhat decomposition
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339:for affine groups.
307:semidirect product
303:Levi decomposition
276:(a consequence of
260:as the product of
167:homogeneous spaces
163:algebraic topology
157:of Lie groups and
270:orthogonal matrix
240:Cartan involution
231:for more details.
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358:Lie groups
352:Categories
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315:semisimple
229:Weyl group
147:Lie groups
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266:nilpotent
151:subgroups
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