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| In mathematics, '''nilpotent orbits''' are generalizations of [[nilpotent]] [[Matrix (mathematics)|matrices]] that play an important role
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| in [[representation theory]] of real and complex [[semisimple Lie group]]s and [[semisimple Lie algebra]]s.
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| == Definition ==
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| An element ''X'' of a [[semisimple Lie algebra]] ''g'' is called '''nilpotent''' if its [[adjoint endomorphism]]
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| : ''ad X'': ''g'' → ''g'', ''ad X''(''Y'') = [''X'',''Y'']
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| is nilpotent, that is, (''ad X'')<sup>''n''</sup> = 0 for large enough ''n''. Equivalently, ''X'' is nilpotent if its [[characteristic polynomial]] ''p''<sub>''ad X''</sub>(''t'') is equal to ''t''<sup>dim ''g''</sup>. | |
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| A semisimple [[Lie group]] or [[algebraic group]] ''G'' acts on its Lie algebra via the [[adjoint endomorphism|adjoint representation]], and the property of being nilpotent is invariant under this action. A '''nilpotent orbit''' is an orbit of the adjoint action such that any (equivalently, all) of its elements is (are) nilpotent.
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| == Examples ==
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| Nilpotent <math>n\times n</math> matrices with complex entries form the main motivating case for the general theory, corresponding to the complex [[general linear group]]. From the [[Jordan normal form]] of matrices we know that each nilpotent matrix is conjugate to a unique matrix with Jordan blocks of sizes <math>\lambda_1\geq \lambda_2\geq\ldots\geq\lambda_r,</math> where <math>\lambda</math> is a [[partition (number theory)|partition]] of ''n''. Thus in the case ''n''=2 there are two nilpotent orbits, the ''zero orbit'' consisting of the [[zero matrix]] and corresponding to the partition (''1'',''1'') and the ''principal orbit'' consisting of all non-zero matrices ''A'' with zero trace and determinant,
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| : <math> A=\begin{bmatrix}x & y\\ z & -x \end{bmatrix}, \quad (x,y,z)\ne (0,0,0)\quad{\;}</math> with <math>x^2+yz=0,</math>
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| corresponding to the partition (''2''). Geometrically, this orbit is a two-dimensional complex quadratic [[cone (geometry)|cone]] in four dimensional vector space of <math>2\times 2</math> matrices minus its apex.
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| The complex [[special linear group]] is a subgroup of the general linear group with the same nilpotent orbits. However, if we replace the ''complex'' special linear group with the ''real'' special linear group, new nilpotent orbits may arise. In particular, for ''n''=2 there are now 3 nilpotent orbits: the zero orbit and two real half-cones (without the apex), corresponding to positive and negative values of <math>y-z</math> in the parametrization above.
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| == Properties ==
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| * Nilpotent orbits can be characterized as those orbits of the adjoint action whose [[Zariski closure]] contains 0.
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| * Nilpotent orbits are finite in number.
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| * The Zariski closure of a nilpotent orbit is a union of nilpotent orbits.
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| * '''Jacobson–Morozov theorem''': over a field of [[characteristic zero]], any nilpotent element ''e'' can be included into an [[Sl2-triple|''sl''<sub>2</sub>-triple]] {''e'',''h'',''f''} and all such triples are conjugate by ''Z''<sub>''G''</sub>(''e''), the [[centralizer]] of ''e'' in ''G''. Together with the representation theory of ''sl''<sub>2</sub>, this allows one to label nilpotent orbits by finite combinatorial data, giving rise to the ''Dynkin–Kostant classification'' of nilpotent orbits.
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| == Poset structure ==
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| Nilpotent orbits form a [[partially ordered set]]: given two nilpotent orbits, ''O''<sub>1</sub> is less than or equal to ''O''<sub>2</sub> if ''O''<sub>1</sub> is contained in the Zariski closure of ''O''<sub>2</sub>. This poset has a unique minimal element, zero orbit, and unique
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| maximal element, the ''regular nilpotent orbit'', but in general, it is not a [[graded poset]].
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| If the ground field is [[algebraically closed]] then the zero orbit is [[covering relation|covered]] by a unique orbit, called the ''minimal orbit'', and the regular orbit covers a unique orbit, called the ''subregular orbit''.
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| In the case of the [[special linear group]] ''SL''<sub>''n''</sub>, the nilpotent orbits are parametrized by the [[partition (number theory)|partitions]] of ''n''. By a theorem of [[Murray Gerstenhaber|Gerstenhaber]], the ordering of the orbits corresponds to the [[dominance order]] on the partitions of ''n''. Moreover, if ''G'' is an [[classical group|isometry group of a bilinear form]], i.e. an orthogonal or symplectic subgroup of ''SL''<sub>''n''</sub>, then its nilpotent orbits are parametrized by partitions of ''n'' satisfying a certain parity condition and the corresponding poset structure is induced by the dominance order on all partitions (this is a nontrivial theorem, due to Gerstenhaber and Hesselink).
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| == See also ==
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| *[[Adjoint endomorphism|Adjoint representation]]
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| == References ==
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| *David Collingwood and William McGovern. ''Nilpotent orbits in semisimple Lie algebras''. Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York, 1993. ISBN 0-534-18834-6
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| [[Category:Lie algebras]]
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The writer's title is Christy. For years he's been living in Alaska and he doesn't strategy on changing it. Credit authorising is how he makes cash. As a woman what she really likes is fashion and she's been doing it for fairly a while.
My site; online psychics - look at these guys -