Nilpotent subspaces of maximal dimension in semisimple lie algebras jan draisma, hanspeter kraft and jochen kuttler abstract we show that a linear subspace of a reductive lie algebra g that consists of nilpotent elements has dimension at most 1 2 dimg. In particular, the situation for nilpotent elements of the lie algebra and unipotent elements of the group is essentially the same, by springers equivariant isomorphism between the two settings the classes orbits and centralizers correspond nicely in good characteristic. The rest of five, called exceptional lie algebras, also admit matrix representation. Let g be a semisimple lie algebra defined over an algebraically closed field. Let g be an adjoint algebraic group of type b, c, or d defined over an algebraically closed field of characteristic 2. These short notes, already wellknown in their original french edition, give the basic theory of semisimple lie algebras over the complex numbers, including classification theorem. We give the number of nilpotent orbits in the lie algebras of orthogonal groups under the adjoint action of the groups over f 2 n. As a sample, we give the tables for lie algebras of type e 6 and e 7. Classifications of nilpotent orbits in lie algebras. In this lecture series, we shall present some basic constructions, connections, and applications of finite w algebras. We construct the springer correspondence for the nilpotent variety in the lie algebra of g. Representations of semisimple lie algebras in the bgg category 0. In 2, hesselink determines the nilpotent orbits in classical lie algebras under the adjoint action of classical lie groups over k.
Hence determine which of these lie algebras are solvable. For each homsl2,c,gc, there uniquely exists an element a. Killing form a lie algebra is semisimple if and only if the killing form is nondegenerate. However, in general, the converse of this property is false. Collingwood through the 1990s, a circle of ideas emerged relating three very different kinds of objects associated to a complex semisimple lie algebra. An introduction on free shipping on qualified orders. A semisimple lie group or algebraic group g acts on its lie algebra via the adjoint representation, and the. On the multiplier of nilpotent nlie algebras sciencedirect.
Pdf nilpotent orbits in simple lie algebras and their. In recent years, the finite w algebras associated to a semisimple lie algebra and its nilpotent element have been studied intensively from different viewpoints. For a semisimple lie algebra over the complex numbers, dynkin 1952 developed an algorithm to classify the regular semisimple subalgebras, up to conjugacy by the inner automorphism group. Abstract this thesis is concerned with the dassitication of 7dimensional nilpotent lie alge bras. Equivalently, x is nilpotent if its characteristic polynomial p ad x t is equal to t dim g. Onishchik, decompositions of reductive lie groups, math. Skjdbred and sund have published in 1977 their method of constructing all nilpotent lie algebras of dimension rr given those algebras of dimension nilpotent lie algebra has an outer automorphism, that is, an automorphism that is not in the image of ad. Nilpotent orbits in semisimple lie algebras pdf free download. Nilpotent subspaces of maximal dimension in semisimple lie. Closure order on nilpotent orbits in exceptional lie algebras. A representation of a lie algebra g is a homomorphism of lie algebras. A semisimple lie group or algebraic group g acts on its lie algebra via the adjoint representation, and the property of being nilpotent is invariant under this action. Invariants of the nilpotent and solvable triangular lie algebras. Nilpotent pairs in semisimple lie algebras and their characteristics dmitri i.
Semisimple and nilpotent elements in this chapter, all vector spaces are. A lie algebra is semisimple if and only if the killing form is nondegenerate. Nilpotent orbits in semisimple lie algebras david h. These notes are a record of a course given in algiers from 10th to 21st may, 1965. Popov, stability criteria for the action of a semisimple group on a factorial manifold. A nilpotent orbit is an orbit of the adjoint action such that any equivalently, all of its elements is are nilpotent. The cartan decomposition of a complex semisimple lie algebra shawn baland university of colorado, boulder november 29, 2007. Classification of nilpotent elements in simple exceptional real lie. In place of item 3 above, we use elashvilias tables in 2, where the reductive parts of centralisers are explicitly indicated. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple lie algebras. Panyushev introduction in a recent article gi99, v. June 7, 1999 nilpotent pairs in semisimple lie algebras. The recent paper 19 has studied spherical nilpotent orbits in complex simple lie algebras from the viewpoint of the notion of strongly visible actions introduced by t. Complex semisimple lie algebras jeanpierre serre springer.
Regular subalgebras and nilpotent orbits of real graded lie. All nonzero nilpotent elements are zero divisors an nbyn matrix a with entries from a field is nilpotent if and only if its characteristic polynomial is t n if x is nilpotent, then 1. Cartan algebras in a reductive lie algebra are all obtained as centralisers. Lie algebras of traceless skew symmetric and hamiltonian matrices. Abundance of nilpotent orbits in real semisimple lie algebras. It is easily seen that the principal nilpotent orbit opr and the minimal nilpotent orbit omin are.
Your setup is essentially equivalent to studying the same problem for a semisimple algebraic group and its lie algebra in arbitrary chaeracteristic, but good. The closure of a sheet is not always a union of sheets, a short note. The first two chapters are a summary, without proofs, of the general properties of nilpotent, solvable, and semisimple lie algebras. In the classical lie algebras, the nilpotent orbits are classi. In this lecture series, we shall present some basic constructions, connections, and. We extend hesselinks method to study the nilpotent orbits in the lie algebras of orthogonal groups over f q. Nilpotent orbits in classical lie algebras over f2n and. This paper proves a stable analog of rossmanns formula for the number of grorbits in g\o,whereo is a nilpotent orbit in g c.
The author begins with a summary of the general properties of nilpotent, solvable, and. Representations of semisimple lie algebras in the bgg category 0 james e. Nilpotent orbits in simple lie algebras and their transverse poisson structures article pdf available june 2008 with 60 reads how we measure reads. The following result provides an upper bound for dimension of multiplier of nilpotent nlie algebras which is less than the bound in corollary 3. Regular subalgebras and nilpotent orbits of real graded lie algebras heiko dietrich, paolo faccin, and willem a. Full text of nilpotent pairs in semisimple lie algebras. We classify the nilpotent orbits of g, in go for the adjoint action by using kostants. Abundance of nilpotent orbits 405 let us denote by homsl2,c,gc the set of all complex lie algebra homomorphisms from sl2,ctogc. This determination of component groups goes back to elashvili and alexeevskii, but has been improved somewhat in a 1998 imrn paper by eric sommers and a later joint paper by him and george mcninch here. As just one example of the analogy between lie algebras and rings, take the following proposition.
The principal aim of this book is to collect together the important results concerning the classification and properties of nilpotent orbits, beginning from. Derived aubalgebras of solvable lie algebras edit the derived subalgebra of a finite dimensional solvable lie algebra over a field of characteristic 0 is nilpotent. Hence determine which of these lie algebras are nilpotent. Questions about maximal torus of semisimple derivations of a. Semisimple lie algebras and the killing form this section follows procesis book on lie groups. Nilpotent and semisimple elements jordan decomposition def. Each walgebra is constructed from a pair g, o of a semisimple lie algebra g and a nilpotent orbit o. The second class of algebras studied below are the.
Let gbe a connected reductive group over r and let g denote its lie algebra. The lie bracket of x and y is denoted by x, y, and the map y. An nbyn matrix a with entries from a field is nilpotent if and only if its characteristic polynomial is t n. Favre discovered the lowest dimensional cnla known until then 35, which additionally was of the same nature as dyers example. A lie algebra is a vector space, l, above, the linear combinations of the ts together with a bilinear operation from l. The lie algebras considered in this chapter are finitedimensional algebras over a field k. Later santharoubane 1979 2829 fiirther generaiizes this idea and establishes a link between nilpotent lie algebras and kacmoody lie algebras. The jacobsonmorozov theorem throughout this paper, let gbe a complex semisimple connected algebraic group of rank l, g its lie algebra, h the cartan subalgebra and n the nilradical of g. Nilpotent orbits in classical lie algebras over f2n and the. Centralizers of nilpotent elements in semisimple lie algebras. Through the 1990s, a circle of ideas emerged relating three very different kinds of objects associated to a complex semisimple lie algebra. Representations of semisimple lie algebras in the bgg.
Full text of nilpotent pairs in semisimple lie algebras and. Semisimple subalgebras of semisimple lie algebras mathnet. The 1dimensional lie algebra is called the trivial simple lie algebra. Proposition every nontrivial simple lie algebra is semisimple. Descargar nilpotent orbits in semisimple lie algebras en. An element x of a semisimple lie algebra g is called nilpotent if its adjoint endomorphism. The cartan decomposition of a complex semisimple lie algebra. On reachable elements and the boundary of nilpotent orbits in. Regular subalgebras and nilpotent orbits of real graded.
Skjelbred and sund 1978 36 reduce the classincation of nilpotent algebras in a given dimension. This is useful in proving the solvability of a lie algebra since, in practice, it is usually easier to prove nilpotency when it holds. Quantizations of regular functions on nilpotent orbits. Questions about maximal torus of semisimple derivations of. This fact seems to be used in varieties of nilpotent elements for simple lie algebras i. On reachable elements and the boundary of nilpotent orbits. Jan 29, 2008 we give the number of nilpotent orbits in the lie algebras of orthogonal groups under the adjoint action of the groups over f 2 n. In 9, spaltenstein gives a parametrization of these nilpotent orbits by pairs of partitions. E springer springer monographs in mathematics springer berlin heidelberg new yor. No nilpotent element can be a unit except in the trivial ring 0, which has only a single element 0 1. Regular semisimple elements form a connected zariskiopen and dense set.
Russian articles, english articles this publication is cited in the following articles. Also, all nilpotent orbits consisting of reachable elements in the ex ceptional lie algebras are listed in 3, theorem 2. Visible actions on spherical nilpotent orbits in complex. The irreducible representations of w, in turn, are closely connected to the representations of g, via the theory of primitive. Regular subalgebras and nilpotent orbits of real graded lie algebras 3 given a carrier algebra, we. Humphreys graduate studies in mathematics volume 94 american mathematical society providence, rhode island. Then g is nilpotent i for each a2g, ad an 0 for some n0.
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