Constructing semisimple subalgebras of semisimple lie. We draw heavily from erdmann and wildons book introduction to lie algebras 3. In this paper, he introduced what are now known as cartan subalgebras, cartan matrices, and root systems 2. On maximal commutative subalgebras of poisson algebras. In ring theory, a branch of mathematics, a semisimple algebra is an associative artinian algebra over a field which has trivial jacobson radical only the zero element of the algebra is in the jacobson radical. For simple finitedimensional lie algebras these classes of subalgebras have been described in the classical works of a. We compare the maximal dimension of abelian subalgebras and the maximal di. Readings introduction to lie groups mathematics mit. Nov 11, 2015 pdf let p be a maximal truncated parabolic subalgebra of a simple lie algebra.
Humphreys graduate studies in mathematics volume 94 american mathematical society providence, rhode island. In this thesis we prove that the maximal toral subalgebras are conjugate when. Masters thesis restricting irreducible representations of. Small semisimple subalgebras of semisimple lie algebras. In his doctoral thesis, elie cartan 18691951 extended killings work on the classi ca tion of simple complex lie algebras. More speci cally, for semisimple lie algebras s the invariant s has been completely determined by malcev 12. This chapter discusses the cartan decompositions and root systems of semisimple lie algebras. It immediately follows that vu is maximal in gu, and thus v is a maximal nilpotent subalgebra, assuming that chark. In this paper, we attack the problem of describing maximal commutative subalgebras of the poisson algebra s k. All maximal graded subalgebras are described completely by a c. It immediately follows that vu is maximal in gu, and thus v is a maximal nilpotent subalgebra, assuming that chark 6 2,3 or gr ug satis.
Cartan subalgebras of simple lie algebras by robert lee wilson abstract. Constructing semisimple subalgebras of semisimple lie algebras. There are functions for computing branching rules, for computing the regular subalgebras, and for working with the database of semisimple subalgebras of the simple lie algebras. Conjugacy of maximal toral subalgebras of direct limits of. Solvable and nilpotent lie algebras, the theorems of engel and lie, representation theory, cartans criteria, weyls theorem, root systems, cartan matrices and dynkin diagrams, the classical lie algebras, representation theory. A lie algebra lis called monolithic if it has a unique minimal ideal w, the monolith of l. Locally semisimple and maximal subalgebras of the finitary lie algebras,, and article in journal of algebra 3226.
It would help to have a modern survey of the entire problem area for both matrix algebras and semisimple lie algebras. A cartan subalgebra of a semisimple lie algebra is a subalgebra which is maximal in regard to the two properties. The question, whether any adsemisimple lie algebra is commutative, is related to the. This will amount to a detailed knowledge of root space decompostions. The study of semisimple maximal subalgebras in complex classical lie algebras, which. This is closely related to the study of the coadjoint representation of k and the set, k reg. Introduction to lie algebras and representation theory.
Dynkintreated theliealgebras ofexceptionaltypein 5. Representations of semisimple lie algebras in this chapter we study a special type of representations of semisimple lie algberas. In fact, if g is semisimple then the cartan subalgebras of g are precisely the maximal abelian subalgebras which consist of semisimple. In 4 dynkin classified the maximal semisimple ssubalgebras of the lie algebras. Exceptional lie algebras, semisimple subalgebra, adjoint group, simple sub. Its irreducible representations are all 1dimensional, hence its orbits consist of one. We describe all locally semisimple subalgebras and all maximal subalgebras of the.
The roots of g which project onto roots of g can be simply. Russian articles, english articles this publication is cited in the following articles. Let l be a finitedimensional semisimple lie algebra over c. A typical lie algebra is a semidirect sum of a semisimple lie algebra and a solvable subalgebra that is invariant. This paper examines lie algebras with particular focus on nitedimensional lie algebras over c, building up to the classi cation of semisimple complex lie algebras. Killing gave a full classi cation of simple complex lie algebras.
A subalgebra bof a semisimple lie algebra lis called regular in l if we. For semisimple lie algebras, cartan subalgebras and the associated root spaces have additional properties. Nilpotent subspaces of maximal dimension in semisimple lie. In this lecture and the few that follow, we will study the structure of nite dimensional semisimple lie algebras with the aim of classifying them. Edges of the form 0, j express that the subalgebra on position j is a maximal semisimple subalgebra of the lie algebra of type type. Minimal faithful representations of reductive lie algebras. They exploit the use of orthogonal bases in weight space, which are related to quark weights. For such a lie algebra, if nonzero, the following conditions are equivalent. Mar 21, 2008 project euclid mathematics and statistics online.
Next we will recall the theorems of morozov and mostow on the maximal subalgebras of the lie algebras. Lecture 12 structure theory of semisimple lie algebras i. We construct a slice for the coadjoint action of p, thus extending a theorem of kostant. We use the classification of finite order automorphisms by kac to characterize all maximal subalgebras, regular, semisimple, reductive or not of a simple complex lie algebra up to conjugacy that we can determine from its dynkin diagram. Let p denote a maximal truncated parabolic subalgebra of a simple lie algebra \ \mathfrakg \. For p 5 and g of type e 8 we give an initial result on non semisimple maximal. Onishchik, decompositions of reductive lie groups, math. Semisimple subalgebras of semisimple lie algebras mathnet.
If the algebra is finitedimensional this is equivalent to saying that it can be expressed as a cartesian product of simple subalgebras. Pdf small semisimple subalgebras of semisimple lie. By inspection of the regular representation in suitable form, we are able to determine the maximal nilpotent and solvable invariant subalgebras of the lie algebra and. Techniques for obtaining generators of nonregular maximal subalgebras of lie algebras, alternative to those of dynkin, are developed. The projection from an algebra g to its nonregular subalgebras g is related to an orthogonal matrix. Since submitting this paper, we have learned that several of our results for associative algebras, in.
Dynkin, semisimple subalgebras of semisimple lie algebras, mat. Lie algebras notes pdf 34p this note explains the following topics. The american mathematical monthly exceptionally well written and ideally suited either for independent reading or as a text for an introduction to lie algebras and their representations. Minimal faithful representations of reductive lie algebras springerlink advertisement. Representations of semisimple lie algebras in the bgg category 0 james e. We describe all locally semisimple subalgebras and all maximal subalgebras of the finitary lie algebras gl. Let g be a finite dimensional semisimple complex lie algebra of rank p. The last chapter discusses the connection between lie algebras, complex groups and compact groups. The lie algebra sl 2 r of 2 by 2 matrices of trace 0 has two nonconjugate cartan subalgebras. The following stability theorem shows that this is indeed the case. Let l be a finitedimensional simple lie algebra over an algebraically closed field of characteristic p 7.
Abelian ideals of maximal dimension for solvable lie algebras dietrich burde and manuel ceballos abstract. Pdf let p be a maximal truncated parabolic subalgebra of a simple lie algebra. Totally geodesic subalgebras in 2step nilpotent lie algebras decoste, rachelle c. Representations of semisimple lie algebras in the bgg. More precisely, 4 contains a procedure by which it is possible for a given lie algebra of classical typeto. For semisimple lie algebras s the invariant s has been completely determined by malcev. The second column displays the number of linear equivalence classes of subalgebras. Pdf supplements to maximal subalgebras of lie algebras. Standard subalgebras of semisimple lie algebras and. The discrete series of semisimple groups peter hochs september 5, 2019. We refer the reader as well to the article of vasiu. A toral subalgebra t is a subalgebra consisting of. Basic definitions and examples, theorems of engel and lie, the killing form and cartans criteria, cartan subalgebras, semisimple lie algebras, root systems, classification and examples of semisimple lie algebras. Maximal subalgebras of simple modular lie algebras request pdf.
Towers department of mathematics, lancaster university lancaster la1 4yf, england abstract. Alekseevskii, maximal finite subgroups of lie groups, funct. 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. The orbits of weyl groups wan of simple an type lie algebras are reduced to the union of orbits of the weyl groups of maximal. Let s be a maximal nonsemisimple subalgebra of a complex semisimple lie algebra l.
The value of for simple lie algebras is reproduced in table 1. Lie algebras are an essential tool in studying both algebraic groups and lie groups. Oct 19, 2007 the result can be used to study dynkins classification of maximal reductive subalgebras of semisimple lie algebras. Subsequent chapters introduce cartan subalgebras, root systems, and linear representations. Since there are no abelian ideals in s, we have s 0. In the case of direct limit lie algebras, the work of salmasian 18 studies sl. A periodisation of semisimple lie algebras larsson, anna, homology, homotopy and applications, 2002. However, i havent gone far enough into the literature to feel confident about exactly what is true. In this chapter we discuss the structure of lie algebras. Introduction to lie algebras and representation theory an excellent introduction to the subject, ideal for a one semester graduate course. A lie algebra is a vector space, l, above, the linear combinations of the ts together with a bilinear operation from l. With 428 exercises sven bodo wirsing textbook mathematics algebra.
This algebra is present in reductive nonsemisimple lie algebras. On the conjugacy of maximal toral subalgebras of certain. Slices for maximal parabolic subalgebras of a semisimple lie algebra. The dimension of a cartan subalgebra is not in general the maximal dimension of an abelian subalgebra, even for complex simple lie algebras. The notion of an rsubalgebra and that of an ssubalgebra are introduced for. The maximal graded subalgebras for four families of lie superalgebras of cartan type over a field of prime characteristic are studied. Nilradicals and cartan subalgebras in associative algebras. Let h be a cartan subalgebra of a semisimple lie algebra g. Popov, stability criteria for the action of a semisimple group on a factorial manifold. Locally semisimple and maximal subalgebras of the finitary.
Maximal abelian subalgebras of pseudoorthogonal lie algebras. Karpelevich 9 rephrased these results in terms of closed subsystems of the root system, which can be stated as follows. Pairs of semisimple lie algebras and their maximal reductive subalgebras boris sirola 1 algebras and representation theory volume 11, article number. Maximal subalgebras for lie superalgebras of cartan type.
We apply these results to exceptional lie algebras of type g 2, and give the complete classi cation of their maximal subalgebras. The weyl group of the oneparameter lie algebra u1 is trivial, consisting of the identity element only. This chapter contains functions for dealing with semisimple subalgebras of semisimple lie algebras. Contents 1 basic definitions and examples 2 2 theorems of engel and lie 4 3 the killing form and cartans criteria 8 4 cartan subalgebras 12 5 semisimple lie algebras 15. In 3, draisma, kraft and kuttler study subspaces of g, rather than subalgebras, consisting of nilpotent elements. The orbits of weyl groups wan of simple an type lie algebras are reduced to the union of orbits of the weyl groups of maximal reductive. In pi1, the author uses a marvellous connection between conjugacy of maximal toral subalgebras and triviality of certain equivariant torsors on a ne schemes to obtain a conjugacy theorem for a general class of algebras which includes toroidal algebras over nite dimensional split simple lie algebras. Maximal subalgebras of semisimple algebras 1 we classify maximal subalgebras of semisimple algebras. If ldoes not contain ad semisimple elements, all its elements are adnilpotent by jordanchevalley theorem and lis nilpotent by.
All solvable extensions of a class of nilpotent lie algebras. The rst step is to use the killing form to understand cartan subalgebras and their actions under the adjoint representation. Of course, primitive lie algebras of types 2 and 3 are semisimple, and those of types 1 and 2 are monolithic. Mar 14, 2016 let p denote a maximal truncated parabolic subalgebra of a simple lie algebra \ \mathfrakg \. Pairs of semisimple lie algebras and their maximal. Borel subalgebra of g is a maximal solvable subalgebra of g, and a parabolic. Dynkin, semisimple subalgebras of semisimple lie algebras. A cartan subalgebra of g certainly has these two properties. A subalgebra b of a semisimple lie algebra l is called regular in l if we. In mathematics, a lie algebra is semisimple if it is a direct sum of simple lie algebras, i.
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