The abstract lie algebra corresponding to the commutation relations of this group have various representations, the 8dimensional, or adjoint representation being one of great interest. Lie triple derivations for the parabolic subalgebras of gl. Is the sum of the index of a parabolic subalgebra of a semisimple lie algebra g and the index of its nilpotent radical always greater than or equal to the rank of g. There are several good books on lie algebras and representation theory available. Show that if l 1 lthen the quotient vector space ll 1 inherits a lie algebra structure from l. Borel, linear algebraic groups, benjamin 1969 mr0251042 zbl 0206.
We give a necessary and sufficient condition for an antichain to determine an adnilpotent ideal of the parabolic. Lie algebra g is a zgrading and to each zgrading of g corresponds a parabolic subalgebra. An important difference with the lie algebra case is that even in the case of a triangular. Parabolic and levi subalgebras of finitary lie algebras. Borel subgroup wikimili, the best wikipedia reader.
A parabolic subalgebra p of g c is a lie algebra satisfying b. On the sum of the index of a parabolic subalgebra and of. K denotes an algebraically closed, characteristic zero. Conversely, any borel subalgebra of l which contains h is standard. Semisimple lie algebras and their representations dover.
In this paper, we give an explicit description of lie triple derivations for the parabolic subalgebra p of \\mboxgln,r\. I have found two different definitions for minimal parabolic subalgebra. Recently, significant work has been done in studying the automorphisms and derivations of matrix algebras and their subalgebras see 17. Queryparabolicsubalgebra check if a list of vectors defines a parabolic subalgebra of a semisimple lie algebra calling sequences query parameters p a list of vectors, defining a subalgebra of a semisimple lie algebra description examples. If is a finitedimensional lie algebra over an arbitrary field, then a subalgebra of it is also called a parabolic subalgebra if is a parabolic subalgebra of, where is the algebraic closure of the field. Parabolic category o for classical lie superalgebras. Borel and parabolic subalgebras of some locally finite lie. For any borel subalgebra b of g there exists a parabolic subalgebra p of g such that b is conjugate to b p.
By definition, any subalgebra of gl n, r containing t is called a parabolic subalgebra of gln, r. We extend the results of cellinipapi on the characterizations of nilpotent and abelian ideals of a borel subalgebra to parabolic subalgebras of a simple lie algebra. Any maximal subalgebra of g, the radical of which is noncompact, is a maximal parabolic subalgebra of g 10. The book is suitable for graduate students and researchers interested in lie groups and related topics. Adnilpotent ideals of a parabolic subalgebra sciencedirect. These characterizations are given in terms of elements of the affine weyl group and faces of alcoves. These are called the standard borel subalgebras of l. A borel subalgebra b is any maximal solvable subalgebra. If is an irreducible linear algebraic group over a field of characteristic 0 and is its lie algebra. Derivations of the parabolic subalgebras of the general.
This article shows that every parabolic subalgebra p of a finitedimensional simple lie algebra defined over an algebraically closed field is always zero product determined. Most of the rest of the course is based on parts of the second chapter of my book \ parabolic geometries i. A lie subalgebra of containing a borel subalgebra is called a parabolic lie algebra. Zgradations of lie algebras and in nitesimal generators. Abstract we study adnilpotent ideals of a parabolic subalgebra of a simple lie algebra. Brice describes the derviations of parabolic subalgebra of a reductive lie algebra over an algebraically closed and characteristics zero field, and proves the zeroproduct determined property of such derivation algebras 1. About reductive levi subalgebra of a parabolic subalgebra. The maximal parabolic subalgebras are interesting because each one of them induces a zgradation of g. If gis a semisimple lie group over r and if p is a parabolic subgroup of. Papi, adnilpotent ideals of a borel subalgebra ii, j. Algebra 258 2002 112121 on the characterizations of adnilpotent and abelian ideals of a borel subalgebra to parabolic subalgebras of a simple lie algebra. Parabolic and levi subalgebras of finitary lie algebras core.
Solid but concise, this account of lie algebra emphasizes the theorys simplicity and offers new approaches to major theorems. Such a lie algebra is a split extension of an abelian lie algebra by a direct sum of copies of. Citeseerx document details isaac councill, lee giles, pradeep teregowda. A parabolic subalgebra of g is any subalgebra which contains a maximal locally solvable that is, borel subalgebra. Let qbe a parabolic subalgebra of a reductive lie algebra gover an algebraically closed, characteristiczero. Differentialgeometryliealgebrasqueryparabolicsubalgebra. However, the study on the derivations or auto morphisms of the parabolic subalgebras of the general linear lie algebra has not been reported. A lie subalgebra of g \displaystyle \mathfrak g containing a borel subalgebra is called a parabolic lie algebra. If the fvector space v is actually an algebra rover f, then the lie algebra glr admits a natural subalgebra. The author finds the famous cartan subalgebra of the lie algebra, shows that it 2dimensional and abelian, and how eigenvectors of the adjoint operator. In mathematics, a toral subalgebra is a lie subalgebra of a general linear lie algebra all of whose elements are semisimple. In algebra, a parabolic lie algebra is a subalgebra of a semisimple lie algebra satisfying one of the following two conditions. Let l be a semisimple lie algebra h a csa, with root system.
Let derr be the subspace of glr consisting of derivations, i. I two semiclosed generalized ags f in v and g in v form a taut couple if the chain f. A parabolic subalgebra of is any subalgebra that contains a maximal locally solvable that is, borel subalgebra. Passing to the lie algebra 5 a primer on the lie group lie algebra correspondence 8. Viiviii mr0682756 mr0573068 mr0271276 mr0240238 mr02805 zbl 0329.
Equivalently, a lie algebra is toral if it contains no nonzero nilpotent elements. The author finds the famous cartan subalgebra of the lie algebra, shows that it 2dimensional and abelian, and how eigenvectors of the adjoint operator can form a basis for the lie algebra, as long as this operator corrresponds to an element of the cartan subalgebra. Note that in the next proposition we do not assume that ris associative. This paper describes an algorithm for computing representatives of conjugacy classes of.
However, the study on the derivations or automorphisms of the parabolic subalgebras of the general linear lie algebra has not been reported. Let lbe the set of all linear transformations mapping qinto its center qz and. Let r be a commutative ring with identity, \\mboxgln,r\ the general linear lie algebra over r, p a parabolic subalgebra of \\mboxgln,r\. In this short note, we investigate the following question of panyushev. Note that it is enough to consider the case of bi parabolic subalgebras of. Biderivations of the parabolic subalgebras of simple lie. For the special case of a lie algebra with a cartan subalgebra, given an ordering of, the borel subalgebra is the direct sum of and the weight spaces of with positive weight. Let g be a complex simple lie algebra and p be a maximal parabolic. Any such ideal determines an antichain in a set of positive roots of the simple lie algebra. The standard example of a borel subalgebra is given as follows. In this paper, we will describe the parabolic subalgebras of gln, r and then give an explicit descrip tion on the derivations of the parabolic subalgebras of gln, r, for r an arbitrary commutative ring. Parabolic subgroups of real direct limit lie groups. Onishchik, contains a collection of articles by participants in the moscow seminar on lie groups and invariant theory headed by e. In this section we will give the classification of the nonlinear primitive lie.
These characterizations are given in terms of elements of. The parabolic subalgebras of sln,c are determined by partitions of n and consist of block lowertriangular. A parabolic subalgebra p is any subalgebra containing a borel subalgebra b. If lis a lie algebra then a kvector subspace l 1 is a lie subalgebra of lif it is closed under the lie bracket. Computing thetastable parabolic subalgebras using lie. These subalgebras are important for studying representations of g. Parabolic and levi subalgebras of nitary lie algebras. Winter, a professor of mathematics at the university of michigan, also presents a general, extensive treatment of cartan and related lie subalgebras over arbitrary fields. Such a lie algebra is a split extension of an abelian lie algebra by a direct sum of copies of sl. Let be a locally reductive complex lie algebra that admits a faithful countabledimensional finitary representation v. Realizations of affine lie algebras verma type modules. The intersection of a maximal toral subalgebra with a simple ideal of a lie algebra is a maximal toral subalgebra of the simple ideal. Ideals in parabolic subalgebras of simple lie algebras.