In this short note, we investigate the following question of panyushev. The intersection of a maximal toral subalgebra with a simple ideal of a lie algebra is a maximal toral subalgebra of the simple ideal. Papi, adnilpotent ideals of a borel subalgebra ii, j. If is an irreducible linear algebraic group over a field of characteristic 0 and is its lie algebra. Ideals in parabolic subalgebras of simple lie algebras. Passing to the lie algebra 5 a primer on the lie group lie algebra correspondence 8. By definition, any subalgebra of gl n, r containing t is called a parabolic subalgebra of gln, r. A parabolic subalgebra p of g c is a lie algebra satisfying b. A parabolic subalgebra of is any subalgebra that contains a maximal locally solvable that is, borel subalgebra. Citeseerx document details isaac councill, lee giles, pradeep teregowda. There are several good books on lie algebras and representation theory available. A parabolic subalgebra p is any subalgebra containing a borel subalgebra b. Borel subgroup wikimili, the best wikipedia reader. I two semiclosed generalized ags f in v and g in v form a taut couple if the chain f.
Let be a locally reductive complex lie algebra that admits a faithful countabledimensional finitary representation v. A biparabolic subalgebra or seaweed subalgebra of a semisimple lie algebra is the intersection of two parabolic subalgebras whose sum is the total lie algebra. The standard example of a borel subalgebra is given as follows. 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\. This paper describes an algorithm for computing representatives of conjugacy classes of. In this paper, we give an explicit description of lie triple derivations for the parabolic subalgebra p of \\mboxgln,r\. We give a necessary and sufficient condition for an antichain to determine an adnilpotent ideal of the parabolic. 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. 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. Parabolic and levi subalgebras of finitary lie algebras. These characterizations are given in terms of elements of. Such a lie algebra is a split extension of an abelian lie algebra by a direct sum of copies of.
Derivations of the parabolic subalgebras of the general. 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. In mathematics, a toral subalgebra is a lie subalgebra of a general linear lie algebra all of whose elements are semisimple. Let qbe a parabolic subalgebra of a reductive lie algebra gover an algebraically closed, characteristiczero. However, the study on the derivations or automorphisms of the parabolic subalgebras of the general linear lie algebra has not been reported. If lis a lie algebra then a kvector subspace l 1 is a lie subalgebra of lif it is closed under the lie bracket. K denotes an algebraically closed, characteristic zero. 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. Solid but concise, this account of lie algebra emphasizes the theorys simplicity and offers new approaches to major theorems. However, the study on the derivations or auto morphisms of the parabolic subalgebras of the general linear lie algebra has not been reported. Zgradations of lie algebras and in nitesimal generators. Parabolic and levi subalgebras of nitary lie algebras. Differentialgeometryliealgebrasqueryparabolicsubalgebra. Abstract we study adnilpotent ideals of a parabolic subalgebra of a simple lie algebra.
Algebra 258 2002 112121 on the characterizations of adnilpotent and abelian ideals of a borel subalgebra to parabolic subalgebras of a simple lie algebra. Let g be a complex simple lie algebra and p be a maximal parabolic. Borel, linear algebraic groups, benjamin 1969 mr0251042 zbl 0206. If gis a semisimple lie group over r and if p is a parabolic subgroup of. 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. 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. About reductive levi subalgebra of a parabolic subalgebra. Parabolic and levi subalgebras of finitary lie algebras core. Any maximal subalgebra of g, the radical of which is noncompact, is a maximal parabolic subalgebra of g 10. Conversely, any borel subalgebra of l which contains h is standard.
Parabolic category o for classical lie superalgebras. A lie subalgebra of g \displaystyle \mathfrak g containing a borel subalgebra is called a parabolic lie algebra. Lie triple derivations for the parabolic subalgebras of gl. 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.
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. Such a lie algebra is a split extension of an abelian lie algebra by a direct sum of copies of sl 1, so 1, sp 1, and nitedimensional simple lie algebras. I have found two different definitions for minimal parabolic subalgebra. If the fvector space v is actually an algebra rover f, then the lie algebra glr admits a natural subalgebra. Note that in the next proposition we do not assume that ris associative. A parabolic subalgebra of g is any subalgebra which contains a maximal locally solvable that is, borel subalgebra. In this section we will give the classification of the nonlinear primitive lie. Borel and parabolic subalgebras of some locally finite lie. In this short note, we investigate the following question of panyushev stated in 10. An important difference with the lie algebra case is that even in the case of a triangular. 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. Parabolic subgroups of real direct limit lie groups. 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. 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.
Lie algebra g is a zgrading and to each zgrading of g corresponds a parabolic subalgebra. The maximal parabolic subalgebras are interesting because each one of them induces a zgradation of g. Realizations of affine lie algebras verma type modules. The parabolic subalgebras of sln,c are determined by partitions of n and consist of block lowertriangular. In algebra, a parabolic lie algebra is a subalgebra of a semisimple lie algebra satisfying one of the following two conditions.
On the sum of the index of a parabolic subalgebra and of. A borel subalgebra b is any maximal solvable subalgebra. 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. Note that it is enough to consider the case of bi parabolic subalgebras of. A lie subalgebra of containing a borel subalgebra is called a parabolic lie algebra. These characterizations are given in terms of elements of the affine weyl group and faces of alcoves. Onishchik, contains a collection of articles by participants in the moscow seminar on lie groups and invariant theory headed by e. Computing thetastable parabolic subalgebras using lie. These are called the standard borel subalgebras of l. Such a lie algebra is a split extension of an abelian lie algebra by a direct sum of copies of sl. The book is suitable for graduate students and researchers interested in lie groups and related topics. Let derr be the subspace of glr consisting of derivations, i.