Showing papers on "Cartan matrix published in 1986"

Graded modules of graded lie algebras of cartan type(ii)——positive and negative gpaded modules

Abstract: The positive and negative graded modules of a graded Lie algebra is under a general discussion. The results are applied to the modules (?) and (?) of a graded Lie algebra L of the Cartan type. Among other things, an intrinsic explanation is obtained for the Tricomi operator representation and the derivation representation, by which L is defined.

The associative forms of the graded Cartan type Lie algebras

Abstract: This paper determines the Cartan type Lie algebras that possess a nonsingular associative form. O. Introduction. The structure of Cartan type Lie algebras has been extensively studied for several years. Derivation algebras and isomorphism classes of Cartan subalgebras are well known and it is the objective of the present paper to determine those graded Cartan type Lie algebras which possess a nonsingular associative form. In contrast with trace forms, general nonsingular associative forms do not play an eminent role in the classification theory of simple Lie algebras of prime characteristic. Their significance primarily rests on cohomology theory. Nonsingular associative forms allow the study of central extensions of Lie algebras by means of derivations. We will employ a slight generalization of this known interrelation in a forthcoming paper. The introductory remarks of the first section are followed by a brief investigation of Cartan subalgebras of graded Lie algebras. Our results generalize those of M. Frank (cf. [8]). §3 establishes general properties of associative forms of graded simple Lie algebras and provides the basic tools for the study of the graded Cartan type Lie algebras. The author would like to express his gratitude to Professor Helmut Strade for many helpful suggestions as well as to the referee for proposing Proposition 3.4 and the resulting simplifications in §4. 1. Graded representations of graded Lie algebras. In the sequel, L = @ ik -r L,, r, k > 1, denotes a graded Lie algebra over an arbitrary field F. A representation T: L gl(V) is said to be graded if V = (13 _ Vj and T(x)(Vj) c Vl+j for every x E Li and -s 1. We consider the subalgebras L+ := eIk l Li and L-:d3 _ Li. Using the P-B-W Theorem it can be readily verified that we have U(L) = U(L-)U(Lo)U(L+) = U(L+)u(Lo)u(L-) for the respective universal enveloping algebras. Received by the editors May 17, 1984 and, in revised form, April 10, 1985. 1980 Mathematics Subject Classification. Primary 17B20. t1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

On the generalized Nakayama conjecture and the Cartan determinant problem

Abstract: For Artin algebras allowing certain filtered module categories, the Generalized Nakayama Conjecture is shown to be true; our result covers all positively graded Artin algebras and those whose radical cube is zero. For the corresponding class of left artinian rings we prove that finite global dimension forces the determinant of the Cartan matrix to be 1.

The BGG resolution, character and denominator formulas, and related results for Kac-Moody algebras

Abstract: Let q be a Kac-Moody algebra defined by a (not necessarily symmetrizable) generalized Cartan matrix. We construct a BGG-type resolution of the irreducible module L(X) with dominant integral highest weight X, and we use this to obtain character and denominator formulas analogous to those of Weyl. We also determine a condition on the algebra which is sufficient for these formulas to take their classical form, and which implies that the set of defining relations is complete.

Lie algebras and related topics : proceedings of a summer seminar held June 26-July 6, 1984

Abstract: Restricted simple Lie algebras by R. E. Block and R. L. Wilson A survey on enveloping algebras of semisimple Lie algebras by W. Borho Three lectures on formal groups by M. Hazewinkel Kac-Moody algebras by I. G. Macdonald A Kac-Moody bibliography and some related references by G. Benkart Structural division algebras and relative rank one simple Lie algebras by B. N. Allison Cartan subalgebras in Lie algebras of Cartan type by G. Benkart The radical conjecture for homogeneous Kahler manifolds by J. Dorfmeister Lie algebras all of whose subalgebras are supersolvable by A. Elduque and V. R. Varea Ad-simple Lie algebras and their applications by R. Farnsteiner Tables of $E_8$ characters and decompositions of plethysms by W. G. McKay, R. V. Moody, and J. Patera The orbit method and primitive ideals for semisimple algebras by D. A. Vogan, Jr. On pointed modules of simple Lie algebras by D. J. Britten and F. W. Lemire A study of subexpressions and its applications by V. V. Deodhar A classification of semisimple symmetric pairs and their restricted root systems by A. G. Helminck Induced modules for semisimple groups and Lie algebras by J. E. Humphreys On elements of finite order and cyclotomic fields by A. Pianzola Odd symplectic groups and combinatorics by R. A. Proctor Beyond Kac-Moody algebras and inside by P. Slodowy Root systems of Lie algebras by D. J. Winter Properness of Lie algebras and enveloping algebras, II by W. Michaelis.

Structure of Semi-Simple Lie Algebras

Abstract: The semi-simple algebras and their representations play a central role in applications to physical examples. For example, so(3), su(2), so(3, 1) and su(3) are all (real) semi-simple algebras. The complex semi-simple algebras were originally classified by Killing and Cartan in his thesis (1894); and later (1914) Cartan classified the real ones. We discuss these matters in some detail in this chapter; but before launching into an abstract algebraic development, we ease into the subject with two simple examples.