Celebratio Mathematica

G. M. Benkart, T. B. Gregory, J. M. Osborn, H. Strade, and R. L. Wilson: “Isomorphism classes of Hamiltonian Lie algebras,” pp. 42–57 in Lie algebras (23–28 August 1987, Madison, WI). Edited by G. Benkart and J. M. Osborn. Lecture Notes in Mathematics 1373. Springer (Berlin), 1989. MR 1007323 Zbl 0677.17012 incollection

Helmut Strade	Related
Thomas Bradford Gregory	Related
Robert Lee Wilson	Related
J. Marshall Osborn	Related

G. Benkart, J. M. Osborn, and H. Strade: “Contributions to the classification of simple modular Lie algebras,” Trans. Am. Math. Soc. 341 : 1 (1994), pp. 227–252. MR 1129435 Zbl 0792.17016 article

We develop results directed towards the problem of classifying the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic \( p > 7 \). A 1-section of such a Lie algebra relative to a torus \( T \) of maximal absolute toral rank possesses a unique subalgebra maximal with respect to having a composition series with factors which are abelian or classical simple. In this paper we show that the sum \( Q \) of those compositionally classical subalgebras is a subalgebra. This extends to the general case a crucial step in the classification by Block and Wilson of the restricted simple Lie algebras. We derive properties of the filtration which can be constructed using \( Q \) and obtain structural information about the 1-sections and 2-sections of \( Q \) relative to \( T \). We further classify all those algebras in which \( Q \) is solvable.

Helmut Strade	Related
J. Marshall Osborn	Related

G. Benkart and J. Feldvoss: “Some problems in the representation theory of simple modular Lie algebras,” pp. 207–228 in Lie algebras and related topics: Workshop on Lie algebras, in honor of Helmut Strade’s 70th birthday (22–24 May 2013, Milan). Edited by M. Avitabile, J. Feldvoss, and T. Weigel. Contemporary Mathematics 652. American Mathematical Society (Providence, RI), 2015. MR 3453057 Zbl 06622485 ArXiv 1503.06762 incollection

The finite-dimensional restricted simple Lie algebras of characteristic \( p > 5 \) are classical or of Cartan type. The classical algebras are analogues of the simple complex Lie algebras and have a well-advanced representation theory with important connections to Kazhdan–Lusztig theory, quantum groups at roots of unity, and the representation theory of algebraic groups. We survey progress that has been made towards developing a representation theory for the restricted simple Cartan-type Lie algebras, discuss comparable results in the classical case, formulate a couple of conjectures, and pose a dozen open problems for further study.

Helmut Strade	Related
Jörg Feldvoss	Related

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