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## Representations and Nilpotent Orbits of Lie Algebraic Systems. In Honour of the 75th Birthday of Tony Joseph

Lie theory, inaugurated through the fundamental work of Sophus Lie during the late

nineteenth century, has proved central in many areas of mathematics and theoretical

physics. Sophus Lie’s formulation was originally in the language of analysis and

geometry; however, by now, a vast algebraic counterpart of the theory has been

developed. As in algebraic geometry, the deepest and most far-reaching results in

Lie theory nearly always come about when geometric and algebraic techniques are

combined.

A core part of Lie theory is the structure and representation theory of complex

semisimple Lie algebras and Lie groups, which is an exemplary harmonious field in

modern mathematics. It has deep ties to physics, and many areas of mathematics,

such as combinatorics, category theory, and others. This field has inspired many

generalizations, among them the representation theories of affine Lie algebras,

vertex operator algebras, locally finite Lie algebras, Lie superalgebras, etc. This

volume originates from a pair of sister conferences titled “Algebraic Modes of

Representations” held in Israel in July 2017. The first conference took place at the

Weizmann Institute of Science, Rehovot, July 16–18, and the second conference

took place at the University of Haifa, July 19–23. Both conferences were dedicated

to the 75th birthday of Anthony Joseph, who has been one of the leading figures

in Lie Theory from the 1970s until today. The conferences were supported by the

United States–Israel Binational Science Foundation and the Chorafas Institute for

Scientific Exchange (Weizmann part) and by the Israel Science Foundation (Haifa

part).

Joseph has had a fundamental influence on both classical representation theory

and quantized representation theory. A detailed description of his work in both

areas has been given in the articles by W. McGovern and D. Farkash–G. Letzter in

the volume “Studies in Lie theory,” Progress in Mathematics, vol. 243, Birkhauser.

Concerning Joseph’s contribution to classical representation theory, it is impossible

not to mention his classification of primitive ideals of the universal enveloping

algebra of sl(n). The essential new ingredient here is the introduction of a partition

of the Weyl group into left cells, corresponding to the Robinson map from the

symmetric group to the standard Young tableaux. Joseph further extended this result to other simple Lie algebras using similar techniques, and this has since then become

a powerful tool in Lie theory.

As for quantized representation theory, Joseph’s monograph “Quantum Groups

and Their Primitive Ideals,” Ergebnisse der Mathematik und Ihrer Grenzgebiete, 3rd

series, vol. 29, has had a fundamental influence over the field since its appearance

in 1995.

The present volume contains 14 original papers covering a broad spectrum of

current aspects of Lie theory. The areas discussed include primitive ideals, invariant

theory, geometry of Lie group actions, crystals, quantum affine algebras, Yangians,

categorification, and vertex algebras.

The authors of this volume are happy to dedicate their works to Anthony Joseph.