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
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
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
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.
We introduce the shifted quantum affine algebras. They map homomor- phically into the quantized K-theoretic Coulomb branches of 3d N = 4 SUSY quiver gauge theories. In type A, they are endowed with a coproduct, and they act on the equivariant K-theory of parabolic Laumon spaces. In type A_1 , they are closely related to the type A open relativistic quantum Toda system.