KILLING, WILHELM.

Die Zusammensetzung der stetigen endlichen Transformationsgruppen.

Leipzig, B. G. Teubner, 1889. 8vo. Bound in recent full black cloth with gilt lettering to spine. In "Mathematische Annalen", Volume 34., 1889. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. Very fine and clean. Pp.57-122 [Entire volume: IV, 600 pp.].

First publication of Killing's important third paper (of a total of four) in which he laid the foundation of a structure theory for Lie algebras.

"In particular he classified all the simple Lie algebras. His method was to associate with each simple Lie algebra a geometric structure known as a root system. He used linear transformation, to study and classify root systems, and then derived the structure of the corresponding Lie algebra from that of the root system."(Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences)
Unfortunately for Killing a myth arose that his work was riddled with error, which later has been proved untrue. "As a result, many key concepts that are actually due to Killing bear names of later mathematicians, including "Cartan subalgebra", "Cartan matrix" and "Weyl group". As mathematician A. J. Coleman says, "He exhibited the characteristic equation of the Weyl group when Weyl was 3 years old and listed the orders of the Coxeter transformation 19 years before Coxeter was born."

The theory of Lie groups, after the Norwegian mathematician Sophus Lie, is a structure having both algebraic and topological properties, the two being related.

Order-nr.: 47131