Why Are There No Triple Affine Hecke Algebras?

What is an intuitive explanation of Kac-Moody algebras?

• Especially in relation to Lie algebras

• Answer:

  Well, you have the theory of Coxeter groups that can be defined in two following ways (see http://www.amazon.com/Reflection-Coxeter-Cambridge-Advanced-Mathematics/dp/0521436133/ref=sr_1_1?ie=UTF8&qid=1398071190&sr=8-1&keywords=humphreys+coxeter+groups for full details): As a finite group generated by reflections acting on Euclidean space [math]R^n[/math]. They are then classified by the http://en.wikipedia.org/wiki/Coxeter-Dynkin_diagram (see Chapter 1 and 2 of above book). Or from a Coxeter-Dynkin diagram [math]D[/math] as a group [math]W(D)[/math] generated by the fundamental reflection of the diagram. The quadratic form [math]q[/math] is obtained from the diagram. If [math]q[/math] is positive definite then [math]W(D)[/math] is finite and we are in preceding case, otherwise [math]W(D)[/math] is infinite (Chapter 5, independent of other chapters). The second way is more general and takes the viewpoint of the diagram as the starting point. The theory of http://en.wikipedia.org/wiki/Lie_algebra is a general way to represent infinitesimal elements of continuous groups. Of special interest are the http://en.wikipedia.org/wiki/Semisimple_Lie_algebra since they describe the general linear group, symplectic group and so on. Those algebras are classified, see for example http://www.amazon.com/Introduction-Algebras-Representation-Graduate-Mathematics/dp/0387900535/ref=sr_1_1?s=books&ie=UTF8&qid=1398071755&sr=1-1&keywords=humphreys+lie+algebra and amazingly enough the classification is very similar to the one of finite Coxeter groups (but not identical, there are a number of little differences) by using diagrams. From a diagram [math]D[/math] we can write a presentation of the Lie algebra that uses only the diagram. What http://en.wikipedia.org/wiki/Victor_Kac did was to take the diagrammatic viewpoint as the starting point and build from that the whole theory, see: http://www.amazon.com/Infinite-Dimensional-Lie-Algebras-Victor-Kac/dp/0521466938/ref=sr_1_1?s=books&ie=UTF8&qid=1398072043&sr=1-1&keywords=kac+infinite+dimensional+lie+algebra. The resulting are the http://en.wikipedia.org/wiki/Kac-Moody_algebra. It is not something that can be explained simply and intuitively, somehow they are the infinite dimensional generalizations of orthogonal, symplectic and general linear group. But their main application seems to be in theoretical physics.