In [Bialgebras of type one, 1978], Warren Nichols introduced a family of algebras that initially received little attention. Perhaps MathSciNet gives a clue as to why, calling the paper, "a fairly technical investigation into the structure of certain sorts of Hopf algebras." No surprise that it took more than a decade, and independent discovery by others, for Nichols' algebras to find purchase in the Hopf algebra community.
Today, Nichols algebras play a critical role in the classification of finite dimensional pointed Hopf algebras. They also have a perfectly natural (and non-technical!) definition. I will start here, then give an overview of the ongoing classification program. The ubiquitous Cartan matrices make an appearance.
Polyhedral geometry has long provided insight into the representations of Lie algebras. An early example of the polyhedral approach was the introduction of the Gelfand--Tsetlin polytopes in the 1950's. More recently, Berenstein and Zelevinsky in 2001 encoded tensor product multiplicities for semi-simple Lie algebras as the number of integer lattice points in certain families of polyhedra. These results have theoretical implications as well as concrete computational applications.
We discuss a particular combinatorial structure on Gelfand--Tsetlin polytopes, which we call their tiling poset. We use this combinatorial structure to compute the degrees of so-called "stretched Kostka coefficients", which express the dimension of a weight space as the parametrizing weights are scaled by an integer parameter. We also discuss applications of polyhedral algorithms to the computational complexity of tensor-product multiplicities. We mention some conjectures motivated by the polyhedral interpretation of these coefficients, including a conjecture that, if true, generalizes the Saturation Theorem of A. Knutson and T. Tao.
Applications of polytopes to the representation theory of Lie algebras
Nov. 19, 2009 2pm (MATH 350)
Lie Theory
Richard Green (CU)
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A Chevalley basis for a Lie algebra over the complex numbers is a basis for the algebra all of whose structure constants are integers. One of the standard constructions of Chevalley bases for simple Lie algebras over the complex numbers involves a certain function taking values in the set {+1, -1}; this function is sometimes called Kac's asymmetry function. Although the definition of this function may appear mysterious at first, I will argue that it has some natural combinatorial interpretations. Along the way, I will present combinatorial constructions of some interesting nonassociative algebras, including the loop algebras associated to simple Lie algebras over the complex numbers.
Chevalley bases for Lie algebras and the combinatorics of Kac's asymmetry function
