Kac-Moody Theory Research Articles

Steinberg groups as amalgams

For any root system and any commutative ring we give a relatively simple presentation of a group related to its Steinberg group St. This includes the case of infinite root systems used in Kac-Moody theory, for which the Steinberg group was defined by Tits and Morita-Rehmann. In most cases our group equals St, giving a presentation with many advantages over the usual presentation of St. This equality holds for all spherical root systems, all irreducible affine root systems of rank>2, and all 3-spherical root systems. When the coefficient ring satisfies a minor condition, the last condition can be relaxed to 2-sphericity. Our presentation is defined in terms of the Dynkin diagram rather than the full root system. It is concrete, with no implicit coefficients or signs. It makes manifest the exceptional diagram automorphisms in characteristics 2 and 3, and their generalizations to Kac-Moody groups. And it is a Curtis-Tits style presentation: it is the direct limit of the groups coming from 1- and 2-node subdiagrams of the Dynkin diagram. Over non-fields this description as a direct limit is new and surprising. Our main application is that many Steinberg and Kac-Moody groups over finitely-generated rings are finitely presented.

Mini-Workshop: Generalizations of Symmetric Spaces

This workshop brought together experts from the areas of algebraic Lie theory, invariant theory, Kac–Moody theory and the theories of Tits buildings and of symmetric spaces. The main focus was on topics related to symmetric spaces in order to stimulate progress in current research projects or trigger new collaboration via comparison, analogy, transfer, generalization, and unification of methods. Specific topics that were covered include Kac–Moody symmetric spaces, double coset decompositions of (groups of rational points of) algebraic groups and Kac–Moody groups, and symmetric/Gelfand pairs in Lie algebras.

A fixed point theorem for Lie groups acting on buildings and applications to Kac–Moody theory

Abstract We establish a fixed point property for a certain class of locally compact groups, including almost connected Lie groups and compact groups of finite abelian width, which act by simplicial isometries on finite rank buildings with measurable stabilisers of points. As an application, we deduce amongst other things that all topological one-parameter subgroups of a real or complex Kac–Moody group are obtained by exponentiating ad-locally finite elements of the corresponding Lie algebra.

Infinite-Dimensional Lie Theory

Nowadays infinite-dimensional Lie theory is a core area of modern mathematics, covering a broad range of branches, such as the structure and classification theory of infinite-dimensional Lie algebras, geometry of infinite-dimensional Lie groups and their homogeneous spaces, and representation theory of infinite-dimensional Lie groups, Lie algebras and Lie-superalgebras. The focus of this workshop was on recent developments in all of these areas with particular emphasis on connections with other branches of mathematics, such as algebraic groups and Galois cohomology. The meeting was attended by 52 participants from many European countries, Canada, the USA, Brazil, Japan and Australia. The meeting was organized around a series of 23 lectures each of 50 minutes duration representing the major recent advances in the area. We feel that the meeting was exciting and highly successful. The quality of the lectures, several of which surveyed recent developments, was outstanding. The exceptional atmosphere of the Oberwolfach Institute provided an optimal environment for bringing people working in algebraically, geometrically or analytically oriented areas of infinite-dimensional Lie theory together, and to create an atmosphere of scientific interaction and cross-fertilization. Without going too much into detail, let us mention some important new developments that were showcased during the workshop. In the structure theory of infinite-dimensional Lie algebras, the classification of extended affine Lie algebras, based on the notion of a Lie torus has now reached a mature state (Neher). In the classification theory of infinite-dimensional Lie algebras, several deep results were obtained recently with Galois cohomology methods exhibiting exciting connections between forms of multiloop algebras and the Galois theory of forms of algebras over rings (Allison, Gille, Chernousov). This branch of structure theory is complemented by the connection between the classification of generalized Kac–Moody algebras and automorphic forms (Scheithauer). In the representation theory of infinite-dimensional Lie algebras, the most exciting new developments concern various kinds of categories of representations of current algebras and Kac–Moody–Lie (super-)algebras (Benkart, Chari, Futorny, Gorelik, Kumar, Littelmann, Serganova). Another interesting, recently very active direction of Kac–Moody theory are Kac–Moody groups over finite fields, which leads to a new class of infinite simple groups (Caprace). On geometric and analytic Lie theory, we had exiting talks on new geometric directions in the representation theory of Banach–Lie groups, related to Banach–Lie–Poisson spaces (Ratiu), and applications of heat kernel measures in the representation theory of loop groups (Pickrell). On the opposite side of the spectrum of Lie group theory, namely direct limit theory, crucial progress has been made on direct limits of unitary representations, as well as in the context of direct limits of infinite-dimensional groups (Wolf, Glöckner). We further had several contributions dealing with geometric aspects such as Chern forms, gerbes and generalized projective geometries (Paycha, Schweigert, Bertram). Finally, we had several exciting talks about several more particular results, dealing with vertex operator algebras, polyzeta values and quantization (Billig, Mathieu, Omori). More specific information is contained in the abstracts which follow in this volume.

From very-extended to overextended gravity and M-theories

The formulation of gravity and M-theories as very-extended Kac-Moody invariant theories encompasses, for each very-extended algebra G+++, two distinct actions invariant under the overextended Kac-Moody subalgebra G++. The first carries a Euclidean signature and is the generalisation to G++ of the E10-invariant action proposed in the context of M-theory and cosmological billiards. The second action carries various Lorentzian signatures revealed through various equivalent formulations related by Weyl transformations of fields. It admits exact solutions, identical to those of the maximally oxidised field theories and of their exotic counterparts, which describe intersecting extremal branes smeared in all directions but one. The Weyl transformations of G++ relates these solutions by conventional and exotic dualities. These exact solutions, common to the Kac-Moody theories and to space-time covariant theories, provide a laboratory for analysing the significance of the infinite set of fields appearing in the Kac-Moody formulations.

Kac–Moody theories for colored phase space (quantum Hall) droplets

We derive the canonical structure and Hamiltonian for arbitrary deformations of a higher-dimensional (quantum Hall) droplet of fermions with spin or color on a general phase space manifold. Gauge fields are introduced via a Kaluza–Klein construction on the phase space. The emerging theory is a nonlinear higher-dimensional generalization of the gauged Kac–Moody algebra. To leading order in ℏ this reproduces the edge state chiral Wess–Zumino–Witten action of the droplets.

Finite-size scaling in one-dimensional quantum liquid with long-range interaction.

Finite-size scaling analysis is performed for one-dimensional Bose and Fermi systems with long-range interaction g/${\mathit{r}}^{2}$, based on the exact solution of Sutherland. The low-energy behavior of this model is shown to be described by the c=1 conformal field theory. Exact formulas for various correlation exponents are obtained. At a special point g=4, the model belongs to the same universality class as the Haldane-Shastry model, the effective theory of which is the level-1 SU(2) Kac-Moody theory.

Gravitational coupling of affine field theories

The coupling of a N=4 affine gauge theory to N=1 supergravity is investigated. For the unbroken Kac-Moody theory the known sum rule for finite Lie multiplets is extended to cover the infinite dimensional Kac-Moody multiplet. For the broken Kac-Moody theory, the infinite Kac-Moody multiplet splits into an infinite set of finite multiplets representing a finite Lie symmetry * the U(1) generated by L0; in this case there is a separate sum rule for each Lie multiplet. The gravitationally coupled theory is more restricted but at the level of this investigation is not qualitatively different from the corresponding global theory.

A current algebra approach to the Kondo effect

The Kondo effect is studied using current algebra techniques which allow a separation of charge and spin degrees of freedom. An algebraic interpretation of the low-temperature fixed point is obtained. The Wilson ratio is shown to be the ratio of the specific heat for the total system to that of the spin sector which is expressed in terms of the conformal anomaly parameter of a Kac-Moody theory.

Fusion rules and (sub-)modular invariant partition functions in non-unitary theories

Fusion rules in non-unitary Kac-Moody and minimal conformal theories are obtained from their modular properties. New modular invariant partition functions in the non-unitary Kac-Moody theories are classified. From the discrete symmetries of the system we obtain new sub-modular invariant partition functions in the Kac-Moody and non-unitary conformal theories.