We consider the automorphic forms which govern the gravitational threshold correction F1 in models of heterotic/IIA duality with N = 2 supersymmetry in four dimensions. In particular we derive the full nonperturbative formula for F1 for the dual pair originally considered by Ferrara, Harvey, Strominger and Vafa (FHSV). The answer involves an interesting automorphic product constructed by Borche...

We consider Nambu-Poisson 3-algebras on three dimensional manifolds M3, such as the Euclidean 3-space R, the 3-sphere S as well as the 3-torus T . We demonstrate that in the Clebsch-Monge gauge, the Lie algebra of volume preserving diffeomorphisms SDiff(M3) is identical to the Nambu-Poisson algebra on M3. Moreover the fundamental identity for the Nambu 3-bracket is just the commutation relation...

In this topic proposal, we discuss the representation theory of semisimple Lie algebras, including results on the structure of these algebras and their modules. In particular, the theory of root systems and Verma modules are very important in this context. We classify all simple Lie algebras, and obtain a presentation for any semisimple Lie algebra. We also encounter several striking formulas t...

There are well-known constructions of the exceptional simple Lie algebras of type E8 and F4 which go back to Witt [Wit41], as Z2-graded algebras g = g0̄⊕g1̄ with even part the orthogonal Lie algebras so16 and so9 respectively, and odd part given by their spin representations (see [Ada96]). Brown [Bro82] found a new simple finite dimensional Lie algebra over fields of characteristic 3 which presen...

(identical representation). The even part G0 is characterized by the condition a0i = ai0 = 0 for i = 1, . . . , n. G0 is isomorphic to gl(n). We choose the Cartan subalgebra H as the set of diagonal matrices of G and G0 resp. Let eij be denote the (n+ 1)× (n+ 1)-matrix with 1 at the place (i, j) and 0 everywhere (i, j = 0, . . . , n) . Then we get by h1 = e00 + e11 , hi = ei−1,i−1 − eii (i = 2,...

The dimensional reduction of heterotic supergravity with gauge fields truncated to the Cartan subalgebra exhibits a continuous O(d, d+ 16;R) global symmetry, related to the O(d, d+16;Z) T-duality of heterotic strings on a d-torus. The O(d, d+16;R) symmetry is not present, however, if the supergravity reduction is done including the full set of E8×E8 or SO(32) gauge fields. We analyze which dual...

Five equivalence classes had been found for systems of two second-order ordinary differential equations, transformable to linear equations linearizable systems by a change of variables. An “optimal or simplest canonical form” of linear systems had been established to obtain the symmetry structure, namely, with 5-, 6-, 7-, 8-, and 15-dimensional Lie algebras. For those systems that arise from a ...

Preface These are the notes of a course on Lie algebras which I gave at the university of Bordeaux in spring 1997. The course was a so-called \Cours PostDEA", and as such had to be held within 12 hours. Even more challenging, no previous knowledge about Lie algebras should be assumed. Nevertheless, I had the goal to reach as peak of the course the character formula for Kac-Moody algebras, and, ...

The notion of multilinear quantum Lie operation appears naturally in connection with a different attempts to generalize the Lie algebras. There are a number of reasons why the generalizations are necessary. First of all this is the demands for a ”quantum algebra” which was formed in the papers by Ju.I. Manin, V.G. Drinfeld, S.L. Woronowicz, G. Lusztig, L.D. Faddeev, and many others. These deman...