Chiral-Yang-Mills theory, non commutative differential geometry, and the need for a Lie super-algebra

In Yang-Mills theory, the charges of the left and right massless Fermions are independent of each other. We propose a new paradigm where we remove this freedom and densify the algebraic structure of Yang-Mills theory by integrating the scalar Higgs field into a new gauge-chiral 1-form which connects Fermions of opposite chiralities. Using the Bianchi identity, we prove that the corresponding covariant differential is associative if and only if we gauge a Lie-Kac super-algebra. In this model, spontaneous symmetry breakdown naturally occurs along an odd generator of the super-algebra and induces a representation of the Connes-Lott non commutative differential geometry of the 2-point finite space.Comment: 17 pages, no figur

Algebraic structure of multi-parameter quantum groups

Multi-parameter versions U_p(g) and C_p[G] of the standard quantum groups U_q(g) and C_q[G] are considered where G is a semi-simple connected complex algebraic group and g is the Lie algebra of G. The primitive spectrum of C_p[G] is calculated, generalizing a result of Joseph for the standard quantum groups. This classification is compared with the classification of symplectic leaves for the associated Poisson structure on G.Comment: AMS Latex, 37 pages, June 1994; to appear in Advances in Mat

Entropy estimates for a class of schemes for the euler equations

In this paper, we derive entropy estimates for a class of schemes for the Euler equations which present the following features: they are based on the internal energy equation (eventually with a positive corrective term at the righ-hand-side so as to ensure consistency) and the possible upwinding is performed with respect to the material velocity only. The implicit-in-time first-order upwind scheme satisfies a local entropy inequality. A generalization of the convection term is then introduced, which allows to limit the scheme diffusion while ensuring a weaker property: the entropy inequality is satisfied up to a remainder term which is shown to tend to zero with the space and time steps, if the discrete solution is controlled in L $\infty$ and BV norms. The explicit upwind variant also satisfies such a weaker property, at the price of an estimate for the velocity which could be derived from the introduction of a new stabilization term in the momentum balance. Still for the explicit scheme, with the above-mentioned generalization of the convection operator, the same result only holds if the ratio of the time to the space step tends to zero