We give an explicit derivation of the Picard-Fuchs equations for N=2 supersymmetric SU(3) Yang-Mills theory with $N_f<6$ massive hypermultiplets in the fundamental representation. We determine the instanton corrections to the prepotential in the weak coupling region using the relation between $\tr<\phi^2>$ and the prepotential. This method can be generalized to other gauge groups.

We explicitly determine the instanton corrections to the prepotential for N = 2 supersymmetric SU(3) Yang-Mills theory with massless hypermultiplets in the weak coupling regions u → ∞ and u å ∞. We construct the Picard-Fuchs equations for Nf < 6 and calculate the monodromies using Picard-Lefschetz theorem for Nf = 2, 4. For all Nf < 6 the instanton corrections to the prepotential are determined using the relation between Tr(Φ2) and the prepotential.

We define quantum matrix groups GL(3) by their coaction on appropriate quantum planes and the requirement that the Poincare series coincides with the classical one. It is shown that this implies the existence of a Yang-Baxter operator. Exploiting stronger equations arising at degree four of the algebra, we classify all quantum matrix groups GL(3). We find 26 classes of solutions, two of which do not admit a normal ordering. The corresponding R-matrices are given.

We construct for all N a solution of the Frenkel Moore N-simplex equation which generalizes the R-matrix for the Jordanian quantum group.

Quantum matrices in two dimensions, admitting left and right quantum spaces, are classified: they fall into two families, the 2-parametric family GLp,q(2) and a 1-parametric family GL infα sup J(2). Phenomena previously found for GLp,q(2) hold in this general situation: (a) powers of quantum matrices are again quantum and (b) entries of the logarithm of a two-dimensional quantum matrix form a Lie algebra.