International Journal of Mathematics

We study the signature pair for certain group-invariant Hermitian polynomials arising in CR geometry. In particular, we determine the signature pair for the finite subgroups of SU(2). We introduce the asymptotic positivity ratio and compute it for cyclic subgroups of U(2). We calculate the signature pair for dihedral subgroups of U(2).

In this paper we shall find a matrix realization of the quantum group 𝔤_p,q introduced in [B. L. Feigin et al., Nucl. Phys. B757 (2006) 303–343]. For this purpose, we construct all primitive idempotents and a basis of 𝔤_p,q. We determine the action of elements of the basis on indecomposable projective modules, that gives rise to a matrix realization of 𝔤_p,q. By using this result, we obtain a basis of the space of symmetric linear functions on 𝔤_p1,p2 and express the symmetric linear functions obtained by the left integral, the balancing element, and the center of 𝔤_p,q in terms of this basis.

We show that the orbit space of Euclidean space, under the action of a closed and connected Lie group of isometries is homeomorphic to a plane or closed half-plane, if the action is of cohomogeneity two.

Let G be a group and let A be the algebra of complex functions on G with finite support. The product in G gives rise to a coproduct Δ on A making the pair (A,Δ) a multiplier Hopf algebra. In fact, because there exist integrals, we get an algebraic quantum group as introduced and studied in [A. Van Daele, Adv. Math.140 (1998) 323].

Now let H be a finite subgroup of G and consider the subalgebra A₁ of functions in A that are constant on double cosets of H. The coproduct in general will not leave this algebra invariant but we can modify Δ and define Δ₁ as

Group-like projections in a *-algebraic quantum group A (as defined and studied in [M. B. Landstad and A. Van Daele, arXiv:math.OA/0702458v1]) give rise, in a natural way, to *-algebraic quantum hypergroups, very much like subgroups do as above for a *-algebraic quantum group associated to a group (again see [M. B. Landstad and A. Van Daele, arXiv:math.OA/0702458v1]). In this paper we push these results further. On the one hand, we no longer assume the *-structure as in [M. B. Landstad and A. Van Daele, arXiv:math.OA/0702458v1] while on the other hand, we allow the group-like projection to belong to the multiplier algebra M(A) of A and not only to A itself. Doing so, we not only get some well-known earlier examples of algebraic quantum hypergroups but also some interesting new ones.

We consider a polynomial map from ℂⁿ to ℂ^n-1 and prove that if there exists a so-called very good projection with respect to the value t⁰, then this value is an atypical value for the map if and only if the Euler characteristic of the fibers are not constant. We describe some topology of the fibers and prove that there is no extension of the characterization of the atypical value via the Lojasiewicz number as in the case n = 2.

We apply the general theory for symplectic torus actions with symplectic or coisotropic orbits to prove that a four-manifold with a symplectic two-torus action admits an invariant complex structure and give an identification of those that do not admit a Kähler structure with Kodaira's class of complex surfaces which admit a nowhere vanishing holomorphic (2,0)-form, but are not a torus nor a K3 surface.