Product (mathematics)

About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.

Periodic Complexes and Group Actions

Abstract: In this paper we show that the cohomology of a connected CW complex is periodic if and only if it is the base space of an orientable spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on a cartesian product of euclidean space and a sphere; we construct non-standard free actions of rank two simple groups on finite complexes Y homotopy equivalent to a product of two spheres and we prove that a finite p-group P acts freely on such a complex if and only if it does not contain a subgroup isomorphic to Z/p X Z/p X Z/p.

Closed orbits for actions of maximal tori on homogeneous spaces

Abstract: Let $G$ be a real algebraic group defined over $\mathbb{Q}$, let $\Gamma$ be an arithmetic subgroup, and let $T$ be any torus containing a maximal $\mathbb{R}$-split torus. We prove that the closed orbits for the action of $T$ on $G/\Gamma$ admit a simple algebraic description. In particular, we show that if $G$ is reductive, an orbit $Tx\Gamma$ is closed if and only if $x^{-1}Tx$ is a product of a compact torus and a torus defined over $\mathbb{Q}$, and it is divergent if and only if the maximal $\mathbb{R}$-split subtorus of $x^{-1}Tx$ is defined over $\mathbb{Q}$ and $\mathbb{Q}$-split. Our analysis also yields the following: · there is a compact $K \subset G/\Gamma$ which intersects every $T$-orbit; · if $\rank_{\mathbb{Q}}G<\rank_{\mathbb{R}} G$, there are no divergent orbits for $T$.

Doubles of Quasi-Quantum Groups

Abstract: Drinfeld showed that any finite dimensional Hopf algebra \G extends to a quasitriangular Hopf algebra \D(\G), the quantum double of \G. Based on the construction of a so--called diagonal crossed product developed by the authors, we generalize this result to the case of quasi--Hopf algebras \G. As for ordinary Hopf algebras, as a vector space the ``quasi--quantum double'' \D(\G) is isomorphic to the tensor product of \G and its dual \dG. We give explicit formulas for the product, the coproduct, the R--matrix and the antipode on \D(\G) and prove that they fulfill Drinfeld's axioms of a quasitriangular quasi--Hopf algebra. In particular \D(\G) becomes an associative algebra containing \G as a quasi--Hopf subalgebra. On the other hand, \dG \otimes 1 is not a subalgebra of \D(\G) unless the coproduct on \G is strictly coassociative. It is shown that the category of finite dimensional representations of \D(\G) coincides with what has been called the double category of \G--modules by S. Majid [M2]. Thus our construction gives a concrete realization of Majid's abstract definition of quasi--quantum doubles in terms of a Tannaka--Krein--like reconstruction procedure. The whole construction is shown to generalize to weak quasi--Hopf algebras with \D(\G) now being linearly isomorphic to a subspace of \dG \otimes \G.

Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker’s first limit formula

Abstract: For a family of compact Riemann surfaces X t of genus g > 1, parameterized by the Schottky space $$\mathfrak{G}_{g} ,$$ we define a natural basis of $$H^{0}\left(X_{t},\omega_{X_{t}}^{n}\right)$$ which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n = 1. We introduce a holomorphic function F(n) on $$\mathfrak{G}_{g}$$ which generalizes the classical product $$\prod_{m=1}^{\infty}\left(1-q^{m}\right)^{2}$$ for n = 1 and g = 1. We prove the holomorphic factorization formula $$ \frac{{\det '\Delta _{n} }} {{\det N_{n} }} = c_{{g,n}} \exp {\left\{ { - \frac{{6n^{2} - 6n + 1}} {{12\pi }}S} \right\}}|F(n)|^{2} , $$ where det'Δ n is the zeta-function regularized determinant of the Laplace operator Δ n in the hyperbolic metric acting on n-differentials, N n is the Gram matrix of the natural basis with respect to the inner product given by the hyperbolic metric, S is the classical Liouville action –a Kahler potential of the Weil–Petersson metric on $$\mathfrak{G}_{g}$$ – and cg,n is a constant depending only on g and n. The factorization formula reduces to Kronecker’s first limit formula when n = 1 and g = 1, and to Zograf’s factorization formula for n = 1 and g > 1.

Ergodic Subequivalence Relations Induced by a Bernoulli Action

Abstract: Let Γ be a countable group and denote by \({\mathcal{S}}\) the equivalence relation induced by the Bernoulli action \({\Gamma\curvearrowright [0, 1]^{\Gamma}}\), where [0, 1]Γ is endowed with the product Lebesgue measure. We prove that, for any subequivalence relation \({\mathcal{R}}\) of \({\mathcal{S}}\), there exists a partition {Xi}i≥0 of [0, 1]Γ into \({\mathcal{R}}\)-invariant measurable sets such that \({\mathcal{R}_{\vert X_{0}}}\) is hyperfinite and \({\mathcal{R}_{\vert X_{i}}}\) is strongly ergodic (hence ergodic and non-hyperfinite), for every i ≥ 1.