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Register a new userRiemann surfaces and AF-algebras
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Authors
Igor Nikolaev
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For a generic set in the Teichmueller space, we construct a covariant functor with the range in a category of the AF-algebras; the functor maps isomorphic Riemann surfaces to the stably isomorphic AF-algebras. As a special case, one gets a categorical correspondence between complex tori and the so-called Effros-Shen algebras.
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Let $A$ be a unital AF-algebra whose Murray-von Neumann order of projections is a lattice. For any two equivalence classes $[p]$ and $[q]$ of projections we write $[p]sqsubseteq [q]$ iff for every primitive ideal $mathfrak p$ of $A$ either $p/mathfrak ppreceq q/mathfrak ppreceq (1- q)/mathfrak p$ or $p/mathfrak psucceq q/mathfrak p succeq (1-q)/mathfrak p.$ We prove that $p$ is central iff $[p]$ is $sqsubseteq$-minimal iff $[p]$ is a characteristic element in $K_0(A)$. If, in addition, $A$ is liminary, then each extremal state of $K_0(A)$ is discrete, $K_0(A)$ has general comparability, and $A$ comes equipped with a centripetal transformation $[p]mapsto [p]^Game$ that moves $p$ towards the center. The number $n(p) $ of $Game$-steps needed by $[p]$ to reach the center has the monotonicity property $[p]sqsubseteq [q]Rightarrow n(p)leq n(q).$ Our proofs combine the $K_0$-theoretic version of Elliotts classification, the categorical equivalence $Gamma$ between MV-algebras and unital $ell$-groups, and L os ultraproduct theorem for first-order logic.
A purely numerical approach to compact Riemann surfaces starting from plane algebraic curves is presented. The critical points of the algebraic curve are computed via a two-dimensional Newton iteration. The starting values for this iteration are obtained from the resultants with respect to both coordinates of the algebraic curve and a suitable pairing of their zeros. A set of generators of the fundamental group for the complement of these critical points in the complex plane is constructed from circles around these points and connecting lines obtained from a minimal spanning tree. The monodromies are computed by solving the defining equation of the algebraic curve on collocation points along these contours and by analytically continuing the roots. The collocation points are chosen to correspond to Chebychev collocation points for an ensuing Clenshaw-Curtis integration of the holomorphic differentials which gives the periods of the Riemann surface with spectral accuracy. At the singularities of the algebraic curve, Puiseux expansions computed by contour integration on the circles around the singularities are used to identify the holomorphic differentials. The Abel map is also computed with the Clenshaw-Curtis algorithm and contour integrals. As an application of the code, solutions to the Kadomtsev-Petviashvili equation are computed on non-hyperelliptic Riemann surfaces.
We introduce and study (strict) Schottky G-bundles over a compact Riemann surface X, where G is a connected reductive algebraic group. Strict Schottky representations are shown to be related to branes in the moduli space of G-Higgs bundles over X, and we prove that all Schottky $G$-bundles have trivial topological type. Generalizing the Schottky moduli map introduced in Florentino to the setting of principal bundles, we prove its local surjectivity at the good and unitary locus. Finally, we prove that the Schottky map is surjective onto the space of flat bundles for two special classes: when G is an abelian group over an arbitrary X, and the case of a general G-bundle over an elliptic curve.
Let $X$ be a compact metric space and let $Lambda$ be a $Z^k$ ($kge 1$) action on $X.$ We give a solution to a version of Voiculescus problem of AF-embedding: The crossed product $C(X)rtimes_{Lambda}Z^k$ can be embedded into a unital simple AF-algebra if and only if $X$ admits a strictly positive $Lambda$-invariant Borel probability measure. Let $C$ be a unital AH-algebra, let $G$ be a finitely generated abelian group and let $Lambda: Gto Aut(C)$ be a monomorphism. We show that $Crtimes_{Lambda} G$ can be embedded into a unital simple AF-algebra if and only if $C$ admits a faithful $Lambda$-invariant tracial state.
An algebra is said to be hopfian if it is not isomorphic to a proper quotient of itself. We describe several classes of hopfian and of non-hopfian unital lattice-ordered abelian groups and MV-algebras. Using Elliott classification and $K_0$-theory, we apply our results to other related structures, notably the Farey-Stern-Brocot AF C$^*$-algebra and all its primitive quotients, including the Behnke-Leptin C$^*$-algebras $mathcal A_{k,q}$.
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