We prove an SL2(Z)-invariance property of multivariable trace functions on modules for a regular VOA. Applying this result, we provide a proof of the inversion transformation formula for Siegel theta series. As another application, we show that if V is a simple regular VOA containing a simple regular subVOA U whose commutant U-c is simple, regular, and satisfies (U-c)(c) = U, then all simple U-modules appear in some simple V-module. (C) 2015 Elsevier Inc. All rights reserved.

In this manuscript, we study the following question raised by Mel Hochster: Let (R, m, K) be a local ring and S be a flat extension with regular closed fiber. Is V(mS) boolean AND Ass(s) H-l(i)(S) finite for every ideal I subset of S and i is an element of N? We prove that the answer is positive when S is either a polynomial or a power series ring over R and dim(R/I boolean AND R) <= 1. In addition, we analyze when this question can be reduced to the case where S is a power series ring over R. An important tool for our proof is the use of Sigma-finite D-modules, which are not necessarily finitely generated as D-modules, but whose associated primes are finite. We give examples of this class of D-modules and applications to local cohomology. (C) 2013 Elsevier Inc. All rights reserved.

In the first part of this paper we study scrollers and linearly Joined varieties Scrollers were introduced in Barile and Morales (2004) [BM4] linearly joined varieties are an extension of scrollers and were defined in Eisenbud et al (2005) [EGHP] there they proved that scrollers are defined by homogeneous ideals having a 2-linear resolution A particular class of varieties of important interest in classical Geometry are Cohen-Macaulay varieties of minimal degree they were classified geometrically by the successive contribution of Del Pezzo (1885) [DP] Bertini (1907) [B] and Xambo (1981) [X] and algebraically in Barile and Morales (2000) [BM2] They appear naturally studying the fiber cone of a codimension two toric Ideals Morales (1995) [M] Gimenez et al (1993 1999) [GMS1 GMS2] Barile and Morales (1998) [BM-1] Ha (2006) [H] Ha and Morales (2009) [HM] Let S be a polynomial ring and I subset of S a homogeneous ideal defining a sequence of linearly joined varieties We compute depth S/I. We prove that c(V) = depth S/I where c(V) is the connectedness dimension of the algebraic set defined by I We characterize sets of generators of I and give an effective algorithm to find equations as an application we prove that ara(I) = projdim(S/I) in the case where V is a union of linear spaces in particular this applies to any square free monomial ideal having a 2 linear resolution In the case where V is a union of linear spaces the ideal I can be characterized by a tableau which is an extension of a Ferrer (or Young) tableau All these results are new and extend results in Barile and Morales (2004) [BM4] Eisenbud et al (2005) [EGHP] (C) 2010 Elsevier Inc All rights reserved

Let P be a Poisson algebra, E a vector space and pi : E -> P an epimorphism of vector spaces with V = Ker(pi). The global extension problem asks for the classification of all Poisson algebra structures that can be defined on E such that pi : E -> P becomes a morphism of Poisson algebras. From a geometrical point of view it means to decompose this groupoid into connected components and to indicate a point in each such component. All such Poisson algebra structures on E are classified by an explicitly constructed classifying set gPH(2)(P, V) which is the coproduct of all non-abelian cohomological objects PH2 (P, (V, .v,[-, -]v)) which are the classifying sets for all extensions of P by (V, .v,[-, -]v). The second classical Poisson cohomology group H-2(P, V) appears as the most elementary piece among all components of gPH(2)(P, V). Several examples are provided in the case of metabelian Poisson algebras or co-flag Poisson algebras over P: the latter being Poisson algebras Q which admit finite chain of epimorphisms of Poisson algebras P-n := Q (pi n)-> Pn-1 . . . P-1 pi(1) -> P-0 := P such that dim(Ker(pi(i))) = 1, for all i = 1, . . . , n. (C) 2014 Elsevier Inc. All rights reserved.

The conformal transformations with respect to the metric defining o(n, C) give rise to an inhomogeneous polynomial representation of o(n + 2, C). Using Shen's technique of mixed product, we generalize the above representation to an inhomogeneous representation of o(n + 2, C) on the tensor space of any finite-dimensional irreducible o(n, C)-module with the polynomial space, where a hidden central transformation is involved. Moreover, we find a condition on the constant value taken by the central transformation such that the generalized representation is irreducible. In our approach, Pieri's formulas, invariant operators and the idea of Kostant's characteristic identities play key roles. The result could be useful in understanding higher-dimensional conformal field theory with the constant value taken by the central transformation as the central charge. Our representations virtually provide natural extensions of the conformal transformations on a Riemannian manifold to its vector bundles. (c) 2012 Elsevier Inc. All rights reserved.

For each separated graph (E, C) we construct a family of branching systems over a set X and show how each branching system induces a representation of the Cohn-Leavitt path algebra associated with (E, C) as homomorphisms over the module of functions in X. We also prove that the abelianized Cohn-Leavitt path algebra of a separated graph with no loops can be written as an amalgamated free product of abelianized Cohn-Leavitt algebras that can be faithfully represented via branching systems. (C) 2014 Elsevier Inc. All rights reserved.