Search

The 4-dimensional Sklyanin algebras, over ℂ., A(E, τ), are constructed from an elliptic curve E and a translation automorphism τ of E. The Klein vierergruppe Γ acts as graded algebra automorphisms of A(E, τ). There is also an action of Γ as automorphisms of the matrix algebra M2(ℂ.) making it isomorphic to the regular representation. The main object of study in this paper is the invariant subalgebra à := A(E, τ)⊗M2(ℂ))Γ. Like A(E, τ), à is noetherian, generated by 4 degree-one elements modulo six quadratic relations, Koszul, Artin-Schelter regular of global dimension 4, has the same Hilbert series as the polynomial ring on 4 variables, satisfies the x condition, and so on. These results are special cases of general results proved for a triple (A, T, H) consisting of a Hopf algebra H, an (often graded) H-comodule algebra A, and an H-torsor T. Those general results involve transferring properties between A, A ⊗ T, and (A ⊗ T)coH. We then investigate à from the point of view of non-commutative projective geometry. We examine its point modules, line modules, and a certain quotient ... := Ã/(Θ,Θ') where Θ and Θ' are homogeneous central elements of degree two. In doing this we show that à differs from A in interesting ways. For example, the point modules for A are parametrized by E and 4 more points, whereas à has exactly 20 point modules. Although ... is not a twisted homogeneous coordinate ring in the sense of Artin and Van den Bergh, a certain quotient of the category of graded ...-modules is equivalent to the category of quasi-coherent sheaves on the curve E/E[2] where E[2] is the 2-torsion subgroup. We construct line modules for à that are parametrized by the disjoint union (E/(ξ1)) ⊔ (E/(ξ2)) ⊔ (E/(ξ3)) of the quotients of E by its three subgroups of order 2. [ABSTRACT FROM AUTHOR]

Representing the conditional independences present in a multivariate random vector via graphs has found widespread use in applications, and such representations are popularly known as graphical models or Markov random fields. These models have many useful properties, but their fundamental attractive feature is their ability to reflect conditional independences between blocks of variables through graph separation, a consequence of the equivalence of the pairwise, local, and global Markov properties demonstrated by Pearl and Paz (1985). Modern-day applications often necessitate working with either an infinite collection of variables (such as in a spatial-temporal field) or approximating a large high-dimensional finite stochastic system with an infinite-dimensional system. However, it is unclear whether the conditional independences present in an infinite-dimensional random vector or stochastic process can still be represented by separation criteria in an infinite graph. In light of the advantages of using graphs as tools to represent stochastic relationships, we undertake in this paper a general study of infinite graphical models. First, we demonstrate that näıve extensions of the assumptions required for the finite case results do not yield equivalence of the Markov properties in the infinite-dimensional setting, thus calling for a more in-depth analysis. To this end, we proceed to derive general conditions which do allow representing the conditional independence in an infinite-dimensional random system by means of graphs, and our results render the result of Pearl and Paz as a special case of a more general phenomenon. We conclude by demonstrating the applicability of our theory through concrete examples of infinite-dimensional graphical models. [ABSTRACT FROM AUTHOR]

Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible G-action. We define and prove finiteness properties for analogues of Hilbert series, systems of parameters, depth, local cohomology, Koszul duality, and regularity. We also show that this category is built out of a simpler, more combinatorial, quiver category which we describe explicitly. Our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions. (For example, the paper by Church, Ellenberg and Farb on the category of FI-modules, which is equivalent to our category.) Along the way, we see several connections with the character polynomials from the representation theory of the symmetric groups. Several examples are given to illustrate that the invariants we introduce are explicit and computable. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the behavior of the unique solution uδ ∈ H01 (Ω), as δ → 0, to the equation div(sδA∇uδ) + k2s0Σuδ = s0 f in Ω, where Ω is a smooth connected bounded open subset of ℝd with d = 2 or 3, f ∈ L2(Ω), k is a non-negative constant, A is a uniformly elliptic matrixvalued function, Σ is a real function bounded above and below by positive constants, and sδ is a complex function whose real part takes the values 1 and −1 and whose imaginary part is positive and converges to 0 as δ goes to 0. This is motivated from a result of Nicorovici, McPhedran, and Milton; another motivation is the concept of complementary media. After introducing the reflecting complementary media, complementary media generated by reflections, we characterize f for which ||uδ|| H1(Ω) remains bounded as δ goes to 0. For such an f, we also show that uδ converges weakly in H1(Ω) and provide a formula to compute the limit. [ABSTRACT FROM AUTHOR]