Pluriassociative algebras I: The pluriassociative operad

Diassociative algebras form a category of algebras recently introduced by Loday. A diassociative algebra is a vector space endowed with two associative binary operations satisfying some very natural relations. Any diassociative algebra is an algebra over the diassociative operad, and, among its most notable properties, this operad is the Koszul dual of the dendriform operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter γ of diassociative algebras, called γ-pluriassociative algebras, so that 1-pluriassociative algebras are diassociative algebras. Pluriassociative algebras are vector spaces endowed with 2γ associative binary operations satisfying some relations. We provide a complete study of the γ-pluriassociative operads, the underlying operads of the category of γ-pluriassociative algebras. We exhibit a realization of these operads, establish several presentations by generators and relations, compute their Hilbert series, show that they are Koszul, and construct the free objects in the corresponding categories. We also study several notions of units in γ-pluriassociative algebras and propose a general way to construct such algebras. This paper ends with the introduction of an analogous generalization of the triassociative operad of Loday and Ronco.     

Koszul Differential Graded Algebras and BGG Correspondence II

The concept of Koszul differential graded （DG for short） algebra is introduced in [8]. Let A be a Koszul DG algebra. If the Ext-algebra of A is finite-dimensional, i.e., the trivial module Ak is a compact object in the derived category of DG A-modules, then it is shown in [8] that A has many nice properties. However, if the Ext-algebra is infinite-dimensional, little is known about A. As shown in [15] （see also Proposition 2.2）, Ak is not compact if H（A） is finite-dimensional. In this paper, it is proved that the Koszul duality theorem also holds when H（A） is finite-dimensional by using Foxby duality. A DG version of the BGG correspondence is deduced from the Koszul duality theorem.     

Deformation theory of objects in homotopy and derived categories II: Pro-representability of the deformation functor

This is the second paper in a series. In part I we developed deformation theory of objects in homotopy and derived categories of DG categories. Here we extend these (derived) deformation functors to an appropriate bicategory of artinian DG algebras and prove that these extended functors are pro-representable in a strong sense.     

Nijenhuis geometry II: Left-symmetric algebras and linearization problem for Nijenhuis operators

A field of endomorphisms R is called a Nijenhuis operator if its Nijenhuis torsion vanishes. In this work we study a specific kind of singular points of R called points of scalar type. We show that the tangent space at such points possesses a natural structure of a left-symmetric algebra (also known as pre-Lie or Vinberg-Kozul algebras). Following Weinstein's approach to linearization of Poisson structures, we state the linearisation problem for Nijenhuis operators and give an answer in terms of non-degenerate left-symmetric algebras. In particular, in dimension 2, we give classification of non-degenerate left-symmetric algebras for the smooth category and, with some small gaps, for the analytic one. These two cases, analytic and smooth, differ. We also obtain a complete classification of two-dimensional real left-symmetric algebras, which may be an interesting result on its own.     

Multiplication operators defined by covering maps on the Bergman space: The connection between operator theory and von Neumann algebras

In this paper, we combine methods of complex analysis, operator theory and conformal geometry to construct a class of Type II factors in the theory of von Neumann algebras, which arise essentially from holomorphic coverings of bounded planar domains. One will see how types of such von Neumann algebras are related to algebraic topology of planar domains. As a result, the paper establishes a fascinating connections to one of the long-standing problems in free group factors. An interplay of analytical, geometrical, operator and group theoretical techniques is intrinsic to the paper.     

Quantum matrix algebras of the GL(m|n) type: The structure and spectral parameterization of the characteristic subalgebra

We continue the study of quantum matrix algebras of the GL(m|n) type. We find three alternative forms of the Cayley-Hamilton identity; most importantly, this identity can be represented in a factored form. The factorization allows naturally dividing the spectrum of a quantum supermatrix into subsets of “even” and “odd” eigenvalues. This division leads to a parameterization of the characteristic subalgebra (the subalgebra of spectral invariants) in terms of supersymmetric polynomials in the eigenvalues of the quantum matrix. Our construction is based on two auxiliary results, which are independently interesting. First, we derive the multiplication rule for Schur functions s _λ (M) that form a linear basis of the characteristic subalgebra of a Hecke-type quantum matrix algebra; the structure constants in this basis coincide with the Littlewood-Richardson coefficients. Second, we prove a number of bilinear relations in the graded ring Λ of symmetric functions of countably many variables. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 14–46, April, 2006.     

Matrix scaling,entropy minimization,and conjugate duality (II): The dual problem

We present a matrix scaling problem calledtruncated scaling and describe applications arising in economics, urban planning, and statistics. We associate a dual pair of convex optimization problems to the scaling problem and prove that the existence of a solution for the truncated scaling problem is characterized by the attainment of the infimum in the dual optimization problem. We show that optimization problems used by Bacharach (1970), Bachem and Korte (1979), Eaves et al. (1985), Marshall and Olkin (1968) and Rothblum and Schneider (1989) to study scaling problems can be derived as special cases of the dual problem for truncated scaling. We present computational results for solving truncated scaling problems using dual coordinate descent, thereby showing that truncated scaling provides a framework for modeling and solving large-scale matrix scaling problems.Research supported in part by NSF grants ECS 8718971 and ECS 8943458.