scholarly journals HIGHER KOSZUL DUALITY FOR ASSOCIATIVE ALGEBRAS

2013 ◽  
Vol 55 (A) ◽  
pp. 55-74 ◽  
Author(s):  
VLADIMIR DOTSENKO ◽  
BRUNO VALLETTE
Keyword(s):  
Algebraic Structure ◽  
Duality Theory ◽  
Koszul Complex ◽  
General Notion ◽  
Koszul Duality ◽  
Associative Algebras ◽  
Yoneda Algebra ◽  
Dual Algebra ◽  
Key Concepts ◽  
Homogeneous Algebras

AbstractWe present a unifying framework for the key concepts and results of higher Koszul duality theory for N-homogeneous algebras: the Koszul complex, the candidate for the space of syzygies and the higher operations on the Yoneda algebra. We give a universal description of the Koszul dual algebra under a new algebraic structure. For that we introduce a general notion: Gröbner bases for algebras over non-symmetric operads.

scholarly journals THE HOMOGENISED ENVELOPING ALGEBRA OF THE LIE ALGEBRA sℓ(2,ℂ)

2014 ◽  
Vol 56 (3) ◽  
pp. 551-568 ◽  
Author(s):  
ROBERTO MARTINEZ-VILLA
Keyword(s):  
Lie Algebra ◽  
Projective Dimension ◽  
Global Dimension ◽  
Verma Modules ◽  
Regular Component ◽  
Koszul Duality ◽  
Koszul Algebra ◽  
Yoneda Algebra ◽  
Enveloping Algebra ◽  
Wild Algebra

AbstractIn this paper, we study the homogenised algebra B of the enveloping algebra U of the Lie algebra sℓ(2,ℂ). We look first to connections between the category of graded left B-modules and the category of U-modules, then we prove B is Koszul and Artin–Schelter regular of global dimension four, hence its Yoneda algebra B! is self-injective of radical five zeros, and the structure of B! is given. We describe next the category of homogenised Verma modules, which correspond to the lifting to B of the usual Verma modules over U, and prove that such modules are Koszul of projective dimension two. It was proved in Martínez-Villa and Zacharia (Approximations with modules having linear resolutions, J. Algebra266(2) (2003), 671–697)] that all graded stable components of a self-injective Koszul algebra are of type ZA∞. Here, we characterise the graded B!-modules corresponding to the Koszul duality to homogenised Verma modules, and prove that these are located at the mouth of a regular component. In this way we obtain a family of components over a wild algebra indexed by ℂ.

scholarly journals The Face Semigroup Algebra of a Hyperplane Arrangement

2009 ◽  
Vol 61 (4) ◽  
pp. 904-929 ◽  
Author(s):  
Franco V. Saliola
Keyword(s):  
Algebraic Structure ◽  
Hyperplane Arrangement ◽  
Semigroup Algebra ◽  
Cohomology Algebra ◽  
Koszul Algebra ◽  
Dual Algebra ◽  
Intersection Lattice ◽  
Projective Resolutions ◽  
The Face ◽  
Cartan Invariants

Abstract.This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on the intersection lattice of the hyperplane arrangement. A complete systemof primitive orthogonal idempotents for the algebra is constructed and other algebraic structure is determined including: a description of the projective indecomposablemodules, the Cartan invariants, projective resolutions of the simple modules, the Hochschild homology and cohomology, and the Koszul dual algebra. A new cohomology construction on posets is introduced, and it is shown that the face semigroup algebra is isomorphic to the cohomology algebra when this construction is applied to the intersection lattice of the hyperplane arrangement.

scholarly journals Matrix factorizations via Koszul duality

2014 ◽  
Vol 150 (9) ◽  
pp. 1549-1578 ◽  
Author(s):  
Junwu Tu
Keyword(s):  
Duality Theory ◽  
Hochschild Homology ◽  
Matrix Factorizations ◽  
Koszul Duality ◽  
Technical Tool ◽  
Graded Algebras ◽  
Differential Graded Algebras ◽  
Homological Perturbation ◽  
The Relationship

AbstractIn this paper we prove a version of curved Koszul duality for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Z}/2\mathbb{Z}$-graded curved coalgebras and their cobar differential graded algebras. A curved version of the homological perturbation lemma is also obtained as a useful technical tool for studying curved (co)algebras and precomplexes. The results of Koszul duality can be applied to study the category of matrix factorizations $\mathsf{MF}(R,W)$. We show how Dyckerhoff’s generating results fit into the framework of curved Koszul duality theory. This enables us to clarify the relationship between the Borel–Moore Hochschild homology of curved (co)algebras and the ordinary Hochschild homology of the category $\mathsf{MF}(R,W)$. Similar results are also obtained in the orbifold case and in the graded case.

scholarly journals Some Remarks on the Application of Bayesian Analysis in Law

2015 ◽  
pp. 31-40
Author(s):  
Bartosz Janik ◽  
Paweł Banaś
Keyword(s):  
Bayesian Analysis ◽  
General Notion ◽  
Probabilistic Framework ◽  
Criminal Cases ◽  
Legal Proceedings ◽  
Key Concepts

This paper discusses the use of Bayesian analysis in law. It introduces the key concepts of Bayesian analysis by giving some common examples of criminal cases. It focuses on the advantages of Bayesian analysis over some other probability interpretations (mainly the frequentist one). The last part of the text discusses the general notion of truth in legal proceedings and its possible interpretations within the probabilistic framework – given the Bayesian subjectivists-objectivists discussion.

scholarly journals Monops, Monoids and Operads: The Combinatorics of Sheffer Polynomials

10.37236/7686 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Miguel A. Méndez ◽  
Rafael Sánchez Lamoneda
Keyword(s):  
Algebraic Structure ◽  
Partially Ordered Sets ◽  
Koszul Duality ◽  
Ordered Sets ◽  
Algebraic Construction ◽  
Partially Ordered ◽  
Whitney Numbers ◽  
The Family ◽  
Sheffer Sequences ◽  
Sheffer Polynomials

We introduce a new algebraic construction, monop, that combines monoids (with respect to the product of species), and operads (monoids with respect to the substitution of species) in the same algebraic structure. By the use of properties of cancellative set-monops we construct a family of partially ordered sets whose prototypical examples are the Dowling lattices. They generalize the enriched partition posets associated to a cancellative operad, and the subset posets associated to a cancellative monoid. Their Whitney numbers of the first and second kind are the connecting coefficients of two umbral inverse Sheffer sequences with the family of powers $\{x^n\}_{n=0}^{\infty}$. Equivalently, the entries of a Riordan matrix and its inverse. This aticle is the first part of a program in progress to develop a theory of Koszul duality for monops.

scholarly journals Curved Koszul duality theory

2012 ◽  
Vol 354 (4) ◽  
pp. 1465-1520 ◽  
Author(s):  
Joseph Hirsh ◽  
Joan Millès
Keyword(s):  
Duality Theory ◽  
Koszul Duality

scholarly journals OPERADIC APPROACH TO COHOMOLOGY OF ASSOCIATIVE TRIPLE AND N-TUPLE SYSTEMS

2019 ◽  
Vol 62 (S1) ◽  
pp. S128-S141 ◽  
Author(s):  
FATEMEH BAGHERZADEH ◽  
MURRAY BREMNER
Keyword(s):  
Associative Algebra ◽  
General Setting ◽  
Koszul Duality ◽  
Associative Algebras ◽  
Explicit Definition ◽  
Basic Properties ◽  
Cup Product ◽  
Cohomology Of Algebras ◽  
Odd Degree ◽  
Definition Of

AbstractThe cup product in the cohomology of algebras over quadratic operads has been studied in the general setting of Koszul duality for operads. We study the cup product on the cohomology of n-ary totally associative algebras with an operation of even (homological) degree. This cup product endows the cohomology with the structure of an n-ary partially associative algebra with an operation of even or odd degree depending on the parity of n. In the cases n=3 and n=4, we provide an explicit definition of this cup product and prove its basic properties.

scholarly journals On Jordan Algebras and Some Unification Results

2020 ◽  
Author(s):  
Florin Nichita
Keyword(s):  
Differential Geometry ◽  
Lie Algebras ◽  
Algebraic Structure ◽  
International Workshop ◽  
Jordan Algebras ◽  
Baxter Equation ◽  
Associative Algebras ◽  
Associative Structures

This paper is based on a talk given at the 14-th International Workshop on Differential Geometry and Its Applications, hosted by the Petroleum Gas University from Ploiesti, between July 9-th and July 11-th, 2019. After presenting some historical facts, we will consider some geometry problems related to unification approaches. Jordan algebras and Lie algebras are the main non-associative structures. Attempts to unify non-associative algebras and associative algebras led to UJLA structures. Another algebraic structure which unifies non-associative algebras and associative algebras is the Yang-Baxter equation. We will review topics relared to the Yang-Baxter equation and Yang-Baxter systems, with the goal to unify constructions from Differential Geometry.

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