Logarithmic Tensor Category Theory for Generalized Modules for a Conformal Vertex Algebra, I: Introduction and Strongly Graded Algebras and Their Generalized Modules

We construct three families of vertex algebras along with their modules from appropriate vertex Lie algebras, using the constructions in [Vertex Lie algebra, vertex Poisson algebras and vertex algebras, in Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory[Formula: see text] Proceedings of an International Conference at University of Virginia[Formula: see text] May 2000, in Contemporary Mathematics, Vol. 297 (American Mathematical Society, 2002), pp. 69–96] by Dong, Li and Mason. These vertex algebras are strongly graded vertex algebras introduced in [Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules, in Conformal Field Theories and Tensor Categories[Formula: see text] Proceedings of a Workshop Held at Beijing International Center for Mathematics Research, eds. C. Bai, J. Fuchs, Y.-Z. Huang, L. Kong, I. Runkel and C. Schweigert, Mathematical Lectures from Beijing University, Vol. 2 (Springer, New York, 2014), pp. 169–248] by Huang, Lepowsky and Zhang in their logarithmic tensor category theory and can also be realized as vertex algebras associated to certain well-known infinite dimensional Lie algebras. We classify irreducible [Formula: see text]-gradable weak modules for these vertex algebras by determining their Zhu’s algebras. We find examples of strongly graded generalized modules for these vertex algebras that satisfy the [Formula: see text]-cofiniteness condition introduced in [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009] by the second author. In particular, by a result of the second author [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009, 26 pp.], the convergence and extension property for products and iterates of logarithmic intertwining operators in [Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, VII: Convergence and extension properties and applications to expansion for intertwining maps, preprint (2011); arXiv:1110.1929 ] among such strongly graded generalized modules is verified.

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Differential equations and logarithmic intertwining operators for strongly graded vertex algebras

Communications in Contemporary Mathematics ◽

10.1142/s0219199716500097 ◽

2017 ◽

Vol 19 (02) ◽

pp. 1650009 ◽

Cited By ~ 2

Author(s):

Jinwei Yang

Keyword(s):

Differential Equations ◽

Category Theory ◽

Tensor Category ◽

Extension Property ◽

Vertex Algebra ◽

Intertwining Operators ◽

Matrix Elements ◽

Systems Of Differential Equations ◽

Convergence And Extension ◽

Logarithmic Intertwining Operators

We derive certain systems of differential equations for matrix elements of products and iterates of logarithmic intertwining operators among strongly graded generalized modules for a strongly graded vertex algebra under a certain finiteness condition and a condition related to the horizontal gradings. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory for such strongly graded generalized modules developed by Huang, Lepowsky and Zhang.

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Graded Medial n-Ary Algebras and Polyadic Tensor Categories

Symmetry ◽

10.3390/sym13061038 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1038

Author(s):

Steven Duplij

Keyword(s):

Tensor Product ◽

Tensor Category ◽

Baxter Equation ◽

Algebraic Structures ◽

Monoidal Categories ◽

Graded Algebras ◽

Tensor Categories ◽

Lie Brackets

Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or ε-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with n−1 associators of the arity 2n−1 satisfying a n2+1-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.

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On the representation theory of the vertex algebra L−5/2(sl(4))

Communications in Contemporary Mathematics ◽

10.1142/s0219199721501042 ◽

2021 ◽

Author(s):

Dražen Adamović ◽

Ozren Perše ◽

Ivana Vukorepa

Keyword(s):

Representation Theory ◽

Tensor Category ◽

Vertex Algebra ◽

Hamiltonian Reduction ◽

Conformal Weight ◽

Conformal Embedding ◽

Fusion Algebra ◽

Minimal Reduction ◽

Simple Quotient ◽

Conformal Embeddings

We study the representation theory of non-admissible simple affine vertex algebra [Formula: see text]. We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra [Formula: see text], and show that it generates the maximal ideal in [Formula: see text]. We classify irreducible [Formula: see text]-modules in the category [Formula: see text], and determine the fusion rules between irreducible modules in the category of ordinary modules [Formula: see text]. It turns out that this fusion algebra is isomorphic to the fusion algebra of [Formula: see text]. We also prove that [Formula: see text] is a semi-simple, rigid braided tensor category. In our proofs, we use the notion of collapsing level for the affine [Formula: see text]-algebra, and the properties of conformal embedding [Formula: see text] at level [Formula: see text] from D. Adamovic et al. [Finite vs infinite decompositions in conformal embeddings, Comm. Math. Phys. 348 (2016) 445–473.]. We show that [Formula: see text] is a collapsing level with respect to the subregular nilpotent element [Formula: see text], meaning that the simple quotient of the affine [Formula: see text]-algebra [Formula: see text] is isomorphic to the Heisenberg vertex algebra [Formula: see text]. We prove certain results on vanishing and non-vanishing of cohomology for the quantum Hamiltonian reduction functor [Formula: see text]. It turns out that the properties of [Formula: see text] are more subtle than in the case of minimal reduction.

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Research brief outlines findings examining the impact on students who took online Algebra I

PsycEXTRA Dataset ◽

10.1037/e534972013-001 ◽

2012 ◽

Keyword(s):

Algebra I ◽

The Impact

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不定自然変換理論に基づく比喩理解モデルの計算論的実装の試み

10.31234/osf.io/bpktw ◽

2020 ◽

Author(s):

Shunsuke Ikeda ◽

Miho Fuyama ◽

Hayato Saigo ◽

Tatsuji Takahashi

Keyword(s):

Category Theory ◽

Analogical Reasoning ◽

Machine Learning Techniques ◽

Huge Amount ◽

Metaphor Comprehension ◽

Weighted Directed Graph ◽

Learning Techniques ◽

Descriptive Validity ◽

Association Data ◽

Human Participants

Machine learning techniques have realized some principal cognitive functionalities such as nonlinear generalization and causal model construction, as far as huge amount of data are available. A next frontier for cognitive modelling would be the ability of humans to transfer past knowledge to novel, ongoing experience, making analogies from the known to the unknown. Novel metaphor comprehension may be considered as an example of such transfer learning and analogical reasoning that can be empirically tested in a relatively straightforward way. Based on some concepts inherent in category theory, we implement a model of metaphor comprehension called the theory of indeterminate natural transformation (TINT), and test its descriptive validity of humans' metaphor comprehension. We simulate metaphor comprehension with two models: one being structure-ignoring, and the other being structure-respecting. The former is a sub-TINT model, while the latter is the minimal-TINT model. As the required input to the TINT models, we gathered the association data from human participants to construct the ``latent category'' for TINT, which is a complete weighted directed graph. To test the validity of metaphor comprehension by the TINT models, we conducted an experiment that examines how humans comprehend a metaphor. While the sub-TINT does not show any significant correlation, the minimal-TINT shows significant correlations with the human data. It suggests that we can capture metaphor comprehension processes in a quite bottom-up manner realized by TINT.

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Category Theory and Foundations

10.1093/oso/9780198748991.003.0005 ◽

2018 ◽

Author(s):

Michael Ernst

Keyword(s):

Category Theory ◽

Mathematical Thinking ◽

Mathematical Practice ◽

Foundations Of Mathematics ◽

Ongoing Debate ◽

Technical Adequacy ◽

Theoretical Foundations

In the foundations of mathematics there has been an ongoing debate about whether categorical foundations can replace set-theoretical foundations. The primary goal of this chapter is to provide a condensed summary of that debate. It addresses the two primary points of contention: technical adequacy and autonomy. Finally, it calls attention to a neglected feature of the debate, the claim that categorical foundations are more natural and readily useable, and how deeper investigation of that claim could prove fruitful for our understanding of mathematical thinking and mathematical practice.

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Enriched Meanings

10.1093/oso/9780198847854.001.0001 ◽

2020 ◽

Author(s):

Ash Asudeh ◽

Gianluca Giorgolo

Keyword(s):

Cognitive Science ◽

Programming Languages ◽

Category Theory ◽

Worst Case ◽

Disciplinary Boundaries ◽

Semantics Of Programming Languages ◽

Basic Meaning ◽

Language Interpretation ◽

Major Branch ◽

Compositional Properties

This book presents a theory of enriched meanings for natural language interpretation. Certain expressions that exhibit complex effects at the semantics/pragmatics boundary live in an enriched meaning space while others live in a more basic meaning space. These basic meanings are mapped to enriched meanings just when required compositionally, which avoids generalizing meanings to the worst case. The theory is captured formally using monads, a concept from category theory. Monads are also prominent in functional programming and have been successfully used in the semantics of programming languages to characterize certain classes of computation. They are used here to model certain challenging linguistic computations at the semantics/pragmatics boundary. Part I presents some background on the semantics/pragmatics boundary, informally presents the theory of enriched meanings, reviews the linguistic phenomena of interest, and provides the necessary background on category theory and monads. Part II provides novel compositional analyses of the following phenomena: conventional implicature, substitution puzzles, and conjunction fallacies. Part III explores the prospects of combining monads, with particular reference to these three cases. The authors show that the compositional properties of monads model linguistic intuitions about these cases particularly well. The book is an interdisciplinary contribution to Cognitive Science: These phenomena cross not just the boundary between semantics and pragmatics, but also disciplinary boundaries between Linguistics, Philosophy and Psychology, three of the major branches of Cognitive Science, and are here analyzed with techniques that are prominent in Computer Science, a fourth major branch. A number of exercises are provided to aid understanding, as well as a set of computational tools (available at the book's website), which also allow readers to develop their own analyses of enriched meanings.

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