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Normed modules, integral sequences, and integrals with variable upper limits

Miantao Liu, Yu-Zhe Liu, Shengda Liu

TL;DR

The article develops a categorification framework for the Lebesgue integral with variable upper limits using normed modules over finite-dimensional algebras and the A^p_Lambda category, culminating in a construction of integral partial ordered sets and a pivotal pmb Game morphism. It proves that the induced images form Lambda-modules and that the pmb Game^natural restricted to Im(+) yields a Lambda-epimorphism, enabling an addition law for integrals with variable upper limits. The authors present two key applications: a categorification of basic elementary functions including (anti-)trigonometric and logarithmic/exponential functions, and a novel representation-theoretic approach to the global dimensions of gentle algebras via multiple integrals and Lebesgue-Stieltjes measures. The work connects category theory, functional analysis, and representation theory to yield new insights into categorified integrals, with potential implications for differential/integral equations and algebraic invariants of gentle algebras.

Abstract

This paper provides a new categorification of the Lebesgue integral with variable upper limits by using normed modules over finite-dimensional $\Bbbk$-algebras $\mathitΛ$ and the category $\mathscr{A}^p_{\mathitΛ}$ associated with $\mathitΛ$. The integration process is redefined through the introduction of an integral partially ordered set and an abstract integral with variable upper limits. Finally, we present two important applications: (1) the categorification of basic elementary functions, including (anti-)trigonometric and logarithmic functions, and (2) a new approach for characterizing the global dimensions of gentle algebras.

Normed modules, integral sequences, and integrals with variable upper limits

TL;DR

The article develops a categorification framework for the Lebesgue integral with variable upper limits using normed modules over finite-dimensional algebras and the A^p_Lambda category, culminating in a construction of integral partial ordered sets and a pivotal pmb Game morphism. It proves that the induced images form Lambda-modules and that the pmb Game^natural restricted to Im(+) yields a Lambda-epimorphism, enabling an addition law for integrals with variable upper limits. The authors present two key applications: a categorification of basic elementary functions including (anti-)trigonometric and logarithmic/exponential functions, and a novel representation-theoretic approach to the global dimensions of gentle algebras via multiple integrals and Lebesgue-Stieltjes measures. The work connects category theory, functional analysis, and representation theory to yield new insights into categorified integrals, with potential implications for differential/integral equations and algebraic invariants of gentle algebras.

Abstract

This paper provides a new categorification of the Lebesgue integral with variable upper limits by using normed modules over finite-dimensional -algebras and the category associated with . The integration process is redefined through the introduction of an integral partially ordered set and an abstract integral with variable upper limits. Finally, we present two important applications: (1) the categorification of basic elementary functions, including (anti-)trigonometric and logarithmic functions, and (2) a new approach for characterizing the global dimensions of gentle algebras.

Paper Structure

This paper contains 31 sections, 24 theorems, 145 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normed modules and Banach modules
  4. The categories $\mathscr{N}\!\!or^p_{\mathit{\Lambda}}$ and $\mathscr{A}^p_{\mathit{\Lambda}}$ and their special objects
  5. The categorification of Lebesgue integrations
  6. Integral partial ordered sets
  7. The definition of integral partial ordered set
  8. An addition induced by integral partial ordered set
  9. Map $\pmb{\Game}$ in the sense of the category $\mathscr{A}^p$
  10. Map $\pmb{\Game}$
  11. Surjection $\pmb{\Game}^{\natural}\circ\pmb{\Game}$
  12. Special case: The map $\pmb{\Game}$ in the case of $\mathscr{A}^1$
  13. Restriction $\pmb{\Game}_{\alpha}$ of $\pmb{\Game}$
  14. Surjection $\pmb{\Game}_{\alpha}^{\natural}\circ\pmb{\Game}_{\alpha}$
  15. Integral with variable upper limit
  16. ...and 16 more sections

Key Result

Theorem 1.1

The restriction of $\pmb{\Game}^{\natural}$ is a $\mathit{\Lambda}$-epimorphism.

Figures (2)

  • Figure 2.1: Order-preserving bijections $\kappa_c$ and $\kappa_d$
  • Figure 2.2: Juxtaposition map (in the case for $\dim_{\mathds{k}}\mathit{\Lambda} = 2$)

Theorems & Definitions (64)

  • Theorem 1.1: Theorem \ref{['thm:spC Lambda-homo']}
  • Theorem 1.2: Theorem \ref{['thm:proper']}
  • Theorem 1.3
  • Example 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 54 more