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The hierarchies of identities and closed products for multiple complexes

Daniel Levin, Alexander Zuevsky

Abstract

We consider infinite $\Z_\Z$-index complexes $\mathcal C$ of spaces with elements depending on a number of parameters, complete with respect to a linear associative regular inseparable multilinear product. The existence of nets of vanishing ideals of orders of and powers of differentials is assumed for subspaces of $\mathcal C$-spaces. In the polynomial case of orders and powers of the differentials, we derive the hierarchies of differential identities and closed multiple products. We prove that a set of maximal orders and powers for differentials, differential conditions, together with coherence conditions on indices of a complex $\mathcal C$ elements generate families of multi-graded differential algebras.

The hierarchies of identities and closed products for multiple complexes

Abstract

We consider infinite -index complexes of spaces with elements depending on a number of parameters, complete with respect to a linear associative regular inseparable multilinear product. The existence of nets of vanishing ideals of orders of and powers of differentials is assumed for subspaces of -spaces. In the polynomial case of orders and powers of the differentials, we derive the hierarchies of differential identities and closed multiple products. We prove that a set of maximal orders and powers for differentials, differential conditions, together with coherence conditions on indices of a complex elements generate families of multi-graded differential algebras.
Paper Structure (20 sections, 3 theorems, 37 equations)

This paper contains 20 sections, 3 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Multiple complexes and the main result
  3. Examples of identities for multiple products
  4. Two differentials, and the total maximal power two for two kinds of elements
  5. Single differential, two kinds of elements
  6. Two differentials, identical elements of the total maximal power
  7. Orders of two differentials of several identical elements
  8. Two differentials, multiple elements
  9. The most general polynomial case: the identities for orders of two differentials and powers of multiple elements
  10. Examples of a single element closed products
  11. A single differential, a single element
  12. Higher orders of a single differential, single element
  13. Orders of a single differential, single element, powers of differentials
  14. Two differentials, single element
  15. Examples of multi-element closed products
  16. ...and 5 more sections

Key Result

Theorem 1

The conditions roda together with a set of maximal orders and powers of differentials for a multiple complex $\mathcal{C}$ result in a hierarchy of closed products in terms of differential identities on elements of $\mathcal{C}$ given by the general formula with $1 \le q_i < q\left(\mathcal{D}_{ {\bm J}_i} \right)$, $1 \le i \le k$, where $q\left(\mathcal{D}_{{\bm J}_i}\right)$ are the maximal po

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Corollary 1
  • Lemma 1
  • proof