[Paper Review] Decorated one-dimensional cobordisms and tensor envelopes of noncommutative recognizable power series

Mikhail Khovanov|arXiv (Cornell University)|Oct 12, 2020

Algebraic structures and combinatorial models39 references4 citations

TL;DR

This paper establishes a categorical framework linking noncommutative recognizable power series to decorated one-dimensional cobordisms, constructing tensor envelopes via universal constructions. It introduces categories of cobordisms with labeled boundary points and floating components, showing that skein categories and their quotients recover syntactic algebras and Deligne-like categories, with equivalences preserved under symmetric evaluations and ideal quotients.

ABSTRACT

The paper explores the relation between noncommutative power series and topological theories of one-dimensional cobordisms decorated by labelled zero-dimensional submanifolds. These topological theories give rise to several types of tensor envelopes of noncommutative recognizable power series, including the categories built from the syntactic algebra and syntactic ideals of the series and the analogue of the Deligne category.

Motivation & Objective

• To formalize a universal construction for noncommutative recognizable power series using decorated one-dimensional cobordisms.
• To define tensor envelopes of such series via skein categories and gligible quotients.
• To generalize the construction to include inner (floating) boundaries and symmetric evaluations.
• To show that the resulting categories recover known algebraic structures such as syntactic algebras and Deligne-type categories.
• To unify the theory via a pair of functionals on a finite-dimensional algebra, generalizing the power series approach.

Proposed method

• Construct categories $\mathcal{C}$, $\mathcal{C}'$, and $\widetilde{\mathcal{C}}$ of one-dimensional cobordisms with labeled boundary points and floating components.
• Define evaluations of closed cobordisms using noncommutative power series $\alpha = (\alpha^\bullet, \alpha^\circ)$, where $\alpha^\bullet$ is a linear functional on words and $\alpha^\circ$ on cyclic words.
• Build the skein category $\mathcal{S}\widetilde{\mathcal{C}}_\alpha$ by quotienting the free category on cobordisms by relations derived from evaluations.
• Introduce tensor envelopes $\mathcal{D}\widetilde{\mathcal{C}}_{(0,\alpha^\circ)}$, $\underline{\mathcal{D}\widetilde{\mathcal{C}}}_{(0,\alpha^\circ)}$, and $\mathcal{D}\mathcal{C}_{\alpha^\circ}$, showing they are isomorphic to their counterparts in the symmetric case.
• Use syntactic ideals $I_{\alpha^\bullet}, I_{\alpha^\circ}, I^{\ell}_{\alpha^\bullet}, I^{r}_{\alpha^\bullet}$ to define the gligible quotient $\mathcal{C}_\alpha$, capturing the algebraic structure of the series.
• Generalize the construction to associative $\mathbf{k}$-algebras $B$ with two functionals $\alpha^\bullet, \alpha^\circ$, replacing generators $S$ with the full algebra.

Experimental results

Research questions

• RQ1 How can noncommutative recognizable power series be categorically represented through topological cobordism theories with decorated boundaries?
• RQ2 What tensor categories arise from the universal construction applied to decorated 1D cobordisms with floating components?
• RQ3 How do the skein categories $\mathcal{S}\widetilde{\mathcal{C}}_\alpha$ and their quotients relate to syntactic algebras and Deligne categories?
• RQ4 What is the role of symmetric functionals $\alpha^\circ$ in defining invariant evaluations and isomorphisms between tensor envelopes?
• RQ5 Can the construction be generalized from free algebras on generators to arbitrary finite-dimensional algebras with trace functionals?

Key findings

• The skein category $\mathcal{S}\widetilde{\mathcal{C}}_\alpha$ associated to a pair $\alpha = (\alpha^\bullet, \alpha^\circ)$ is equivalent to the category built from the syntactic algebra and ideals of the series.
• The tensor envelopes $\mathcal{D}\widetilde{\mathcal{C}}_{(0,\alpha^\circ)}$, $\underline{\mathcal{D}\widetilde{\mathcal{C}}}_{(0,\alpha^\circ)}$, and $\mathcal{D}\mathcal{C}_{\alpha^\circ}$ are isomorphic, showing consistency across different constructions.
• When $\alpha^\circ = 0$, the category $\widetilde{\mathcal{C}}_\alpha$ allows non-zero $U$-turns, and the evaluation of a circle is zero, reflecting the absence of cyclic symmetry.
• For $S = \emptyset$, the unoriented skein category $\mathcal{S}\widetilde{\mathcal{C}}_\alpha$ is isomorphic to the partial Brauer category (or rook-Brauer category), recovering a known structure.
• When $S = \{s\}$, the series $\alpha$ is encoded by two rational functions $Z_{\alpha^\bullet}(T)$ and $Z_{\alpha^\circ}(T)$, linking the construction to rational generating functions.
• The construction generalizes to finite-dimensional $\mathbf{k}$-algebras $B$ with two functionals $\alpha^\bullet, \alpha^\circ$, where the resulting categories are equivalent to those from a power series once a generating set $S$ is chosen.