Decorated one-dimensional cobordisms and tensor envelopes of noncommutative recognizable power series. (English) Zbl 1541.18023
In the universal construction approach to low-dimensional topological theories, start with an evaluation of closed \(n\)-dimensional objects \(M\) taking values in a ground commutative ring or a field and then define state spaces \(A(N)\) for \((n - 1)\)-dimensional objects \(N\) via the bilinear pairing on \(n\)-dimensional objects \(M\) with a given boundary, \(\partial M \cong N\), by coupling two such objects \(M_1\) and \(M_2\) along the boundary and evaluating the resulting closed object \(M_1 \cup_N M_2\) by gluing along \(N\). M. Khovanov 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.
Reviewer: Mee Seong Im (Annapolis)
MSC:
| 18M05 | Monoidal categories, symmetric monoidal categories |
| 18M30 | String diagrams and graphical calculi |
| 57K16 | Finite-type and quantum invariants, topological quantum field theories (TQFT) |
| 16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |
| 15A63 | Quadratic and bilinear forms, inner products |
Keywords:
universal construction; low-dimensional cobordism; topological theories; TQFT; noncommutative power series; noncommutative recognizable power series; Deligne categoryReferences:
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