The geometry of tensor calculus. I. (English) Zbl 0738.18005
The goal of this paper is to formalize the use of certain diagrams for a wide variety of situations in pure and applied mathematics. The main examples are the Feynman diagrams. Other examples are given by circuit diagrams, networks, Petri nets, flow charts, and planar diagrams of knots or links.
R. Penrose was the first to use the graphical notation for calculations with tensors. It is now currently used by theoretical physicists.
The paper consists of four chapters. In the first one, the authors recall the basic algebraic structure, namely that of a tensor category (also called “monoidal category”) which is simply a category with an associative (up to coherent isomorphism) tensor product operation. The authors introduce the concept of a graph appropriate for both this paper and the next one. They define the concept of a valuation which labels the nodes of a graph with arrows from a tensor category \(\mathcal V\) and labels the edges with objects of \(\mathcal V\). A planar graph \(\Gamma\) equipped with a valuation \(v\) is called a plane diagram in \(\mathcal V\). Then the authors define an arrow \(v(\Gamma)\) in \(\mathcal V\), called the value of the diagram. The main result of this chapter is that the value is invariant under continuous deformation of plane diagrams. In Section 4, the authors check that free tensor categories can be described in terms of isotopy classes of plane diagrams.
In the second chapter the authors consider the case of symmetric tensor categories for which the tensor product is equipped with an extra structure of symmetry. In this situation they show that the value \(v(\Gamma)\) of a diagram can be defined even when \(\Gamma\) is abstract (no planarity is needed). In the second section of this chapter the authors construct free symmetric tensor categories using isomorphism classes of abstract diagrams.
In Chapter 3 the authors consider the case of braided tensor categories [the authors, Braided monoidal categories, Macquarie Math. Reports (November, 1986)]. In this situation the diagram \(\Gamma\) is embedded in 3-space. They prove that the value of a diagram is invariant under deformation. Then they describe free braided tensor categories using isotopy classes of embedded diagrams in 3-space.
In Chapter 4 the authors introduce the concept of balanced tensor category. In this case the embedded graphs \(\Gamma\) are framed, or made of ribbons. Again they prove the invariance of the value of a ribbon diagram under continuous deformation and construct the free balanced tensor category from isotopy classes of ribbon diagrams.
The authors announce that their second paper will deal with tensor categories in which the objects are duals.
R. Penrose was the first to use the graphical notation for calculations with tensors. It is now currently used by theoretical physicists.
The paper consists of four chapters. In the first one, the authors recall the basic algebraic structure, namely that of a tensor category (also called “monoidal category”) which is simply a category with an associative (up to coherent isomorphism) tensor product operation. The authors introduce the concept of a graph appropriate for both this paper and the next one. They define the concept of a valuation which labels the nodes of a graph with arrows from a tensor category \(\mathcal V\) and labels the edges with objects of \(\mathcal V\). A planar graph \(\Gamma\) equipped with a valuation \(v\) is called a plane diagram in \(\mathcal V\). Then the authors define an arrow \(v(\Gamma)\) in \(\mathcal V\), called the value of the diagram. The main result of this chapter is that the value is invariant under continuous deformation of plane diagrams. In Section 4, the authors check that free tensor categories can be described in terms of isotopy classes of plane diagrams.
In the second chapter the authors consider the case of symmetric tensor categories for which the tensor product is equipped with an extra structure of symmetry. In this situation they show that the value \(v(\Gamma)\) of a diagram can be defined even when \(\Gamma\) is abstract (no planarity is needed). In the second section of this chapter the authors construct free symmetric tensor categories using isomorphism classes of abstract diagrams.
In Chapter 3 the authors consider the case of braided tensor categories [the authors, Braided monoidal categories, Macquarie Math. Reports (November, 1986)]. In this situation the diagram \(\Gamma\) is embedded in 3-space. They prove that the value of a diagram is invariant under deformation. Then they describe free braided tensor categories using isotopy classes of embedded diagrams in 3-space.
In Chapter 4 the authors introduce the concept of balanced tensor category. In this case the embedded graphs \(\Gamma\) are framed, or made of ribbons. Again they prove the invariance of the value of a ribbon diagram under continuous deformation and construct the free balanced tensor category from isotopy classes of ribbon diagrams.
The authors announce that their second paper will deal with tensor categories in which the objects are duals.
Reviewer: I.Pop (Iaşi)
MSC:
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 57M15 | Relations of low-dimensional topology with graph theory |
| 18A10 | Graphs, diagram schemes, precategories |
Keywords:
Feynman diagrams; circuit diagrams; networks; Petri nets; flow charts; planar diagrams of knots or links; tensor category; monoidal category; valuation; plane diagram; free tensor categories; symmetric tensor categories; abstract diagrams; braided tensor categories; isotopy classes; embedded diagrams; balanced tensor category; ribbons; continuous deformationReferences:
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