Document Zbl 1193.18009 - zbMATH Open
Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.
Fields
| any | anywhere |
| an | internal document identifier |
| au | author, editor |
| ai | internal author identifier |
| ti | title |
| la | language |
| so | source |
| ab | review, abstract |
| py | publication year |
| rv | reviewer |
| cc | MSC code |
| ut | uncontrolled term |
| dt | document type (j: journal article; b: book; a: book article) |
Operators
| a & b | logic and |
| a | b | logic or |
| !ab | logic not |
| abc* | right wildcard |
| "ab c" | phrase |
| (ab c) | parentheses |
See also our General Help.
Cohomology of categorical self-distributivity. (English) Zbl 1193.18009
A binary operation on a set X is called self-distributive if \((ab)c=(ac)(bc)\) for all \(a, b, c\) in \(X\). Such an \(X\) is called a rack if for each \(a\) and \(b\) in \(X\), there is a unique \(c\) in \(X\) such that \(a=cb\). A rack \(X\) is called a quandle if \(aa=a\) for all \(a\) in \(X\). These notions generalize to objects in a category \(\mathcal K\) with finite products. A shelf in \(\mathcal K\) is a pair \((X, q), X\) an object of \({\mathcal K}, q\) a morphism of \(X \times X\) to \(X\) which satisfies the self-distributive law. This paper studies self-distrubitive structures in the categories of coalgebras and of commutative coalgebras over a field. Examples come from vector spaces whose bases are the elements of a finite quandle, the direct sum of a Lie algebra and its base field, and Hopf algebras. Such self-distibutive structures provide solutions of the Yang-Baxter equation and conversely.
The authors study how quandles and racks and their cohomology theories are related to other algebraic systems and their cohomology theories. Self-distributive maps are described by internalization, which means roughly viewing everything in terms of commutative diagrams interpreted in an appropriate ambient category. The cohomology theory presented encompasses both Lie algebra and quandle cohomology, is analogous to Hochschild cohomology, and can be used to study deformations of these self-distributive structures. All the calculations are informed by diagrammatic computations.