nLab gebra theory

Idea

Gebras denote algebras, coalgebras and their combinations. They generalize algebras over operads and similar notions by possibly involving cooperations. Common frameworks include properads.

In modern algebra, dealing with co-operations $A\to A^{\otimes n}$ instead of only with operations $A^{\otimes n}\to A$ for some monoidal product $\otimes$, is a specialty of a group of people who are often centered around the study of Hopf algebras, but the subject is much wider, including internal co(al)gebras, cogroups in cartesian categories, Tannaka reconstruction, PROPs (which have even operations taking $m$ arguments to $k$ arguments, hence generalizing both operations and co-operations), co-operads, bi(al)gebroids, Hopf algebroids, entwining structures, mixed distributive laws and so on. Common tools are graphical calculations (extending also to planar algebras and alike), Sweedler notation and so on.

Historical note

We may call this area the theory of gebras as in the article Serre 1993.

Bourbaki complained that in Arabic ‘al’ is an article so classical ‘bi-’ for two is incompatible with it. Thus Bourbaki and much of the French school makes it a point to say ‘cogebra, bigebra’ instead of ‘coalgebra, bialgebra’.

Literature

The terminology is from

• Jean-Pierre Serre, Gèbres (in French; Engl. Gebras) Enseign. Math. (2) 39 (1993), no. 1-2, 33–85, MR1225256 (94h:16074)

A monograph on generalizations of Hopf algebras in categorical framework

• Gabriella Böhm, Hopf algebras and their generalizations from a category theoretical point of view, Lec. Notes Math. 2226, Springer 2018

The $n$Lab entries about or closely related to gebras and co-operations in $n$lab so far include bialgebra, Hopf algebra, comonoid, bimonoid, coalgebra, comonad, coring, Sweedler coring, cocategory, coaction, corepresentation, comodule, contramodule, [coinvariant]?, comodule algebra, cotensor product, induced comodule, module algebra, Hopf module, crossed product algebra, Hopf-Galois extension, measuring, cleft extension, quantum group, matrix Hopf algebra, Hopf envelope, bialgebroid, Hopf algebroid, distributive law, convolution algebra, Heisenberg double, noncommutative thin scheme, Lie bialgebra, Tannaka duality, quasi-Hopf algebra, Takeuchi product, cooperad, properad, prop…

category: algebra