Abstract
Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or ε-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with n−1 associators of the arity 2n−1 satisfying a n2+1-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.
Highlights
The commutativity property and its “breaking” are quite obvious and unique for binary algebraic structures, because the permutation group S2 has only one non-identity element
We introduce arity-nonreducible n-ary braidings and find the equation for them that in the binary case turns into the Yang–Baxter equation in the tensor product form
Instead we propose to consider a new concept, polyadic mediality, as a polyadic inductive generalization of commutativity
Summary
The commutativity property and its “breaking” are quite obvious and unique for binary algebraic structures, because the permutation group S2 has only one non-identity element. In the first part of our paper we consider n-ary graded algebras and propose the following idea: instead of considering the non-unique commutativity property and its “breaking”, to investigate the unique property of mediality and its “breaking”we exploit this “commutativity-to-mediality” ansatz to introduce and study almost medial n-ary graded algebras by analogy with almost commutative algebras. The peculiarities of polyadic semigroupal and monoidal categories are studied and the differences from the corresponding binary tensor categories are outlined. We introduce a new kind of tensor categories, polyadic nonunital “groupal” categories, which contain a “querfunctor” and “quertors” (similar to querelements in n-ary groups [15,16]). The proposed “commutativity-to-mediality” ansatz can lead to medial n-ary superalgebras and Lie superalgebras as well as to a medial analog of noncommutative geometry
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