论文标题
分级内侧$ n $ -ary代数和多辅助张量类别
Graded medial $n$-ary algebras and polyadic tensor categories
论文作者
论文摘要
引入了内侧特性代替的代数结构。我们考虑（关联）分级代数，而不是几乎是通勤性（通用的通勤或$ \ varepsilon $ -cmutativity），我们引入了几乎是内侧（“通勤性到中等性” ANSATZ）。定义了较高分级的扭曲产品和“变形”括号（是谎言括号的内侧类似物）。 Toyoda的定理将（通用）内侧代数与Abelian代数连接起来，已被证明是此处介绍的几乎内侧分级代数。以类似的方式，我们概括了张量类别和编织张量类别。多核（非图案）张量类别具有$ n $ ar的张量产品，作为额外的乘法，与$ n-1 $ arity $ 2n-1 $满足$ \ left（n^{2} +1 \ right）$ -GON关系的乘积是五角形Axiom的多核类似物。多层单体类别可能包含几个单位对象，并且所有对象都是单位。定义了一种新型的多形类别（称为“群体”）：它们接近单体类别，但可能不包含单位：相反，Querfunctor和（自然）函数异态性异态性（Quertors）被考虑（通过类似于$ n $ ary-ary组中的querleylement）。引入了Arity-Nonredubible $ n $ ar-ary编织，并得出了其方程式，该方程式$ n = 2 $与Yang-Baxter方程相吻合。然后，类似于本文的第一部分，我们引入了“内侧”，而不是编织和构建“内侧”多核张量类别。
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 $\varepsilon$-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 $\left( n^{2}+1\right) $-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.
