论文标题
非缔合代数
Non-associative algebras
论文作者
论文摘要
$ \ mathbb {k} $上的非求和代数是$ \ mathbb {k} $ - vector space $ a $ a $配备了双线性操作\ [{a \ times a \ to a \ colon \; （x，y）\ mapsto x \ cdot y = xy}。 \]在$ \ mathbb {k} $上的所有非相关代数的集合以及它们之间的产品保护线性地图一起形成了多种代数：类别$ \ mathsf {alg} _ \ mathbb {k} $。乘法不必满足任何其他属性，例如关联或单位的存在。熟悉的类别，例如$ \ mathsf {alg} _ \ mathbb {k {k {k} $的诸如联想代数，谎言代数等的品种，可以通过$ \ mathsf {alg} _ \ Mathbb {k} $，在此处通过$ x（yz）=（yz）=（xy）z $（acsociativitivitive）或$ xy = - yx $ x $ x $ x（yz）= yz（yz） （分别是反交换性和雅各比身份）。 这些讲座的目的是解释一些分类代数的基本概念，从非相关代数的角度来看，反之亦然。通常，矢量空间结构的存在使这里的事物比其他结构不那么丰富的类别更容易理解。 我们探索诸如正常的亚物体和商，共同体和杂种之类的概念。另一方面，我们讨论了（非缔合性）多项式，均匀方程以及其他方程如何导致反射子类别的作用。
A non-associative algebra over a field $\mathbb{K}$ is a $\mathbb{K}$-vector space $A$ equipped with a bilinear operation \[ {A\times A\to A\colon\; (x,y)\mapsto x\cdot y=xy}. \] The collection of all non-associative algebras over $\mathbb{K}$, together with the product-preserving linear maps between them, forms a variety of algebras: the category $\mathsf{Alg}_\mathbb{K}$. The multiplication need not satisfy any additional properties, such as associativity or the existence of a unit. Familiar categories such as the varieties of associative algebras, Lie algebras, etc. may be found as subvarieties of $\mathsf{Alg}_\mathbb{K}$ by imposing equations, here $x(yz)=(xy)z$ (associativity) or $xy =- yx$ and $x(yz)+z(xy)+ y(zx)=0$ (anti-commutativity and the Jacobi identity), respectively. The aim of these lectures is to explain some basic notions of categorical algebra from the point of view of non-associative algebras, and vice versa. As a rule, the presence of the vector space structure makes things easier to understand here than in other, less richly structured categories. We explore concepts like normal subobjects and quotients, coproducts and protomodularity. On the other hand, we discuss the role of (non-associative) polynomials, homogeneous equations, and how additional equations lead to reflective subcategories.
