Non-associative algebras
(2021) New Perspectives in Algebra, Topology and Categories — ISBN: [978-3-030-84318-2], p. 225–258, published
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- Abstract
- A non-associative algebra over a field 𝕂 is a 𝕂-vector space A equipped with a bilinear operation A×A→A:(x,y)↦x⋅y=xy. The collection of all non-associative algebras over 𝕂, together with the product-preserving linear maps between them, forms a variety of algebras: the category 𝖠𝗅𝗀_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 𝖠𝗅𝗀_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.
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Citations
Van der Linden, T. (2021). Non-associative algebras. In Maria Manuel Clementino, Alberto Facchini, Marino Gran, editors (ed.), New Perspectives in Algebra, Topology and Categories (p. p. 225–258). https://hdl.handle.net/2078.5/231911