{{more footnotes|date=February 2023}}
{{Short description|Algebraic structure in mathematical physics}} In mathematics and mathematical physics, a '''factorization algebra''' is an algebraic structure first introduced by Beilinson and Drinfel'd in an algebro-geometric setting as a reformulation of chiral algebras<ref name="BD">{{cite book |last1=Beilinson |first1=Alexander |last2=Drinfeld |first2=Vladimir |title=Chiral algebras |date=2004 |publisher=American Mathematical Society |location=Providence, R.I. |isbn=978-0-8218-3528-9 |url=https://books.google.com/books?id=yHZh3p-kFqQC |access-date=21 February 2023}}</ref> and applied in a more general setting by Costello and Gwilliam to formalize quantum field theory.<ref name="CG">{{cite book |last1=Costello |first1=Kevin |last2=Gwilliam |first2=Owen |title=Factorization algebras in quantum field theory, Volume 1 |date=2017 |location=Cambridge |isbn=9781316678626}}</ref>
== Definition == === Prefactorization algebras === A factorization algebra is a prefactorization algebra satisfying some properties, similar to sheafs being a presheaf with extra conditions.
If <math>M</math> is a topological space, a '''prefactorization algebra''' <math>\mathcal{F}</math> of vector spaces on <math>M</math> is an assignment of vector spaces <math>\mathcal{F}(U)</math> to open sets <math>U</math> of <math>M</math>, along with the following conditions on the assignment: * For each inclusion <math>U \subset V</math>, there's a linear map <math>m_V^U: \mathcal{F}(U) \rightarrow \mathcal{F}(V)</math> * There is a linear map <math>m_V^{U_1, \cdots, U_n}: \mathcal{F}(U_1)\otimes \cdots \otimes \mathcal{F}(U_n) \rightarrow \mathcal{F}(V)</math> for each finite collection of open sets with each <math>U_i \subset V</math> and the <math>U_i</math> pairwise disjoint. * The maps compose in the obvious way: for collections of opens <math>U_{i, j}</math>, <math>V_i</math> and an open <math>W</math> satisfying <math>U_{i,1}\sqcup \cdots \sqcup U_{i, n_i} \subset V_i</math> and <math>V_1 \sqcup \cdots V_n \subset W</math>, the following diagram commutes. <math> \begin{array}{lcl} & \bigotimes_i \bigotimes_j \mathcal{F}(U_{i,j}) & \rightarrow & \bigotimes_i \mathcal{F}(V_i) & \\ & \downarrow & \swarrow & \\ & \mathcal{F}(W) & & & \\ \end{array} </math>
So <math>\mathcal{F}</math> resembles a precosheaf, except the vector spaces are tensored rather than (direct-)summed.
The category of vector spaces can be replaced with any symmetric monoidal category.
=== Factorization algebras === To define factorization algebras, it is necessary to define a Weiss cover. For <math>U</math> an open set, a collection of opens <math>\mathfrak{U} = \{U_i | i \in I\}</math> is a '''Weiss cover''' of <math>U</math> if for any finite collection of points <math>\{x_1, \cdots, x_k\}</math> in <math>U</math>, there is an open set <math>U_i \in \mathfrak{U}</math> such that <math>\{x_1, \cdots, x_k\} \subset U_i</math>.
Then a '''factorization algebra''' of vector spaces on <math>M</math> is a prefactorization algebra of vector spaces on <math>M</math> so that for every open <math>U</math> and every Weiss cover <math>\{U_i | i \in I\}</math> of <math>U</math>, the sequence <math display = block> \bigoplus_{i,j} \mathcal{F}(U_i \cap U_j) \rightarrow \bigoplus_k \mathcal{F}(U_k) \rightarrow \mathcal{F}(U) \rightarrow 0</math> is exact. That is, <math>\mathcal{F}</math> is a factorization algebra if it is a cosheaf with respect to the Weiss topology.
A factorization algebra is ''multiplicative'' if, in addition, for each pair of disjoint opens <math>U, V \subset M</math>, the structure map <math display=block> m^{U, V}_{U\sqcup V} : \mathcal{F}(U)\otimes \mathcal{F}(V) \rightarrow \mathcal{F}(U \sqcup V)</math> is an isomorphism.
=== Algebro-geometric formulation === While this formulation is related to the one given above, the relation is not immediate.
Let <math>X</math> be a smooth complex curve. A '''factorization algebra''' on <math>X</math> consists of * A quasicoherent sheaf <math>\mathcal{V}_{X, I}</math> over <math>X^{I}</math> for any finite set <math>I</math>, with no non-zero local section supported at the union of all partial diagonals * Functorial isomorphisms of quasicoherent sheaves <math>\Delta^*_{J/I}\mathcal{V}_{X, J} \rightarrow \mathcal{V}_{X, I}</math> over <math>X^I</math> for surjections <math>J \rightarrow I</math>. * (''Factorization'') Functorial isomorphisms of quasicoherent sheaves <math display = block> j^*_{J/I}\mathcal{V}_{X, J} \rightarrow j^*_{J/I}(\boxtimes_{i \in I} \mathcal{V}_{X, p^{-1}(i)})</math> over <math>U^{J/I}</math>. * (''Unit'') Let <math>\mathcal{V} = \mathcal{V}_{X, \{1\}}</math> and <math>\mathcal{V}_2 = \mathcal{V}_{X, \{1, 2\}}</math>. A global section (the ''unit'') <math>1 \in \mathcal{V}(X)</math> with the property that for every local section <math>f \in \mathcal V(U)</math> (<math>U \subset X</math>), the section <math>1 \boxtimes f</math> of <math>\mathcal{V}_2|_{U^2\Delta}</math> extends across the diagonal, and restricts to <math>f \in \mathcal{V} \cong \mathcal{V}_2|_\Delta</math>.
== Example == === Associative algebra === {{See also| associative algebra}} Any associative algebra <math>A</math> can be realized as a prefactorization algebra <math>A^{f}</math> on <math>\mathbb{R}</math>. To each open interval <math>(a,b)</math>, assign <math>A^f((a,b)) = A</math>. An arbitrary open is a disjoint union of countably many open intervals, <math>U = \bigsqcup_i I_i</math>, and then set <math>A^f(U) = \bigotimes_i A</math>. The structure maps simply come from the multiplication map on <math>A</math>. Some care is needed for infinite tensor products, but for finitely many open intervals the picture is straightforward.
== See also == * Algebraic quantum field theory * Vertex algebra
== References == {{reflist}}
Category:Abstract algebra