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The \(L_\infty\)-deformation complex of diagrams of algebras. (English) Zbl 1183.14004

The authors construct the deformation complex \((C^\ast_P(T;T),\delta_P)\) of an algebra \(T\) over a colored PROP \(P\). The authors prove that this complex has the structure of an \(L_\infty\)-algebra.
Considering colored PROPs is necessary if one wants to study \(L_\infty\)-deformations of diagrams of algebras over a PROP, module-algebras, modules over an associative algera, and Yetter-Drinfel’d and Hopf modules over a bialgebra.
For this to make sense, a definition of PROP’s is necessary: Let \(\mathfrak C\) be a nonempty set whose elements are called colors. A \(\sigma\)-bimodule is a collection \(E=\{E(m,\}_{m,n\geq 0}\) of \(k\)-modules over a fixed ground field \(k\), in which each \(E(m,n)\) is equipped with a left \(\Sigma_m\) and a right \(\Sigma_n\) action that commute with each other. A \(\mathfrak C\)-colored \(\Sigma\)-bimodule is a \(\Sigma\)-bimodule \(E\) in which \(E(m,n)\) admits a \(\mathfrak C\)-colored decomposition into submodules \[ E(m,n)=\underset{c_i,d_j\in\mathfrak C}\bigoplus_E\left(\begin{matrix} d_1,\dots,d_m\\ c_1,\dots,c_n\end{matrix}\right) \] that is compatible with the \(\Sigma_m\)-\(\Sigma_n\)-actions. A colored PROP is a \(\mathcal C\)-colored \(\Sigma\)-bimodule \(P=\{P(m,n)\}\) that is equipped with a horizontal and a vertical composition required to satisfy some associativity-type axioms. Colored Operads are particluar cases of colored PROPs such that \[ P\left(\begin{matrix} d_1,\dots,d_m\\ a_1,\dots,a_k\end{matrix}\right)=0 \] for \(m\geq 2\).
The deformation complex \((C^\ast_P(T;T),\delta_P)\) is constructed by first taking a minimal model \(\alpha:(F(E),\partial)\rightarrow P\) of the colored PROP \(P,\) which should be thought of as a resolution of \(P\). Given a \(P\)-algebra \(\rho:P\rightarrow\text{End}_T,\) \(C^\ast_P(T;T)=\text{Der}(F(E),\mathcal E)\), in which \(\mathcal E=\text{End}_T\) is considered as an \(F(E)\)-module via the morphism \(\beta=\rho\alpha,\) and \(\text{Der}(F(E),\mathcal E)\) denotes the vector space of derivations \(F(E)\rightarrow\mathcal E.\) The latter has a natural differential \(\delta\) that sends \(\theta\in\text{Der}(F(E),\mathcal E)\) to \(\theta\partial.\) This construction applies to more general cofibrant resolutions as well.
The \(L_\infty\)-operations on \(C^\ast_P(T;T)\) are constructed using graph substitutions. For a morphism \(g:U\rightarrow V\) of associative algebras, considered as an algebra over the \(2\)-colored PROP \(As_{B\rightarrow W}\), the authors write down explicitly all the \(L_\infty\)-operations \(l_k\) on the deformation complex of \(g\). The underlying cochain complex of the deformation complex of \(g\) is isomorphic to the Gerstenhaber-Schack cochain complex of \(g\). Therefore, the latter also has an explicit \(L_\infty\)-structure. The \(2\)-colored operad \(Iso\) whose algebras are of the form \(F:U\leftrightarrows V:G\), in which \(U\) and \(V\) are chain complexes and \(F\) and \(G\) are mutually inverse chain maps, is another example of the \(L_\infty\)-deformation complex. The authors write down explicitly the \(L_\infty\)-operations on the deformation complex of a typical \(Iso\)-algebra \(T\).
The article is rather detailed and technical, which is necessary to get deformation theoretical results. The authors explain the relation between the deformation complex and deformations of colored PROP algebras. The main focus of the article is the explicit expressions of the \(L_\infty\)-operations \(l_k\) in as many particular cases as possible, rather than explaining the possible overall applications. The results however, are nice and important.

MSC:

14A22 Noncommutative algebraic geometry
14B15 Local cohomology and algebraic geometry
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