Recent zbMATH articles in MSC 18https://zbmath.org/atom/cc/182024-02-15T19:53:11.284213ZWerkzeugAdmissible subsets and completions of ordered algebrashttps://zbmath.org/1526.060032024-02-15T19:53:11.284213Z"Laan, Valdis"https://zbmath.org/authors/?q=ai:laan.valdis"Feng, Jianjun"https://zbmath.org/authors/?q=ai:feng.jianjun"Zhang, Xia"https://zbmath.org/authors/?q=ai:zhang.xiaOrdered \(\Omega\)-algebras of fixed type \(\Omega\) are considered. \textit{Linear functions} on an ordered algebra \(\mathcal{A}\) are defined iteratively: 1) identity mapping on \(\mathcal{A}\) is a linear function and 2) if \(p: \mathcal{A}\to\mathcal{A}\) is a linear function and \(\omega\in \Omega_n\), \(a_i\in \mathcal{A}\), then \(x\mapsto \omega(a_1,\ldots, a_{i-1},p(x), a_{i+1},\ldots, a_n)\) defines a linear function as well. Linear functions with \(p=\operatorname{id}_A\) are called \textit{elementary translations}. \(A\) is a \textit{sup-algebra} if it as a poset is a complete lattice and all elementary translations preserve joins. A subset \(M\) of an ordered algebra \(\mathcal{A}\) is called \textit{admissible} if \(\bigvee p(M)\) exists and \(p(\bigvee M)=\bigvee p(M)\) for all \(p\in L_{\mathcal{A}}\), where \( L_{\mathcal{A}}\) is the set of all linear functions on \(\mathcal{A}\). A lower subset \(S\) of an ordered algebra \(\mathcal{A}\) is called a \(\mathcal{D}\)-\textit{ideal} if for any admissible subset \(M\) of \(S\), one has that \(\bigvee M\in S\).
The authors prove that for an ordered algebra \(\mathcal{A}\), for every sup-algebra \(\mathcal{Q}\) and every \texttt{OAlg}\(^*\)-morphism \(f: \mathcal{A}\to \mathcal{Q}\), there exists a unique \texttt{SupAlg}-morphism \(g: \mathcal{D}(A)\to \mathcal{Q}\) such that \(gr=f\), where \(r: \mathcal{A}\to \mathcal{D}(A), a\mapsto a\!\downarrow\) (\texttt{OAlg}\(^*\) is a category of all ordered algebras with admissible-join preserving homomorphisms, \texttt{SupAlg} is a category of all sup-algeras with join-preserving homomorphisms and \(\mathcal{D}(A)\) is the set of all \(\mathcal{D}\)-ideals of \(A\)). It turnes out that \(\mathcal{D}(A)\) is a join-completion of \(\mathcal{A}\) and that \texttt{SupAlg} is a reflective subcategory of \texttt{OAlg}\(^*\).
Reviewer: Peeter Normak (Tallinn)Periodic cyclic homology of crossed productshttps://zbmath.org/1526.160092024-02-15T19:53:11.284213Z"Puschnigg, Michael"https://zbmath.org/authors/?q=ai:puschnigg.michaelThis paper studies the cyclic homology of crossed product algebras from the Cuntz-Quillen point of view. First, the author recalls the work of Burghelea and Nistor, then he adapts Nistor's approach to Banach crossed products. He also describes the periodic cyclic homology of a crossed product of a complex algebra \(A\) by the action of an abstract group \(G\). He proves that this periodic cyclic homology is determined by a family of cyclic bicomplexes of the crossed products of \(A\) by the cyclic subgroups of \(G\) and the \(G\)-action. Finally, the author obtains explicit formulas. Topological remarks are also discussed.
For the entire collection see [Zbl 1507.19001].
Reviewer: Angela Gammella-Mathieu (Metz)Some remarks on Avella-Alaminos-Geiss invariants of gentle algebrashttps://zbmath.org/1526.160172024-02-15T19:53:11.284213Z"Nakaoka, Hiroyuki"https://zbmath.org/authors/?q=ai:nakaoka.hiroyukiSummary: This is a report on the talk given at the 51st Symposium on Ring Theory and Representation Theory 2018. We recall how the original definition of Avella-Alaminos-Geiss invariants can be rephrased by using blossoming, following the article by Asashiba. These are derived invariants calculable combinatorially from their bound quivers. Ladkani has given a formula which describes the dimensions of the Hochschild cohomologies of a gentle algebra in terms of its Avella-Alaminos-Geiss invariants. \par In the latter part we introduce a `repetitive' construction of gentle algebras out of a gentle algebra in a similar manner as that of the usual repetitive algebras. We show how the Avella-Alaminos-Geiss invariants of the resulting algebras are related to those of the original one.
For the entire collection see [Zbl 1415.16002].Finite dimensional algebras arising from dimer models and their derived equivalenceshttps://zbmath.org/1526.160252024-02-15T19:53:11.284213Z"Nakajima, Yusuke"https://zbmath.org/authors/?q=ai:nakajima.yusukeSummary: The notion of \(n\)-representation infinite algebra is a higher dimensional analogue of representation infinite hereditary algebra. It is known that this algebra can be obtained as the degree zero part of an \((n+1)\)-Calabi-Yau algebra with a particularly nice grading. On the other hand, some 3-Calabi-Yau algebras are obtained from consistent dimer models which are bipartite graphs on the real-two torus. In this article, we first explain how to give a grading that induces a 2-representation infinite algebra to the 3-Calabi-Yau algebra arising from a consistent dimer model. Then, we study derived equivalent classes of 2-representation infinite algebras using perfect matchings of dimer models and their mutations.
For the entire collection see [Zbl 1415.16002].Polynomial functors and two-parameter quantum symmetric pairshttps://zbmath.org/1526.170242024-02-15T19:53:11.284213Z"Buciumas, Valentin"https://zbmath.org/authors/?q=ai:buciumas.valentin"Ko, Hankyung"https://zbmath.org/authors/?q=ai:ko.hankyungSummary: We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups \(\mathrm{GL}_n\), the two-parameter polynomial functors give a new interpretation of (polynomial) representations of the quantum symmetric pair \(({U}_{Q,q}^B (\mathfrak{gl}_n), U_{q}(\mathfrak{gl}_{n}))\) which specializes to type AIII/AIV quantum symmetric pairs. The coideal subalgebra \({U}_{Q,q}^B (\mathfrak{gl}_n)\) appears in a Schur-Weyl duality with the type B Hecke algebra \(\mathcal{H}_{Q,q}^B (d)\). We endow two-parameter polynomial functors with a cylinder braided structure which we use to construct the two-parameter Schur functors. Our polynomial functors can be precomposed with the quantum polynomial functors of type \textsf{A} producing new examples of action pairs.Trace decategorification of categorified quantum \(\mathrm{sl}(3)\)https://zbmath.org/1526.170282024-02-15T19:53:11.284213Z"Živković, Marko"https://zbmath.org/authors/?q=ai:zivkovic.markoSummary: We prove that the trace of categorified quantum \(\mathfrak{sl}_3\) introduced by Khovanov and Lauda can also be identified with quantum \(\mathfrak{sl}_3\), thus providing an alternative way of decategorification. This is the second step of trace decategorification of quantum \(\mathfrak{sl}_n\) groups over the integers, the first being the \(\mathfrak{sl}_2\) case. The main technique used is decoupling of categorified quantum group into its positive and negative part. This technique can be used for more general categorified quantum groups to reduce the problem to the trace decategorification of its positive part. In the case of quantum \(\mathfrak{sl}_3\), there is an explicit form of the canonical basis of the positive (and isomorphically negative) part of it based on indecomposables found by Stošić, leading to the full result in this case.Exact sequences in the cohomology of a Lie superalgebra extensionhttps://zbmath.org/1526.170332024-02-15T19:53:11.284213Z"Hazra, Samir Kumar"https://zbmath.org/authors/?q=ai:hazra.samir-kumar"Habib, Amber"https://zbmath.org/authors/?q=ai:habib.amberSummary: Let \(0\to \mathfrak{a}\to\mathfrak{e}\to\mathfrak{g}\to 0\) be an abelian extension of the Lie superalgebra \(\mathfrak{g}\). In this article we consider the problems of extending endomorphisms of \(\mathfrak{a}\) and lifting endomorphisms of \(\mathfrak{g}\) to certain endomorphisms of \(\mathfrak{e}\). We connect these problems to the cohomology of \(\mathfrak{g}\) with coefficients in \(\mathfrak{a}\) through construction of two exact sequences which are our main results. The first exact sequence is obtained using the Hochschild-Serre spectral sequence corresponding to the above extension while to prove the second we rather take a direct approach. As an application of our results we obtain descriptions of certain automorphism groups of semidirect product Lie superalgebras.Injective generation of derived categories and other applications of cohomological invariants of infinite groupshttps://zbmath.org/1526.200102024-02-15T19:53:11.284213Z"Biswas, Rudradip"https://zbmath.org/authors/?q=ai:biswas.rudradipSummary: In the study of the representation theory of infinite groups, cohomological invariants play a very useful role. In a recent paper, we proved a number of properties regarding how these invariants interact with each other, extending the scope of some results in the literature. In this short article, we look into several ways in which the behavior of these invariants can be applied in various areas.Quantum symmetries of Cayley graphs of abelian groupshttps://zbmath.org/1526.200752024-02-15T19:53:11.284213Z"Gromada, Daniel"https://zbmath.org/authors/?q=ai:gromada.danielThe paper is concerned with the problem of determining quantum automorphism group \(\Aut^{+}\mathrm{Cay}(\Gamma,S)\) of Cayley graphs \(\mathrm{Cay}(\Gamma,S)\) of an abelian group \(\Gamma\) with generating set \(S\). To formulate the main result formulated as the algorithm to compute these quantum groups, the author uses techniques based on the intertwiner formalism.
The proposed method can be summarized as follows. First, one has to find all irreducible representations of \(\Gamma\), use them as the eigenbasis of the adjacency matrix to determine its spectrum, and describe the corresponding eigenspaces. Since these subspaces are invariant under the fundamental representation \(u\) of \(\Aut^{+}\mathrm{Cay}(\Gamma,S)\), one defines the Fourier transform of \(u\), and decomposes as a direct sum with respect to the eigenspace decomposition. Choosing some of the eigenspaces such that their direct sum \(W\) generates \(C(\Gamma)\) as an algebra and taking the corresponding direct sum of representations, one finds a faithful representation of \(\Aut^{+}\mathrm{Cay}(\Gamma,S)\). Finally, one has to define certain intertwiners and study their behavior on \(W\).
The author illustrates the general algorithm on several examples: halved hypercube graphs \(\frac{1}{2}Q_{n+1}\), folded hypercube graphs \(FQ_{n+1}\) and the class of Hamming graphs \(H(n,m)\); the results are:
\begin{itemize}
\item \(\Aut^{+}\frac{1}{2}Q_{n+1}=SO^{-1}_{n+1}\), for \(n\in\mathbb{N}\setminus\{1,3\}\),
\item \(\Aut^{+}FQ_{n+1}=PO^{-1}_{n+1}\), for \(n\in\mathbb{N}\setminus\{1,3\}\),
\item \(\Aut^{+}H(n,m)=S_m^+\wr S_n\), for \(n\in\mathbb{N},\ m\in\mathbb{N}\setminus\{1,2\}\).
\end{itemize}
Reviewer: Arkadiusz Bochniak (Garching)Interpolated family of non-group-like simple integral fusion rings of Lie typehttps://zbmath.org/1526.460402024-02-15T19:53:11.284213Z"Liu, Zhengwei"https://zbmath.org/authors/?q=ai:liu.zhengwei"Palcoux, Sebastien"https://zbmath.org/authors/?q=ai:palcoux.sebastien"Ren, Yunxiang"https://zbmath.org/authors/?q=ai:ren.yunxiangSummary: This paper is motivated by the quest of a non-group irreducible finite index depth 2 maximal subfactor. We compute the generic fusion rules of the Grothendieck ring of Rep(PSL\((2,q))\), \(q\) prime-power, by applying a Verlinde-like formula on the generic character table. We then prove that this family of fusion rings \((\mathcal{R}_q)\) interpolates to all integers \(q\geq 2\), providing (when \(q\) is not prime-power) the first example of infinite family of non-group-like simple integral fusion rings. Furthermore, they pass all the known criteria of (unitary) categorification. This provides infinitely many serious candidates for solving the famous open problem of whether there exists an integral fusion category which is not weakly group-theoretical. We prove that a complex categorification (if any) of an interpolated fusion ring \(\mathcal{R}_q\) (with \(q\) non-prime-power) cannot be braided, and so its Drinfeld center must be simple. In general, this paper proves that a non-pointed simple fusion category is non-braided if and only if its Drinfeld center is simple; and also that every simple integral fusion category is weakly group-theoretical if and only if every simple integral modular fusion category is pointed.Maurer-Cartan methods in deformation theory. The twisting procedurehttps://zbmath.org/1526.530012024-02-15T19:53:11.284213Z"Dotsenko, Vladimir"https://zbmath.org/authors/?q=ai:dotsenko.vladimir-v"Shadrin, Sergey"https://zbmath.org/authors/?q=ai:shadrin.sergey"Vallette, Bruno"https://zbmath.org/authors/?q=ai:vallette.brunoThe seminal work of \textit{L. Maurer} [Münch. Ber. 103--150 (1888; JFM 20.0102.02)] and \textit{E. Cartan} [Ann. Sci. Éc. Norm. Supér. (3) 21, 153--206 (1904; JFM 35.0176.04)] investigating the integrability of Lie algebras to Lie groups introduced what differential geometers now call the Maurer-Cartan 1-form, for which the Maurer-Cartan equation
\[
d\omega+\frac{1}{2}\left[ \omega,\omega\right] =0
\]
becomes the flatness condition for the connection defined by that form. In general, a flat connection in a vector bundle \(E\rightarrow M\)\ allows one to define a twisted de Rham differential on the sheaf of \(E\)-valued differential forms. In the case of principal bundles, one actually deals with differential forms with values in the structure Lie algebra, which form a differential graded Lie algebra, that is, a Lie algebra structure in the category of chain complexes. This is the conceptual framework for the Maurer-Cartan equation, its solutions being called the Maurer-Cartan elements and coinciding with flat connections in the case of principal bundles. The Maurer-Cartan equation, the twisting procedure, and the gauge group action constitute the Maurer-Cartan methods, which lie at the core of gauge theory.
Around 1960, the Maurer-Cartan equation started to be understood as the structural equation in deformation theory (see [\textit{A. Frölicher} and \textit{A. Nijenhuis}, Proc. Natl. Acad. Sci. USA 43, 239--241 (1957; Zbl 0078.14201); \textit{K. Kodaira} et al., Ann. Math. (2) 68, 450--459 (1958; Zbl 0088.38004); \textit{K. Kodaira} and \textit{D. C. Spencer}, Ann. Math. (2) 67, 328--401 (1958; Zbl 0128.16901); \textit{K. Kodaira} and \textit{D. C. Spencer}, Ann. Math. (2) 67, 403--466 (1958; Zbl 1307.14016); \textit{M. Kuranishi}, Ann. Math. (2) 75, 536--577 (1962; Zbl 0106.15303); \textit{M. Gerstenhaber}, Ann. Math. (2) 79, 59--103 (1964; Zbl 0123.03101)]). A few years later, \textit{A. Nijenhuis} and \textit{R. W. Richardson jun.} [Bull. Am. Math. Soc. 72, 1--29 (1966; Zbl 0136.30502)] noticed the omnipresence of differential graded Lie algebras in deformation theory, which was one of the inspirations behind the work of \textit{V. P. Palamodov} [Usp. Mat. Nauk 31, No. 3(189), 129--194 (1976; Zbl 0332.32013)], where deformation theories of complex structures and of commutative algebras were brought together. These examples and the unifying role played by the conceptual notion of a differential Lie algebra eventually led \textit{P. Deligne} [\url{https://mathoverflow.net/questions/249979/delignes-letter-to-millson}] and \textit{V. Drinfeld} [EMS Surv. Math. Sci. 1, No. 2, 241--248 (2014; Zbl 1327.18029)] to formulate the general principle of deformation theory claiming that over a field of characteristic 0, any deformation problem is to be encoded by a differential graded Lie algebra. Maurer-Cartan elements in the twisted differential graded Lie algebra should correspond to deformations of the original structure, while the Maurer-Cartan elements lying in the same orbit of the gauge group action should correspond to equivalent structures.
This monograph guides the reader through various versions of the twisting procedure, aiming to provide an extensive toolbox, including new properties, various applications and an elaborate survey. The guiding principle is that Maurer-Cartan elements should be studied through their symmetries. Over the recent years, the authors have pursued a research program to develop all functorial procedures producing new homotopy algebra structures from a given one, using suitable gauge symmetries. In this way, they recovered the homotopy transfer theorem [\textit{V. Dotsenko} et al., Mosc. Math. J. 16, No. 3, 505--543 (2016; Zbl 1386.18054)], the Koszul hierarchy [\textit{V. Dotsenko} et al., Mosc. Math. J. 16, No. 3, 505--543 (2016; Zbl 1386.18054); \textit{M. Markl}, J. Homotopy Relat. Struct. 10, No. 3, 637--667 (2015; Zbl 1330.13032)] and the Givental action [\textit{V. Dotsenko} et al., J. Éc. Polytech., Math. 2, 213--246 (2015; Zbl 1331.18010); ``Deformation theory of cohomological field theories'', Preprint, \url{arXiv:2006.01649}].
The synopsis of the book is as follows.
Chapter 1 aims to give a rather exhaustive survey on the Maurer-Cartan equation and its related topics, which lie at the core of this book.
Chapter 2 establishes various properties for the monoidal structures on categories of filtered modules and complete modules and for their associated monoidal functors. The principal goal is to develop the theory of operads and operadic algebras in this context.
Chapter 3 recalls the necessary definitions and results concerning pre-Lie algebras and the symmetries of their Maurer-Cartan elements, generalizing the approach of \textit{M. Lazard} [Ann. Sci. Éc. Norm. Supér. (3) 71, 101--190 (1954; Zbl 0055.25103)] of Lie theory to develop the integration theory of complete pre-Lie algebras.
Chapter 4 provides a first application of the complete operadic deformation theory developed in the previous section, dealing with the easiest example of gauge action.
Chapter 5 lifts the twisting procedure to the level of complete differential graded operads, applying it in detail to the example of multiplicative nonsymmetric operads. In the symmetric case, this is the seminal theory of \textit{V. Dolgushev} and \textit{T. Willwacher} [J. Pure Appl. Algebra 219, No. 5, 1349--1428 (2015; Zbl 1305.18032)], inspired by \textit{J. Chuang} and \textit{A. Lazarev} [Lett. Math. Phys. 103, No. 1, 79--112 (2013; Zbl 1271.18009)].
Chapter 6 discusses instances of the operadic twisting leading to various graph complexes, outlining the corresponding homology computations and their applications.
Chapter 7 provides details on seminal applications of the twisting procedure in several domains of mathematics such as deformation theory, higher Lie theory, rational homotopy theory, higher category theory and symplectic topology.
Reviewer: Hirokazu Nishimura (Tsukuba)A model for configuration spaces of pointshttps://zbmath.org/1526.550112024-02-15T19:53:11.284213Z"Campos, Ricardo"https://zbmath.org/authors/?q=ai:campos.ricardo"Willwacher, Thomas"https://zbmath.org/authors/?q=ai:willwacher.thomasIn this paper, the authors provide combinatorial models for the real homotopy type of configuration spaces of closed smooth oriented manifolds of dimension at least 2. Their model for the configuration spaces \(\mathrm{Conf}_r(M)\), as \(r\) ranges over the natural numbers, is a dg Hopf collection consisting of dg commutative \(\mathbb{R}\)-algebras with symmetric group actions, \({}^*\mathrm{Graphs}_M(r)\). Roughly, \({}^*\mathrm{Graphs}_M(r)\) consists of linear combinations of graphs in which the vertices are decorated by elements of the symmetric algebra on the reduced cohomology of \(M\). This is a generalization of Kontsevich's dg cooperad \({}^* \mathrm{Graphs}_D\), which models the little disks operad. If \(M\) is parallelizable, \({}^* \mathrm{Graphs}_M\) is furthermore a right cooperadic comodule over \({}^* \mathrm{Graphs}_D\), just as configuration spaces in a parallelizable manifold form a right operadic module over the little disks operad. The differential on \({}^* \mathrm{Graphs}_M\) is defined using the partition function \(Z_M\) of the perturbative AKSZ \(\sigma\)-model on \(M\). As a result of this construction, the real homotopy type of \(\mathrm{Conf}_r(M)\) depends only on \(Z_M\).
The authors then study these models, and the partition function \(Z_M\), and show that for \(M\) simply connected, \({}^* \mathrm{Graphs}_M\) depends only on the real homotopy type of \(M\). This settles the \(\mathbb{R}\)-based version of a long-standing question in algebraic topology, showing that for a simply-connected closed smooth oriented manifold \(M\), the real homotopy type of \(\mathrm{Conf}_r(M)\) depends only on the real homotopy type of \(M\).
Reviewer: Inbar Klang (New York)Diagrammatic algebrahttps://zbmath.org/1526.570012024-02-15T19:53:11.284213Z"Carter, J. Scott"https://zbmath.org/authors/?q=ai:carter.j-scott"Kamada, Seiichi"https://zbmath.org/authors/?q=ai:kamada.seiichiJ.S. Carter and S. Kamada present in this book techniques and results on diagrammatic algebras, which represent an algebraic quantity or a topological quality arranged in the plane. The authors begin by giving an overview of Penrose's algebraic tensors. The definition of Frobenius algebra is given and then diagrammatic methods are used to show that the set of \(n\times n\) matrices forms a Frobenius algebra.
The authors then abstractify the algebraic axioms of a Frobenius algebra, i.e., multiplication, co-multiplication, pairing, co-pairing, and unit and co-unit, providing a category in which arrows are diagrams. They also define the notion of an exchanger and discuss the naturality of exchangers. Frobenius algebra axioms are then transformed into surface-like diagrams and next into 2-arrows in a multi-category. A thorough discussion with plentiful diagrams is provided throughout the text.
Kauffman's diagrammatic description of the Temperley-Lieb algebra is discussed, which is a tour of relationships among associated products and triangulations of manifolds with certain counting of objects given by Catalan numbers. A representation of the Temperley-Lieb algebra giving rise to the Jones polynomial is presented.
Oriented framed tangles are described categorically, and Reidemeister moves, the fundamental group, the braid group and the fundamental quandle are defined. An example of a calculation in Khovanov homology is presented. Isotopies of surfaces embedded in the 4-space are given and relations among embedded foams are explained, which are related by means of the naturality of exchangers.
Reviewer: Mee Seong Im (Annapolis)Finite symmetries of quantum character stackshttps://zbmath.org/1526.570092024-02-15T19:53:11.284213Z"Keller, Corina"https://zbmath.org/authors/?q=ai:keller.corina"Müller, Lukas"https://zbmath.org/authors/?q=ai:muller.lukasLet \(k\) be an algebraically closed field of characteristic \(0\), \(G < \mathrm{GL}(n)\) a reductive algebraic group over \(k\), and \(\mathrm{Out}(G)\) the outer automorphism group of \(G\). For any group \(D\), denote by \(BD\) the classifying space of \(D\). Let \(\Sigma\) be an orientable surface with non-empty boundary (circles). Then the fundamental group \(\pi = \pi_1(\Sigma)\) is free on \(m\) generators.
The \(G\)-character variety \(\mathcal{M}(\Sigma)\) is the coarse moduli space of equivalent representations \(\Hom(\pi, G)/G \cong G^m/G\) and is of fundamental importance in algebraic, symplectic and mathematical physics. In particular, it corresponds to principal flat \(G\)-bundles \(P\) on \(\Sigma\).
Denote by \(\mathsf{Man}_2\) the \(\infty\)-category of 2-manifolds with open embedding as morphisms. \(\mathsf{Man}_2\) contains the subcategory of \(\mathsf{Disk}_2\) with objects homeomorphic to \(\mathbb{R}^2\). The frame bundle on \(\Sigma\) corresponds to a classifying map \(\Sigma \to B\mathrm{GL}(n)\) and this gives rise to the functor \(\tau\) and Cartesian product and defines \(\mathsf{Man}_2^G\):
\[
\begin{array}{ccc}
\mathsf{Man}_2^\mathcal{G} & \longrightarrow & \mathrm{Space}_{/BG} \\
\Big\downarrow & \boxtimes & \Big\downarrow \\
\mathsf{Man}_2 & \stackrel{\tau}\longrightarrow & \mathrm{Space}_{/B\mathrm{GL}(n)}.
\end{array}
\]
Let \((\mathcal{V}, \otimes)\) be a monoidal (and other minor adjectives) category. A \(\mathsf{Disk}_2\)-algebra is a functor \(\mathcal{A} : \mathsf{Disk}_2 \to \mathcal{V}\). The factorization homology \(\int_{\bullet} \mathcal{A}\) is the left Kan extension (colimit):
\[
\begin{array}{ccc}
\mathsf{Disk}_2^\mathcal{G} & \longrightarrow & \mathrm{Space}_{/BG} \\
\Big\downarrow & \nearrow _{\int_{\bullet} \mathcal{A}} & \\
\mathsf{Man}_2^\mathcal{G} & &
\end{array}
\]
It is the main result of [\textit{D. Ben-Zvi} et al., Sel. Math., New Ser. 24, No. 5, 4711--4748 (2018; Zbl 1456.17010)] that
\[
\mathrm{QCoh}(\mathcal{M}(\Sigma)) \cong \int_\Sigma \mathsf{Rep}(G).
\]
This is an important result as \(\mathrm{QCoh}(\mathcal{M}(\Sigma))\) essentially determines \(\mathcal{M}(\Sigma)\). Consequently, this equivalence naturally defines a quantum deformation of \(\mathcal{M}(\Sigma)\) by replacing \(\mathsf{Rep}(G)\) with \(\mathsf{Rep}_q(G)\) of locally finite \(U_q(\mathfrak{g})\)-modules.
Let \(D = \mathrm{Out}(G)\). The current paper under review extends the result by studying the \(D\)-action on \(D\)-twisted character variety: Let \(\varphi : \Sigma \to BD\). Since \(D\) is discrete, the corresponding principal bundle \(P \to \Sigma\) is flat; hence, \(\varphi\) gives rise to a \(\rho : \pi \to D\). Then \(\mathrm{Hom}_\rho(\pi, G)\) consists of representations \(\sigma\) making the following diagram commute
\[
\begin{array}{ccc}
& & G \rtimes D \\
& ^{\sigma} \nearrow & \Big\downarrow ^{\mathrm{pr}_2} \\
\pi & \stackrel{\rho}{\longrightarrow} & D
\end{array}
\]
There is a twisted \(G\)-conjugate action on \(\mathrm{Hom}_\rho(\pi, G)\) and the quotient is the twisted character variety \(\mathcal{M}_\rho(\Sigma)\). The paper then gives an explicit expression for \(\int_{\Sigma \stackrel{\varphi}{\to} BD} \mathsf{Rep}(G)\) and proves that it is equivalent to \(\mathrm{QCoh}(\mathcal{M}_\rho(\Sigma))\). Following [Ben-Zvi et al., loc. cit.], it then defines its quantum \(q\)-deformation as \(\int_{\Sigma \stackrel{\varphi}{\to} BD} \mathsf{Rep}_q(G)\)
After introducing the basic setting, the paper extends the \(D\)-action to \(\mathsf{Rep}(G)\) and \(\mathsf{Rep}_q(G)\). It then introduces the \(D\)-\(\mathsf{Man}_2\) in place of \(\mathsf{Man}_2\) to incorporate the \(D\)-action. After that the proof goes along the line of [D. Ben-Zvi et al., loc. cit.] which is combinatorial: It works out the case of \(S^1 \times \mathbb{R}\) and finally uses the excision axiom to complete the proof.
Reviewer: Eugene Xia (Tainan)Homotopy theory of net representationshttps://zbmath.org/1526.810352024-02-15T19:53:11.284213Z"Anastopoulos, Angelos"https://zbmath.org/authors/?q=ai:anastopoulos.angelos"Benini, Marco"https://zbmath.org/authors/?q=ai:benini.marco|benini.marco.1Summary: The homotopy theory of representations of nets of algebras over a (small) category with values in a closed symmetric monoidal model category is developed. We illustrate how each morphism of nets of algebras determines a change-of-net Quillen adjunction between the model categories of net representations, which is furthermore, a Quillen equivalence when the morphism is a weak equivalence. These techniques are applied in the context of homotopy algebraic quantum field theory with values in cochain complexes. In particular, an explicit construction is presented that produces constant net representations for Maxwell \(p\)-forms on a fixed oriented and time-oriented globally hyperbolic Lorentzian manifold.Category theory and theory of evolutionhttps://zbmath.org/1526.920392024-02-15T19:53:11.284213Z"Kozyrev, S. V."https://zbmath.org/authors/?q=ai:kozyrev.sergei-vSummary: The genome as a program and biological evolution as a computational process are discussed from the point of view of functional programming and the application of category theory in programming. Epigenetic code is considered as an imperative superstructure over genetics (functional programming). Biological evolution is discussed in relation to machine learning and non-deterministic computations.