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Published: 9 March 2023

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1

Positselski, Leonid, and Alexander Vishik. "Koszul duality and Galois cohomology." Mathematical Research Letters 2, no. 6 (1995): 771–81. http://dx.doi.org/10.4310/mrl.1995.v2.n6.a8.

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2

Hirsh, Joseph, and Joan Millès. "Curved Koszul duality theory." Mathematische Annalen 354, no. 4 (January 3, 2012): 1465–520. http://dx.doi.org/10.1007/s00208-011-0766-9.

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3

Mirković, Ivan, and Simon Riche. "Linear Koszul duality." Compositio Mathematica 146, no. 1 (December 16, 2009): 233–58. http://dx.doi.org/10.1112/s0010437x09004357.

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AbstractIn this paper we construct, for F1 and F2 subbundles of a vector bundle E, a ‘Koszul duality’ equivalence between derived categories of 𝔾m-equivariant coherent(dg-)sheaves on the derived intersection $F_1 \rcap _E F_2$, and the corresponding derived intersection $F_1^{\perp } \rcap _{E^*} F_2^{\perp }$. We also propose applications to Hecke algebras.
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4

Riche, Simon, Wolfgang Soergel, and Geordie Williamson. "Modular Koszul duality." Compositio Mathematica 150, no. 2 (December 13, 2013): 273–332. http://dx.doi.org/10.1112/s0010437x13007483.

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AbstractWe prove an analogue of Koszul duality for category$ \mathcal{O} $of a reductive group$G$in positive characteristic$\ell $larger than$1$plus the number of roots of$G$. However, there are no Koszul rings, and we do not prove an analogue of the Kazhdan–Lusztig conjectures in this context. The main technical result is the formality of the dg-algebra of extensions of parity sheaves on the flag variety if the characteristic of the coefficients is at least the number of roots of$G$plus$2$.
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5

Chuang, Joseph, Andrey Lazarev, and Wajid Mannan. "Koszul–Morita duality." Journal of Noncommutative Geometry 10, no. 4 (2016): 1541–57. http://dx.doi.org/10.4171/jncg/265.

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6

Beilinson, Alexander, Victor Ginzburg, and Wolfgang Soergel. "Koszul Duality Patterns in Representation Theory." Journal of the American Mathematical Society 9, no. 2 (1996): 473–527. http://dx.doi.org/10.1090/s0894-0347-96-00192-0.

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7

Etgü, Tolga, and Yankı Lekili. "Koszul duality patterns in Floer theory." Geometry & Topology 21, no. 6 (August 31, 2017): 3313–89. http://dx.doi.org/10.2140/gt.2017.21.3313.

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8

Tu, Junwu. "Matrix factorizations via Koszul duality." Compositio Mathematica 150, no. 9 (July 17, 2014): 1549–78. http://dx.doi.org/10.1112/s0010437x14007295.

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AbstractIn this paper we prove a version of curved Koszul duality for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Z}/2\mathbb{Z}$-graded curved coalgebras and their cobar differential graded algebras. A curved version of the homological perturbation lemma is also obtained as a useful technical tool for studying curved (co)algebras and precomplexes. The results of Koszul duality can be applied to study the category of matrix factorizations $\mathsf{MF}(R,W)$. We show how Dyckerhoff’s generating results fit into the framework of curved Koszul duality theory. This enables us to clarify the relationship between the Borel–Moore Hochschild homology of curved (co)algebras and the ordinary Hochschild homology of the category $\mathsf{MF}(R,W)$. Similar results are also obtained in the orbifold case and in the graded case.
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9

Bodzenta, Agnieszka, and Julian Külshammer. "Ringel duality as an instance of Koszul duality." Journal of Algebra 506 (July 2018): 129–87. http://dx.doi.org/10.1016/j.jalgebra.2018.03.025.

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10

DOTSENKO, VLADIMIR, and BRUNO VALLETTE. "HIGHER KOSZUL DUALITY FOR ASSOCIATIVE ALGEBRAS." Glasgow Mathematical Journal 55, A (October 2013): 55–74. http://dx.doi.org/10.1017/s0017089513000505.

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AbstractWe present a unifying framework for the key concepts and results of higher Koszul duality theory for N-homogeneous algebras: the Koszul complex, the candidate for the space of syzygies and the higher operations on the Yoneda algebra. We give a universal description of the Koszul dual algebra under a new algebraic structure. For that we introduce a general notion: Gröbner bases for algebras over non-symmetric operads.
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11

Chuang, Joseph, Andrey Lazarev, and W. H. Mannan. "Cocommutative coalgebras: homotopy theory and Koszul duality." Homology, Homotopy and Applications 18, no. 2 (2016): 303–36. http://dx.doi.org/10.4310/hha.2016.v18.n2.a17.

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12

Bellier, Olivia. "Koszul duality theory for operads over Hopf algebras." Algebraic & Geometric Topology 14, no. 1 (January 9, 2014): 1–35. http://dx.doi.org/10.2140/agt.2014.14.1.

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13

Martínez-Villa, Roberto, and Manuel Saorín. "A duality theorem for generalized Koszul algebras." Journal of Algebra 315, no. 1 (September 2007): 121–33. http://dx.doi.org/10.1016/j.jalgebra.2006.05.041.

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14

Madsen, Dag Oskar. "Quasi-hereditary algebras and generalized Koszul duality." Journal of Algebra 395 (December 2013): 96–110. http://dx.doi.org/10.1016/j.jalgebra.2013.08.005.

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15

Krause, Henning. "Koszul, Ringel and Serre duality for strict polynomial functors." Compositio Mathematica 149, no. 6 (March 18, 2013): 996–1018. http://dx.doi.org/10.1112/s0010437x12000814.

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AbstractThis is a report on recent work of Chałupnik and Touzé. We explain the Koszul duality for the category of strict polynomial functors and make explicit the underlying monoidal structure which seems to be of independent interest. Then we connect this to Ringel duality for Schur algebras and describe Serre duality for strict polynomial functors.
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16

MAZORCHUK, VOLODYMYR. "KOSZUL DUALITY FOR STRATIFIED ALGEBRAS II. STANDARDLY STRATIFIED ALGEBRAS." Journal of the Australian Mathematical Society 89, no. 1 (August 2010): 23–49. http://dx.doi.org/10.1017/s1446788710001497.

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AbstractWe give a complete picture of the interaction between the Koszul and Ringel dualities for graded standardly stratified algebras (in the sense of Cline, Parshall and Scott) admitting linear tilting (co)resolutions of standard and proper costandard modules. We single out a certain class of graded standardly stratified algebras, imposing the condition that standard filtrations of projective modules are finite, and develop a tilting theory for such algebras. Under the assumption on existence of linear tilting (co)resolutions we show that algebras from this class are Koszul, that both the Ringel and Koszul duals belong to the same class, and that these two dualities on this class commute.
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17

Maunder, James. "Koszul duality and homotopy theory of curved Lie algebras." Homology, Homotopy and Applications 19, no. 1 (2017): 319–40. http://dx.doi.org/10.4310/hha.2017.v19.n1.a16.

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18

Calaque, Damien, Giovanni Felder, Andrea Ferrario, and Carlo A. Rossi. "Bimodules and branes in deformation quantization." Compositio Mathematica 147, no. 1 (August 11, 2010): 105–60. http://dx.doi.org/10.1112/s0010437x10004847.

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AbstractWe prove a version of Kontsevich’s formality theorem for two subspaces (branes) of a vector space X. The result implies, in particular, that the Kontsevich deformation quantizations of S(X*) and ∧(X) associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet’s recent paper on Koszul duality in deformation quantization.
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19

Drozd, Yuriy, and Volodymyr Mazorchuk. "Koszul duality for extension algebras of standard modules." Journal of Pure and Applied Algebra 211, no. 2 (November 2007): 484–96. http://dx.doi.org/10.1016/j.jpaa.2007.01.014.

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20

Villa, Roberto Martínez, and Alex Martsinkovsky. "Stable Projective Homotopy Theory of Modules, Tails, and Koszul Duality." Communications in Algebra 38, no. 10 (September 30, 2010): 3941–73. http://dx.doi.org/10.1080/00927870903339980.

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21

Chen, Xiaojun, Youming Chen, Farkhod Eshmatov, and Song Yang. "Poisson cohomology, Koszul duality, and Batalin–Vilkovisky algebras." Journal of Noncommutative Geometry 15, no. 3 (October 15, 2021): 889–918. http://dx.doi.org/10.4171/jncg/425.

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22

Medvedev, Alice, and Ramin Takloo-Bighash. "An Invitation to Model-Theoretic Galois Theory." Bulletin of Symbolic Logic 16, no. 2 (June 2010): 261–69. http://dx.doi.org/10.2178/bsl/1286889126.

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AbstractWe carry out some of Galois' work in the setting of an arbitrary first-order theory T. We replace the ambient algebraically closed field by a large model M of T, replace fields by definably closed subsets of M, assume that T codes finite sets, and obtain the fundamental duality of Galois theory matching subgroups of the Galois group of L over F with intermediate extensions F ≤ K ≤ L. This exposition of a special case of [10] has the advantage of requiring almost no background beyond familiarity with fields, polynomials, first-order formulae, and automorphisms.
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23

Masuoka, Akira, and Tadashi Yanai. "Hopf module duality applied to X-outer Galois theory." Journal of Algebra 265, no. 1 (July 2003): 229–46. http://dx.doi.org/10.1016/s0021-8693(03)00130-3.

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24

Dascalescu, S., S. Raianu, and Y. H. Zhang. "Finite Hopf-Galois Coextensions, Crossed Coproducts, and Duality." Journal of Algebra 178, no. 2 (December 1995): 400–413. http://dx.doi.org/10.1006/jabr.1995.1356.

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25

Blumberg, Andrew J., and Michael A. Mandell. "Derived Koszul duality and involutions in the algebraic K -theory of spaces." Journal of Topology 4, no. 2 (2011): 327–42. http://dx.doi.org/10.1112/jtopol/jtr003.

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26

DAVEY, BRIAN A., JANE G. PITKETHLY, and ROSS WILLARD. "THE LATTICE OF ALTER EGOS." International Journal of Algebra and Computation 22, no. 01 (February 2012): 1250007. http://dx.doi.org/10.1142/s021819671100673x.

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We introduce a new Galois connection for partial operations on a finite set, which induces a natural quasi-order on the collection of all partial algebras on this set. The quasi-order is compatible with the basic concepts of natural duality theory, and we use it to turn the set of all alter egos of a given finite algebra into a doubly algebraic lattice. The Galois connection provides a framework for us to develop further the theory of natural dualities for partial algebras. The development unifies several fundamental concepts from duality theory and reveals a new understanding of full dualities, particularly at the finite level.
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27

Idrissi, Najib. "Curved Koszul duality of algebras over unital versions of binary operads." Journal of Pure and Applied Algebra 227, no. 3 (March 2023): 107208. http://dx.doi.org/10.1016/j.jpaa.2022.107208.

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28

Guan, Ai, and Andrey Lazarev. "Koszul duality for compactly generated derived categories of second kind." Journal of Noncommutative Geometry 15, no. 4 (December 7, 2021): 1355–71. http://dx.doi.org/10.4171/jncg/438.

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29

MONJARDET, BERNARD. "SOME ORDER DUALITIES IN LOGIC, GAMES AND CHOICES." International Game Theory Review 09, no. 01 (March 2007): 1–12. http://dx.doi.org/10.1142/s0219198907001242.

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We first present the concept of duality appearing in order theory, i.e. the notions of dual isomorphism and of Galois connection. Then, we describe two fundamental dualities, the duality extension/intention associated with a binary relation between two sets, and the duality between implicational systems and closure systems. Finally, we present two "concrete" dualities occuring in social choice and in choice functions theories.
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30

Gazaki, Evangelia. "A finer Tate duality theorem for local Galois symbols." Journal of Algebra 509 (September 2018): 337–85. http://dx.doi.org/10.1016/j.jalgebra.2018.05.007.

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31

Marti´nez Villa, Roberto, and Alex Martsinkovsky. "Cohomology of tails, Tate–Vogel cohomology, and noncommutative Serre duality over Koszul quiver algebras." Journal of Algebra 280, no. 1 (October 2004): 58–83. http://dx.doi.org/10.1016/j.jalgebra.2004.05.017.

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32

Gazaki, Evangelia. "A Tate duality theorem for local Galois symbols II; The semi-abelian case." Journal of Number Theory 204 (November 2019): 532–60. http://dx.doi.org/10.1016/j.jnt.2019.04.017.

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33

Stalder, Nicolas. "The semisimplicity conjecture for A-motives." Compositio Mathematica 146, no. 3 (March 18, 2010): 561–98. http://dx.doi.org/10.1112/s0010437x09004448.

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AbstractWe prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V𝔭(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G𝔭(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G𝔭(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V𝔭(M). The second states that the connected component of G𝔭(M) is reductive if M is semisimple and has a separable endomorphism algebra.
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34

Cais, Bryden. "The geometry of Hida families II: -adic -modules and -adic Hodge theory." Compositio Mathematica 154, no. 4 (March 8, 2018): 719–60. http://dx.doi.org/10.1112/s0010437x17007680.

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We construct the $\unicode[STIX]{x1D6EC}$-adic crystalline and Dieudonné analogues of Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology, and employ integral $p$-adic Hodge theory to prove $\unicode[STIX]{x1D6EC}$-adic comparison isomorphisms between these cohomologies and the $\unicode[STIX]{x1D6EC}$-adic de Rham cohomology studied in Cais [The geometry of Hida families I:$\unicode[STIX]{x1D6EC}$-adic de Rham cohomology, Math. Ann. (2017), doi:10.1007/s00208-017-1608-1] as well as Hida’s $\unicode[STIX]{x1D6EC}$-adic étale cohomology. As applications of our work, we provide a ‘cohomological’ construction of the family of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules attached to Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology by Dee [$\unicode[STIX]{x1D6F7}$–$\unicode[STIX]{x1D6E4}$modules for families of Galois representations, J. Algebra 235 (2001), 636–664], and we give a new and purely geometric proof of Hida’s finiteness and control theorems. We also prove suitable $\unicode[STIX]{x1D6EC}$-adic duality theorems for each of the cohomologies we construct.
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35

Gehrmann, Lennart, and Giovanni Rosso. "Big principal series, -adic families and -invariants." Compositio Mathematica 158, no. 2 (February 2022): 409–36. http://dx.doi.org/10.1112/s0010437x2200731x.

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In earlier work, the first named author generalized the construction of Darmon-style $\mathcal {L}$ -invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial coefficient system and Steinberg at a fixed prime. In this paper, assuming that the Archimedean component of the group has discrete series we show that these automorphic $\mathcal {L}$ -invariants can be computed in terms of derivatives of Hecke eigenvalues in $p$ -adic families. Our proof is novel even in the case of modular forms, which was established by Bertolini, Darmon and Iovita. The main new technical ingredient is the Koszul resolution of locally analytic principal series representations by Kohlhaase and Schraen. As an application of our results we settle a conjecture of Spieß: we show that automorphic $\mathcal {L}$ -invariants of Hilbert modular forms of parallel weight $2$ are independent of the sign character used to define them. Moreover, we show that they are invariant under Jacquet–Langlands transfer and, in fact, equal to the Fontaine–Mazur $\mathcal {L}$ -invariant of the associated Galois representation. Under mild assumptions, we also prove the equality of automorphic and Fontaine–Mazur $\mathcal {L}$ -invariants for representations of definite unitary groups of arbitrary rank. Finally, we study the case of Bianchi modular forms to show how our methods, given precise results on eigenvarieties, can also work in the absence of discrete series representations.
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36

Yamashita, Hiroshi. "On the rank of the first radical layer of a p-class group of an algebraic number field." Nagoya Mathematical Journal 156 (1999): 85–108. http://dx.doi.org/10.1017/s0027763000007078.

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Let p be a prime number. Let M be a finite Galois extension of a finite algebraic number field k. Suppose that M contains a primitive pth root of unity and that the p-Sylow subgroup of the Galois group G = Gal(M/k) is normal. Let K be the intermediate field corresponding to the p-Sylow subgroup. Let = Gal(K/k). The p-class group C of M is a module over the group ring ZpG, where Zp is the ring of p-adic integers. Let J be the Jacobson radical of ZpG. C/JC is a module over a semisimple artinian ring Fp. We study multiplicity of an irreducible representation Φ apperaring in C/JC and prove a formula giving this multiplicity partially. As application to this formula, we study a cyclotomic field M such that the minus part of C is cyclic as a ZpG-module and a CM-field M such that the plus part of C vanishes for odd p.To show the formula, we apply theory of central extensions of algebraic number field and study global and local Kummer duality between the genus group and the Kummer radical for the genus field with respect to M/K.
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37

DAVID, MARIE-CLAUDE, and NICOLAS M. THIÉRY. "EXPLORATION OF FINITE-DIMENSIONAL KAC ALGEBRAS AND LATTICES OF INTERMEDIATE SUBFACTORS OF IRREDUCIBLE INCLUSIONS." Journal of Algebra and Its Applications 10, no. 05 (October 2011): 995–1106. http://dx.doi.org/10.1142/s0219498811005099.

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We study the four infinite families KA(n), KB(n), KD(n), and KQ(n) of finite-dimensional Hopf (in fact Kac) algebras constructed, respectively, by A. Masuoka and L. Vainerman: isomorphisms, automorphism groups, self-duality, lattices of coideal sub-algebras. We reduce the study to KD(n) by proving that the others are isomorphic to KD(n), its dual, or an index 2 subalgebra of KD(2n). We derive many examples of lattices of intermediate subfactors of the inclusions of depth 2 associated to those Kac algebras, as well as the corresponding principal graphs, which is the original motivation. Along the way, we extend some general results on the Galois correspondence for depth 2 inclusions, and develop some tools and algorithms for the study of twisted group algebras and their lattices of coideal subalgebras. This research was driven by heavy computer exploration, whose tools and methodology we describe.
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38

Francis, John. "The tangent complex and Hochschild cohomology of -rings." Compositio Mathematica 149, no. 3 (December 10, 2012): 430–80. http://dx.doi.org/10.1112/s0010437x12000140.

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AbstractIn this work, we study the deformation theory of${\mathcal {E}}_n$-rings and the${\mathcal {E}}_n$analogue of the tangent complex, or topological André–Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence$A[n-1] \rightarrow T_A\rightarrow {\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)[n]$, relating the${\mathcal {E}}_n$-tangent complex and${\mathcal {E}}_n$-Hochschild cohomology of an${\mathcal {E}}_n$-ring$A$. We give two proofs: the first is direct, reducing the problem to certain stable splittings of configuration spaces of punctured Euclidean spaces; the second is more conceptual, where we identify the sequence as the Lie algebras of a fiber sequence of derived algebraic groups,$B^{n-1}A^\times \rightarrow {\mathrm {Aut}}_A\rightarrow {\mathrm {Aut}}_{{\mathfrak B}^n\!A}$. Here${\mathfrak B}^n\!A$is an enriched$(\infty ,n)$-category constructed from$A$, and${\mathcal {E}}_n$-Hochschild cohomology is realized as the infinitesimal automorphisms of${\mathfrak B}^n\!A$. These groups are associated to moduli problems in${\mathcal {E}}_{n+1}$-geometry, a less commutative form of derived algebraic geometry, in the sense of the work of Toën and Vezzosi and the work of Lurie. Applying techniques of Koszul duality, this sequence consequently attains a nonunital${\mathcal {E}}_{n+1}$-algebra structure; in particular, the shifted tangent complex$T_A[-n]$is a nonunital${\mathcal {E}}_{n+1}$-algebra. The${\mathcal {E}}_{n+1}$-algebra structure of this sequence extends the previously known${\mathcal {E}}_{n+1}$-algebra structure on${\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)$, given in the higher Deligne conjecture. In order to establish this moduli-theoretic interpretation, we make extensive use of factorization homology, a homology theory for framed$n$-manifolds with coefficients given by${\mathcal {E}}_n$-algebras, constructed as a topological analogue of Beilinson and Drinfeld’s chiral homology. We give a separate exposition of this theory, developing the necessary results used in our proofs.
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39

Li, Haonan, and Quanshui Wu. "Generalized Koszul Algebra and Koszul Duality." Journal of Algebra, December 2022. http://dx.doi.org/10.1016/j.jalgebra.2022.12.023.

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40

Cassella, Alberto, and Claudio Quadrelli. "Right-angled Artin groups and enhanced Koszul properties." Journal of Group Theory, August 25, 2020. http://dx.doi.org/10.1515/jgth-2020-0049.

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AbstractLet 𝔽 be a finite field. We prove that the cohomology algebra H^{\bullet}(G_{\Gamma},\mathbb{F}) with coefficients in 𝔽 of a right-angled Artin group G_{\Gamma} is a strongly Koszul algebra for every finite graph Γ. Moreover, H^{\bullet}(G_{\Gamma},\mathbb{F}) is a universally Koszul algebra if, and only if, the graph Γ associated to the group G_{\Gamma} has the diagonal property. From this, we obtain several new examples of pro-𝑝 groups, for a prime number 𝑝, whose continuous cochain cohomology algebra with coefficients in the field of 𝑝 elements is strongly and universally (or strongly and non-universally) Koszul. This provides new support to a conjecture on Galois cohomology of maximal pro-𝑝 Galois groups of fields formulated by J. Mináč et al.
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41

Ayala, David, and John Francis. "ZERO-POINTED MANIFOLDS." Journal of the Institute of Mathematics of Jussieu, July 2, 2019, 1–74. http://dx.doi.org/10.1017/s1474748019000343.

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We formulate a theory of pointed manifolds, accommodating both embeddings and Pontryagin–Thom collapse maps, so as to present a common generalization of Poincaré duality in topology and Koszul duality in${\mathcal{E}}_{n}$-algebra.
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42

Asensouyis, Hassan, Jilali Assim, Zouhair Boughadi, and Youness Mazigh. "Poitou–Tate duality for totally positive Galois cohomology." Communications in Algebra, April 25, 2022, 1–22. http://dx.doi.org/10.1080/00927872.2022.2060995.

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43

Hartonas, Chrysafis. "Duality for normal lattice expansions and sorted residuated frames with relations." Algebra universalis 84, no. 1 (January 29, 2023). http://dx.doi.org/10.1007/s00012-023-00802-y.

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AbstractWe revisit the problem of Stone duality for lattices with quasioperators, presenting a fresh duality result. The new result is an improvement over that of our previous work in two important respects. First, the axiomatization of frames is now simplified, partly by incorporating Gehrke’s proposal of section stability for relations. Second, morphisms are redefined so as to preserve Galois stable (and co-stable) sets and we rely for this, partly again, on Goldblatt’s recently proposed definition of bounded morphisms for polarities. In studying the dual algebraic structures associated to polarities with relations we demonstrate that stable/co-stable set operators result as the Galois closure of the restriction of classical (though sorted) image operators generated by the frame relations to Galois stable/co-stable sets. This provides a proof, at the representation level, that non-distributive logics can be regarded as fragments of sorted residuated (poly)modal logics, a research direction recently initiated by this author.
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44

Tamaroff, Pedro. "Resolutions of operads via Koszul (bi)algebras." Journal of Homotopy and Related Structures, March 3, 2022. http://dx.doi.org/10.1007/s40062-022-00302-1.

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AbstractWe introduce a construction that produces from each bialgebra H an operad $$\mathsf {Ass}_H$$ Ass H controlling associative algebras in the monoidal category of H-modules or, briefly, H-algebras. When the underlying algebra of this bialgebra is Koszul, we give explicit formulas for the minimal model of this operad depending only on the coproduct of H and the Koszul model of H. This operad is seldom quadratic—and hence does not fall within the reach of Koszul duality theory—so our work provides a new rich family of examples where an explicit minimal model of an operad can be obtained. As an application, we observe that if we take H to be the mod-2 Steenrod algebra $${\mathscr {A}}$$ A , then this notion of an associative H-algebra coincides with the usual notion of an $$\mathscr {A}$$ A -algebra considered by homotopy theorists. This makes available to us an operad $$\mathsf {Ass}_{{\mathscr {A}}}$$ Ass A along with its minimal model that controls the category of associative $${\mathscr {A}}$$ A -algebras, and the notion of strong homotopy associative $${\mathscr {A}}$$ A -algebras.
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45

Méndez, Miguel A., and Rafael Sánchez Lamoneda. "Monops, Monoids and Operads: The Combinatorics of Sheffer Polynomials." Electronic Journal of Combinatorics 25, no. 3 (August 10, 2018). http://dx.doi.org/10.37236/7686.

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We introduce a new algebraic construction, monop, that combines monoids (with respect to the product of species), and operads (monoids with respect to the substitution of species) in the same algebraic structure. By the use of properties of cancellative set-monops we construct a family of partially ordered sets whose prototypical examples are the Dowling lattices. They generalize the enriched partition posets associated to a cancellative operad, and the subset posets associated to a cancellative monoid. Their Whitney numbers of the first and second kind are the connecting coefficients of two umbral inverse Sheffer sequences with the family of powers $\{x^n\}_{n=0}^{\infty}$. Equivalently, the entries of a Riordan matrix and its inverse. This aticle is the first part of a program in progress to develop a theory of Koszul duality for monops.
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46

Dyckerhoff, Tobias, Gustavo Jasso, and Yankι Lekili. "The symplectic geometry of higher Auslander algebras: Symmetric products of disks." Forum of Mathematics, Sigma 9 (2021). http://dx.doi.org/10.1017/fms.2021.2.

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AbstractWe show that the perfect derived categories of Iyama’sd-dimensional Auslander algebras of type${\mathbb {A}}$are equivalent to the partially wrapped Fukaya categories of thed-fold symmetric product of the$2$-dimensional unit disk with finitely many stops on its boundary. Furthermore, we observe that Koszul duality provides an equivalence between the partially wrapped Fukaya categories associated to thed-fold symmetric product of the disk and those of its$(n-d)$-fold symmetric product; this observation leads to a symplectic proof of a theorem of Beckert concerning the derived Morita equivalence between the corresponding higher Auslander algebras of type${\mathbb {A}}$. As a by-product of our results, we deduce that the partially wrapped Fukaya categories associated to thed-fold symmetric product of the disk organise into a paracyclic object equivalent to thed-dimensional Waldhausen$\text {S}_{\bullet }$-construction, a simplicial space whose geometric realisation provides thed-fold delooping of the connective algebraicK-theory space of the ring of coefficients.
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