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Monoidal structures arising from $B_\infty$-algebras with applications to Hopf algebras

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Pith's one-line read A B∞-structure gives derived right-module categories a tensor product.

desk verdict Theorem A gives a genuine new monoidal construction, but Theorem 5.9's monoidal equivalence rests on an unproved compactness assertion that the authors need to justify or replace. read the letter →

arxiv 2608.08511 v1 pith:XCH7GHQL submitted 2026-08-09 math.RT math.QA

Gongxiang Liu , Zhengfang Wang , Mengdie Zhang This is my paper · ORCID
classification math.RTmath.QA MSC 16E0513D0316G2016E60
keywords B∞-algebraA∞-modulederivedcategorymonoidaltriangulatedHopfalgebraKoszuldualityYonedadgbrace
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

An ordinary noncommutative algebra has no tensor product on right modules alone, since tensoring over the algebra needs a compatible left action on the second factor. This paper proves that a $B_\infty$-structure—an $A_\infty$-algebra whose bar construction is a dg bialgebra—repairs exactly that missing action: the antipode of the cofree dg Hopf algebra induces an induction functor $\iota$ that turns every right $A_\infty$-module into an $A_\infty$-bimodule, and the formula $M\boxtimes_A N = M\otimes^\infty_A \iota(N)$ makes the derived category of right $A_\infty$-modules a monoidal triangulated category. Applied to a finite-dimensional Hopf algebra $H$, the Yoneda dg algebra $\mathcal{Y}(\Bbbk,\Bbbk)$ of the trivial module carries a brace $B_\infty$-structure coming from the coproduct, and the Koszul duality functor from injective $H$-complexes to modules over that Yoneda algebra becomes a monoidal triangulated equivalence wherever the trivial module's injective resolution generates. A reader should care because this supplies an explicit, purely algebraic construction of a tensor structure in settings where none was available, and it gives a new proof of a monoidal equivalence previously obtained through classifying spaces.

What carries the argument

The load-bearing object is the induction functor $\iota$ from right $A_\infty$-modules to $A_\infty$-bimodules, produced by the antipode of the dg Hopf algebra $T^c(sA)$ associated to a $B_\infty$-algebra. Explicitly, $\varphi=\widehat{\mu}(S\otimes 1)$ is a dg coalgebra morphism $T^c(sA)^{\mathrm{op}}\otimes T^c(sA)\to T^c(sA)$, and $\iota$ sends a right module $(M,\rho)$ to the bimodule whose left structure is obtained by feeding the right action through $\varphi$; for brace $B_\infty$-algebras a comparison theorem identifies $\iota(M)$ with the original dg bimodule when $M$ is a brace module. These explicit quasi-isomorphisms, verified by string diagrams, carry the monoidal axioms: they supply the unit constraint, the pentagon, and the triangle identity, and in the Hopf application they identify the derived tensor product over the Yoneda algebra with the diagonal tensor product of injective $H$-complexes.

What would settle it

Test compactness of $\mathbf{1}=\mathcal{Y}(H,\Bbbk)$ in $K(\mathrm{Inj}\text{-}H)$: for a family of injective complexes $\{I_i\}$, check whether the natural map $\bigoplus_i \mathrm{Hom}_{K(\mathrm{Inj}\text{-}H)}(\mathbf{1}, I_i)\to \mathrm{Hom}_{K(\mathrm{Inj}\text{-}H)}(\mathbf{1}, \bigoplus_i I_i)$ is an isomorphism. A single family where it is not disproves the unstated assumption used in the proof; conversely, proving it for all families in a non-local example would show the monoidal equivalence extends beyond the local case.

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Extended reading notes

Core claim

The central claim is that if $A$ is a $B_\infty$-algebra, the derived category $\mathcal{D}^\mathrm{r}_\infty(A)$ of right $A_\infty$-modules is a monoidal triangulated category with unit $A$ and tensor product $M\boxtimes_A N = M\otimes^\infty_A \iota(N)$. The functor $\iota$ is built from the antipode $S$ of the cofree dg Hopf algebra $T^c(sA)$ via the coalgebra map $\varphi = \widehat{\mu}(S\otimes 1)$, and the hard part is that $\iota$ is faithful but not full, so the unit and associativity constraints must be explicit $A_\infty$-bimodule quasi-isomorphisms $\iota(A)\simeq A$ and $\iota(M)\otimes^\infty_A \iota(N)\simeq \iota(M\otimes^\infty_A \iota(N))$. The paper further claims that for a finite-dimensional Hopf algebra $H$, the Yoneda dg algebra $\mathcal{Y}(\Bbbk,\Bbbk)$ is a brace $B_\infty$-algebra and the Koszul duality functor $F=\mathrm{Hom}_H(\mathcal{Y}(H,\Bbbk),-)$ is triangulated lax monoidal; restricted to the localizing subcategory generated by $\mathcal{Y}(H,\Bbbk)$ it is a monoidal triangulated equivalence, and if $H$ is local then $F$ itself is such an equivalence.

Load-bearing premise

The proof of Theorem 5.9 assumes without proof or citation that the injective resolution $\mathcal{Y}(H,\Bbbk)$ is compact in the homotopy category $K(\mathrm{Inj}\text{-}H)$; if this object fails to be compact, the derived Morita argument that identifies the localizing subcategory it generates with $\mathcal{D}(\mathcal{Y}(\Bbbk,\Bbbk))$ no longer applies.

Editorial extensions

If this is right

  • Every $B_\infty$-algebra $A$ yields a monoidal triangulated derived category of right $A_\infty$-modules, with a tensor product defined by a concrete formula rather than by abstract transfer.
  • For a finite-dimensional Hopf algebra $H$, the derived category of the Yoneda dg algebra $\mathcal{D}(\mathcal{Y}(\Bbbk,\Bbbk))$ is a closed monoidal triangulated category, with brace operations inherited from the coproduct.
  • The Koszul duality functor $K(\mathrm{Inj}\text{-}H)\to \mathcal{D}(\mathcal{Y}(\Bbbk,\Bbbk))$ is triangulated and lax monoidal on all of $K(\mathrm{Inj}\text{-}H)$, and becomes a true monoidal equivalence on the localizing subcategory generated by the injective resolution of the trivial module.
  • When $H$ is local, the Koszul duality functor is a monoidal triangulated equivalence, reproducing the Benson–Krause monoidal equivalence by purely algebraic means.
  • For graded-commutative algebras with trivial braces the construction recovers the usual derived tensor product; for non-local Hopf algebras the laxness is real, so the localizing restriction is essential.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If compactness of $\mathcal{Y}(H,\Bbbk)$ in $K(\mathrm{Inj}\text{-}H)$ holds beyond the local case, the monoidal equivalence would extend from the localizing subcategory to all injective complexes, making Koszul duality a monoidal invariant of arbitrary finite-dimensional Hopf algebras.
  • The examples show the product remembers the coproduct, not just the underlying algebra or the Ext-algebra; this suggests the monoidal structure can distinguish Hopf structures that ordinary cohomological invariants cannot.
  • The same induction construction should work for dg bialgebras or homotopy-coherent Hopf objects, transferring the monoidal structure to derived categories of modules in settings where antipodes are only defined up to homotopy; this is testable because the only ingredient needed is a dg coalgebra morphism $\varphi$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. This paper constructs monoidal triangulated structures on derived categories of right A-infinity modules over a B-infinity algebra. For a B-infinity algebra A, the authors define an induction functor iota from right A-infinity modules to A-infinity bimodules using the antipode of the associated dg Hopf algebra T^c(sA), and set M ⊠_A N = M ⊗^∞_A iota(N). They prove Theorem A (Theorem 4.1) that (D^r_∞(A), ⊠_A, A) is a monoidal triangulated category, with unit and associativity constraints given by explicit bimodule quasi-isomorphisms. For brace B-infinity algebras they prove a comparison theorem (Theorem 4.14) identifying iota(M) with the original dg bimodule. The main application is to finite-dimensional Hopf algebras: the Yoneda dg algebra E = Y(k,k) is shown to carry a brace B-infinity structure, and the Koszul duality functor F = Hom_H(Y(H,k), -) : K(Inj-H) -> D(E) is triangulated lax monoidal; its restriction to the localizing subcategory generated by Y(H,k) is claimed to be a monoidal triangulated equivalence (Theorem 5.9), with a monoidal equivalence in the local case (Corollary 5.11), recovering the Krause/Benson-Krause equivalence. The final section treats graded-commutative algebras, a non-local Hopf algebra where laxness is strict, and elementary 2-groups with coproduct-dependent brace operations.

Significance. If the proofs are completed, this paper provides a concrete A-infinity-algebraic framework for tensor products on right module categories and a purely algebraic proof of a monoidal equivalence previously obtained through classifying spaces. The paper's strengths are its explicit formulas for iota, the unit and associativity quasi-isomorphisms, and the informative examples, especially the demonstration that the monoidal structure can depend on the Hopf coproduct. The construction is not an abstract transfer along a fully faithful embedding; the authors explicitly address the non-fullness of iota. However, the load-bearing compactness assertion in the proof of Theorem 5.9 and the omitted verifications in Proposition 5.1 and Theorem 4.14 currently prevent the manuscript from being fully convincing as written.

major comments (3)
  1. [Section 5.4, proof of Theorem 5.9] In the proof of Theorem 5.9, immediately after Lemma 5.8, the sentence 'The object 1 is compact' is asserted without proof or citation. Compactness of 1 = Y(H,k) in K(Inj-H) is load-bearing: it is the hypothesis that makes the 'standard derived Morita argument' identify F|Loc(1) with a triangulated equivalence, and without it F need not preserve coproducts. Since 1 is an unbounded complex of finite-dimensional injectives, compactness is not automatic and must be proved. Please supply a proof or a precise reference (for instance, via the equivalence K(Inj-H) ≃ D(H) and compactness of k in D(H)). The related generation assertion used in the proof of Corollary 5.11 also needs justification.
  2. [Section 5.2, Proposition 5.1] Proposition 5.1 asserts that E = Y(k,k) is a brace B-infinity algebra and that Y(X,Y) is a brace B-infinity module over E, but the proof ends with 'We omit the routine sign verification' for the module identities, and the algebra part is transferred from the standard brace structure on C*(H,H). The module identities are load-bearing: Theorem 5.3 and Lemma 5.8 use the brace module structure to identify iota(F(X)) with F(X), and hence the lax monoidal structure in Theorem 5.9 depends on it. The authors should write out the verification of the brace-module identities (Definition 4.11) or cite a source that covers exactly this module case.
  3. [Section 4.1, Theorem 4.14] The proof of Theorem 4.14 says that the verification that I_M is an A-infinity-bimodule morphism is 'the same graphical computation' as in Theorem 3.8, with the brace-module identities replacing the corresponding brace-algebra identities. These computations are not literally the same: the brace-module identities of Definition 4.11 (higher pre-Jacobi, distributivity, higher homotopy) are involved in a different configuration from the brace-algebra identities used in Theorem 3.8. Since Theorem 4.14 is the bridge between the abstract construction and the Hopf-algebra applications, the proof should indicate which axioms are used at which step, or provide the computation.
minor comments (3)
  1. [References] Reference [3] contains the typo 'constrcutions' and reference [28] contains the typo 'strcutre'; both should be corrected.
  2. [Section 5.2, formula (5.4)] The sign epsilon_q in (5.4) is presented without derivation; a short explanation of how it follows from the Koszul convention stated in the introduction would improve reproducibility.
  3. [Section 5.3] The symbol 1 is used both for the unit object of the monoidal structure and for the injective resolution Y(H,k); given how central this object is, a distinct notation such as ℒ or Ι might reduce ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the monoidal structure is explicitly constructed from the B-infinity data, and self-citations are background results rather than imported conclusions.

full rationale

The derivation chain is self-contained against its own inputs. Theorem 4.1 defines M⊠_A N = M⊗^∞_A iota(N), but it does not assume monoidality: the unit constraint is proved in Theorem 3.8 via explicit quasi-isomorphisms iota(A)≃A, and associativity is proved in Lemma 4.3 and Proposition 4.5 using explicit A∞-bimodule maps, not by transferring a known monoidal structure. For the Hopf application, E=Y(k,k) is a concrete dg algebra, and its brace B∞-structure is constructed from the coproduct through the embedding into Hochschild cochains (Proposition 5.1), citing the standard brace B∞-structure on C*(H,H); this is external and independent. The identification E≃Hom_H(1,1) is cited from Chen-Wang [9, Prop. 3.11] and is a Yoneda dg algebra fact, not equivalent to the monoidal equivalence being proved; even though [9] shares an author, it is not the target result and does not smuggle in the conclusion. The lax monoidal morphism η comes from the multiplicative injective resolution (Propositions 5.6 and 5.7), and the localizing-subcategory argument checks that the structure maps are isomorphisms, using exactness and coproduct preservation rather than a fitted parameter. The one unproved assertion, compactness of 1 in the proof of Theorem 5.9, is a support/correctness concern, not a circular reduction: no equation of the paper reduces to its own input. No fitted input is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted parameters or invented entities appear; the input is a B-infinity structure on A or a Hopf algebra H. The main unproved premise is the compactness of 1 = Y(H,k), which is asserted in the proof of Theorem 5.9. The omitted sign checks in Proposition 5.1 and the deferred verification in Theorem 4.14 are additional structural gaps that a full proof should close.

assumptions (6)
  • domain assumption k is a field and all A-infinity algebras, modules, and bimodules are strictly unital.
    Stated at the end of Section 1 and throughout Section 2. Standard for the intended applications and not restrictive.
  • standard math The dg Hopf algebra associated to a B-infinity algebra admits an antipode satisfying the Hopf identities (3.2) through (3.8).
    The tensor coalgebra T^c(sA) is connected with respect to tensor length, so an antipode exists; used in Proposition 3.5 to construct the induction functor.
  • domain assumption For a finite-dimensional Hopf algebra, the tensor product of injective modules is injective, so K(Inj-H) is a monoidal triangulated category.
    Invoked at the start of Section 5; follows from H being a Frobenius algebra.
  • ad hoc to paper The object 1 = Y(H,k) is compact in K(Inj-H).
    Asserted in Section 5.4 without proof or citation; needed for the standard derived Morita argument that identifies Loc(1) with D(E). This is the weakest load-bearing premise.
  • ad hoc to paper The brace-module identities for Y(X,Y), including the brace B-infinity structure on E = Y(k,k), hold as stated.
    Proposition 5.1 states that the identities follow by the same computation and omits the routine sign verification.
  • standard math Every A-infinity module over a dg algebra is A-infinity quasi-isomorphic to a dg module, and the A-infinity tensor product agrees with the derived tensor product for dg bimodules.
    Used in Remark 4.13 and Theorem 5.9 to identify F(X) tensor F(Y) with the derived tensor product over E.

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Pith. "Pith review of Monoidal structures arising from $B_\infty$-algebras with applications to Hopf algebras." pith.science (2026). https://pith.science/paper/XCH7GHQL

@misc{pith2026260808511,
  author       = {Pith},
  title        = {Pith review of: Monoidal structures arising from $B_\infty$-algebras with applications to Hopf algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XCH7GHQL}},
  note         = {Machine review of arXiv:2608.08511}
}
abstract

We give an explicit construction of monoidal structures on derived categories of right $A_\infty$-modules over an $A_\infty$-algebra $A$ equipped with a $B_\infty$-structure. Given such a $B_\infty$-algebra $A$, we construct an induction functor \[\iota\colon \mathcal{D}^{\rm{r}}_\infty(A) \longrightarrow \mathcal{D}^{\rm{bi}}_\infty(A)\] from right $A_\infty$-modules to $A_\infty$-bimodules and define \[M\boxtimes_A N=M\overset{\infty}{\otimes}_A\iota(N).\] We prove that $(\mathcal{D}^{\rm{r}}_\infty(A),\boxtimes_A,A)$ is a monoidal triangulated category: the unit and associativity constraints are induced by explicit quasi-isomorphisms of $A_\infty$-bimodules, including \[\iota(A)\simeq A \qquad\text{and}\qquad \iota(M)\overset{\infty}{\otimes}_A\iota(N)\simeq \iota(M\boxtimes_A N).\] We apply this construction to finite-dimensional Hopf algebras $H$, the Yoneda dg algebra $\mathcal{Y}(\Bbbk,\Bbbk)$ of the trivial $H$-module carries a natural brace $B_\infty$-structure, and hence its derived category carries the monoidal structure constructed above. We show that the Koszul duality functor \[\rm{Hom}_H(\mathcal{Y}(H,\Bbbk),-)\colon \mathcal{K}(\rm{Inj}\text{-}H)\longrightarrow \mathcal{D}(\mathcal{Y}(\Bbbk,\Bbbk))\] is triangulated lax monoidal, and that its restriction to the localizing subcategory generated by the injective resolution $\mathcal{Y}(H,\Bbbk)$ is a monoidal triangulated equivalence. If $H$ is local, this localizing subcategory is all of $\mathcal{K}(\rm{Inj}\text{-}H)$. In particular, this gives an alternative, purely algebraic proof of the monoidal equivalence conjectured by Krause and established by Benson--Krause through the classifying space $BG$. Finally, our examples recover the usual tensor product for graded commutative algebras and show that the resulting brace $B_\infty$ and monoidal structures can depend essentially on the chosen Hopf structure.

Figures

Figures reproduced from arXiv: 2608.08511 by the authors.

Figure 1
Figure 1. The left-hand side of (4.2) when q = 0 [PITH_FULL_IMAGE:figures/full_fig_p030_1.png] view at source ↗
Figure 2
Figure 2. The right-hand side of (4.2) when q = 0. We now compare the summands in Figures 1 and 2. Observe that: • The first two summands in [PITH_FULL_IMAGE:figures/full_fig_p031_2.png] view at source ↗

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Works this paper leans on

118 extracted references · 73 canonical work pages

  1. [1]

    Abrams, and G

    G. Abrams, and G. Aranda Pino , The Leavitt path algebra of a graph , J. Algebra 293 (2) (2005), 319--334

    2005

  2. [2]

    Abrams, and G

    G. Abrams, and G. Aranda Pino , The Leavitt path algebras of arbitrary graphs , Houston J. Math. 34 (2) (2008), 423--442

    2008

  3. [3]

    Alahmadi, H

    A. Alahmadi, H. Alsulami, S.K. Jain and E. Zelmanov, Leavitt path algebras of finite Gelfand-Kirillov dimension, J. Algebra Appl. 11 (6) (2012), 1250--225

    2012

  4. [4]

    Abbaspour , On algebraic structures of the Hochschild complex , Free loop spaces in geometry and topology, 165--222, IRMA Lect

    H. Abbaspour , On algebraic structures of the Hochschild complex , Free loop spaces in geometry and topology, 165--222, IRMA Lect. Math. Theor. Phys. 24 , Eur. Math. Soc. Z\"urich, 2015

    2015

  5. [5]

    Abrams, A

    G. Abrams, A. Louly, E. Pardo, and C. Smith, Flow invariants in the classification of Leavitt path algebras , J. Algebra 333 (2011), 202--231

    2011

  6. [6]

    Abrams, F

    G. Abrams, F. Mantese, and A. Tonolo , Extensions of simple modules over Leavitt path algebras , J. Algebra 431 (2014), 78--106

    2014

  7. [7]

    Abrams, and K.M

    G. Abrams, and K.M. Rangaswamy , Row-finite equivalents exist only for row-countable graphs , Contemp. Math. 562 (2012), 1--10

    2012

  8. [8]

    Alahmadi, H

    A. Alahmadi, H. Alsulami, S.K. Jain, and E. Zelmanov , Leavitt path algebras of finite Gelfand-Kirillov dimension , J. Algebra Appl. 11 (6) (2012), 6pp

    2012

Show all 118 references
  1. [9]

    Ara, M.A

    P. Ara, M.A. Gonzalez-Barroso, K.R. Goodearl, and E. Pardo , Fractional skew monoid rings , J. Algebra 278 (1) (2004), 104--126

    2004

  2. [10]

    Ara, M.A

    P. Ara, M.A. Moreno, and E. Pardo , Nonstable K -theory for graph algebras , Algebr. Represent. Theor. 10 (2) (2007), 157--178

    2007

  3. [11]

    Aranda Pino, and K

    G. Aranda Pino, and K. Crow , The center of a Leavitt path algebra , Rev. Mat. Iberoam. 27 (2) (2011), 621--644

    2011

  4. [12]

    Avramov, and R.-O

    L.L. Avramov, and R.-O. Buchweitz , Support varieties and cohomology over complete intersections , Invent. Math. 142 (2) (2000), 285--318

    2000

  5. [13]

    Balmer , The spectrum of prime ideals in tensor triangulated categories, J

    P. Balmer , The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005), 149–168

    2005

  6. [14]

    Balmer , Spectra, spectra, spectra - Tensor triangular spectra versus Zariski spectra of endomorphism rings, Algebr

    P. Balmer , Spectra, spectra, spectra - Tensor triangular spectra versus Zariski spectra of endomorphism rings, Algebr. Geom. Topol. 10 (2010), 1521–1563

    2010

  7. [15]

    Baues , The double bar and cobar constrcutions, Compos

    H.J. Baues , The double bar and cobar constrcutions, Compos. Math. 43 (1981), 331-341

    1981

  8. [16]

    Benson, S.B

    D.J. Benson, S.B. Iyengar and H. Krause , Local cohomology and support for triangulated categories , Ann. Sci. \' E c. Norm. Sup\' e r. 41 (2008), 573–619

    2008

  9. [17]

    Benson, S.B

    D.J. Benson, S.B. Iyengar and H. Krause , Stratifying modular representations of finite groups , Ann. of Math. 174 (2011), 1643-1684

    2011

  10. [18]

    Benson and H

    D.J. Benson and H. Krause , Complexes of injective kG -modules , Algebra Number Theory 2 (2008), 1-30

    2008

  11. [19]

    Barmeier, and Z

    S. Barmeier, and Z. Wang , Deformations of path algebras of quivers with relations , arXiv:2002.10001v2

    2002 arXiv

  12. [20]

    Biglari , A K\"unneth formula in tensor triangulated categories , J

    S. Biglari , A K\"unneth formula in tensor triangulated categories , J. Pure Appl. Algebra. 210 (2007), 645–650

    2007

  13. [21]

    Blanc, M

    A. Blanc, M. Robalo, B. T\" o en, and G. Vezzosi , Motivic realizations of singularity categories and vanishing cycles , J. \' E c. polytech. Math. 5 (2018), 651--747

    2018

  14. [22]

    Brou\'e , Equivalences of blocks of group algebras , Finite dimensional algebras and related topics

    M. Brou\'e , Equivalences of blocks of group algebras , Finite dimensional algebras and related topics. Proceedings of the NATO Advanced Research Workshop on Representations of algebras and related topics. Ottawa, Canada, August 10-18, 1992, 1994, 1--26

    1992

  15. [23]

    A.B. Buan, H. Krause, and . Solberg ,

  16. [24]

    Buchweitz , Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings , http://hdl.handle.net/1807/16682, Univ

    R.O. Buchweitz , Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings , http://hdl.handle.net/1807/16682, Univ. Hannover, 1986

    1986

  17. [25]

    Buchweitz, and H

    R.O. Buchweitz, and H. Flenner , Global Hochschild (co-)homology of singular spaces , Adv. Math. 217 (2008), 205--242

    2008

  18. [26]

    Chen , The singularity category of an algebra with radical square zero , Doc

    X.-W. Chen , The singularity category of an algebra with radical square zero , Doc. Math. 16 (2011), 921--936

    2011

  19. [27]

    Chen, H.H

    X.-W. Chen, H.H. Li and Z. Wang , Leavitt path algebras, B_ -algebras and Keller's conjecture for singular Hochschild cohomology , Mem. Amer. Math. Soc. 313 (2025)

    2025

  20. [28]

    Chen and Z

    X.-W. Chen and Z. Wang , The singular Yoneda category and the stabilization functor , Math. Z. 308 (2024)

    2024

  21. [29]

    Chen and Z

    X.-W. Chen and Z. Wang , The dg Leavitt path algebra, singular Yoneda category and singularity category, with an appendix by Bernhard Keller and Yu Wang , Adv. Math. 440 (2024)

    2024

  22. [30]

    Chen, and D

    X.-W. Chen, and D. Yang , Homotopy categories, Leavitt path algebras and Gorenstein projective modules , Inter. Math. Res. Not. IMRN 10 (2015), 2597--2633

    2015

  23. [31]

    Clark, C

    L.O. Clark, C. Farthing, A. Sims, and M. Tomforde , A groupoid generalisation of Leavitt path algebras , Semigroup Forum 89 (2014), 501--517

    2014

  24. [32]

    Clark, D

    L.O. Clark, D. Martin Barguero, C. Martin Gonzales, and M. Siles Molina , Using the Steinberg algebra model to determine the center of any Leavitt path algebra , arXiv: 1604.01079v1, 2016

    2016 arXiv

  25. [33]

    Cuntz, and D

    J. Cuntz, and D. Quillen , Algebra extensions and nonsingularity , J. Amer. Math. Soc. 8 (2) (1995), 251--289

    1995

  26. [34]

    de Thanhoffer de Volcsey, and M

    L. de Thanhoffer de Volcsey, and M. Van den Bergh , Explicit models for some stable categories of maximal Cohen-Macaulay modules , Math. Res. Lett. 23 (5) (2016), 1507--1526

    2016

  27. [35]

    Drinfeld , DG quotients of DG categories , J

    V. Drinfeld , DG quotients of DG categories , J. Algebra 272 (2) (2004), 643--691

    2004

  28. [36]

    Dyckerhoff , Compact generator in categories of matrix factorizations , Duke Math

    T. Dyckerhoff , Compact generator in categories of matrix factorizations , Duke Math. J. 159 (2) (2011), 223--274

    2011

  29. [37]

    Elagin, V.A

    A. Elagin, V.A. Lunts, and O.M. Schn\"urer , Smoothness of derived categories of algebras, Mosc. Math. J. 20 (2) (2020), 277--309

    2020

  30. [38]

    Dyckerhoff, and D

    T. Dyckerhoff, and D. Murfet , The Kapustin-Li formula revisited , Adv. Math. 231 (3-4) (2012), 1858--1885

    2012

  31. [39]

    Dyckerhoff, and D

    T. Dyckerhoff, and D. Murfet , Pushing forward matrix factorizations , Duke Math. J. 162 (7) (2013), 1249--1311

    2013

  32. [40]

    Etingof, S

    P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik , Tensor Categories (Mathematical Surveys and Monographs) , American Mathematical Society. 205 (2015)

    2015

  33. [41]

    Erdmann, M

    K. Erdmann, M. Holloway, R. Taillefer, N. Snashall, and . Solberg , Support varieties for selfinjective algebras , K -theory 33 (1) (2004), 67--87

    2004

  34. [42]

    Farinati and A.L

    M.A. Farinati and A.L. Solotar , G -structure on the cohomology of Hopf algebras , Proc. Amer. Math. Soc. 132 (2004), 2859-2865

    2004

  35. [43]

    Ferrario , A_ -bimodules in deformation quantization, Ph.D

    A. Ferrario , A_ -bimodules in deformation quantization, Ph.D. thesis, Eidgenössische Technische Hochschule Zürich, 2012

    2012

  36. [44]

    Gálvez-Carrillo, M

    I. Gálvez-Carrillo, M. Ronco and A. Tonks , On differential Hopf algebra and B_ -algebras, Mediterr. J. Math. 22 (2025), No. 93

    2025

  37. [45]

    Ganatra , Symplectic Cohomology and Duality for the Wrapped Fukaya Category, arXiv:1304.7312, 2013

    S. Ganatra , Symplectic Cohomology and Duality for the Wrapped Fukaya Category, arXiv:1304.7312, 2013

    2013 arXiv

  38. [46]

    Gerstenhaber , The cohomology structure of an associative ring, Ann

    M. Gerstenhaber , The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267--288

    1963

  39. [47]

    Gerstenhaber, and S.D

    M. Gerstenhaber, and S.D. Schack , Algebraic Cohomology and Deformation Theory , in M. Hazewinkel and M. Gerstenhaber (eds), Deformation Theory of Algebras and Structures and Applications, Kluwer, Dordrecht, 1988, 11--264

    1988

  40. [48]

    Gerstenhaber and A.A

    M. Gerstenhaber and A.A. Voronov , Homotopy G -algebra and moduli space operad , Inter. Math. Res. Not. IMRN 1995 (3) (1995), 141-153

    1995

  41. [49]

    Getzler , Lie theory for nilpotent L_ -algebra , Ann

    E. Getzler , Lie theory for nilpotent L_ -algebra , Ann. of Math. 170 (2009) 271--301

    2009

  42. [50]

    Getzler, and J.D.S

    E. Getzler, and J.D.S. Jones , Operads, homotopy algebra and iterated integrals for double loop spaces , arXiv:hep-th/9403055, 1994

    1994 arXiv

  43. [51]

    Gutt , An explicit * -product on the cotangent bundle of a Lie group , Lett

    S. Gutt , An explicit * -product on the cotangent bundle of a Lie group , Lett. Math. Phys. 7 (1983), 249--258

    1983

  44. [52]

    Hazrat , The dynamics of Leavitt path algebras , J

    R. Hazrat , The dynamics of Leavitt path algebras , J. Algebra 384 (2013), 242--266

    2013

  45. [53]

    Hogancamp , Idempotents in triangulated monoidal categories , arXiv: 1703.01001, 2017

    M. Hogancamp , Idempotents in triangulated monoidal categories , arXiv: 1703.01001, 2017

    2017 arXiv

  46. [54]

    Hovey, J.H

    M. Hovey, J.H. Palmieri, and N.P. Strickland , Axiomatic stable homotopy theory , Mem. Amer. Math. Soc. 128 (610) (1997), x+114 pp

    1997

  47. [55]

    Hua, and B

    Z. Hua, and B. Keller , Cluster categories and rational curves , arXiv: 1810.00749v3, 2018

    2018 arXiv

  48. [56]

    Guo, and F

    L. Guo, and F. Li , Structure of Hochschild cohomology of path algebras and differential formulation of Euler's polyhedron formula, Asian J. Math. 18 (2014), 545--572

    2014

  49. [57]

    Hovey , Additive closed symmetric monoidal structures on R -modules , J

    M. Hovey , Additive closed symmetric monoidal structures on R -modules , J. Pure Appl. Algebra. 215 (2011), 789--805

    2011

  50. [58]

    Joyal and R

    A. Joyal and R. Street , The geometry of tensor calculus, I , Adv. Math. 88 (1991), no. 1, 55--112

    1991

  51. [59]

    Kadeishvili , On the cobar construction of a bialgebra , Homology, Homotopy and Appl

    T. Kadeishvili , On the cobar construction of a bialgebra , Homology, Homotopy and Appl. 7 (2005), 109--122

    2005

  52. [60]

    Kaufmann , On spineless cacti, Deligne's conjecture and Connes-Kreimer's Hopf algebra , Topology 46 (1) (2007), 39--88

    R. Kaufmann , On spineless cacti, Deligne's conjecture and Connes-Kreimer's Hopf algebra , Topology 46 (1) (2007), 39--88

    2007

  53. [61]

    Keller , Deriving DG-categories , Ann

    B. Keller , Deriving DG-categories , Ann. Sci. \'Ecole Norm. Sup. 27 (4) (1994), 63--102

    1994

  54. [62]

    Keller , Introduction to A-infinity algebras and modules, Homology, Homotopy and Appl

    B. Keller , Introduction to A-infinity algebras and modules, Homology, Homotopy and Appl. 3 (2001), 1--35

    2001

  55. [63]

    Keller , A-infinity algebras in representation theory , Representations of Algebra, vols

    B. Keller , A-infinity algebras in representation theory , Representations of Algebra, vols. I, II, Beijing Norm. Univ. Press, Beijing, 2002, 74--86

    2002

  56. [64]

    Keller , Derived invariance of higher structures on the Hochschild complex , available as https://webusers.imj-prg.fr/bernhard.keller/publ/index.html

    B. Keller , Derived invariance of higher structures on the Hochschild complex , available as https://webusers.imj-prg.fr/bernhard.keller/publ/index.html

  57. [65]

    Keller , On triangulated orbit categories , Documenta

    B. Keller , On triangulated orbit categories , Documenta. Math. 10 (2005), 551--591

    2005

  58. [66]

    Keller , On differential graded categories , International Congress of Mathematicians

    B. Keller , On differential graded categories , International Congress of Mathematicians. Vol. II, 151--190, Eur. Math. Soc., Z\"urich, 2006

    2006

  59. [67]

    Keller , Singular Hochschild cohomology via the singularity category , arXiv: 1809.05121

    B. Keller , Singular Hochschild cohomology via the singularity category , arXiv: 1809.05121

    arXiv

  60. [68]

    Kontsevich and Y

    M. Kontsevich and Y. Soibelman , Deformations of algebras over operads and the Deligne conjecture , Conf\'erence Mosh\'e Flato 1999, Volume I, 255-307, Math. Phys. Stud. 22, Kluwer Acad. Publ., Dordrecht, 2000

    1999

  61. [69]

    Krause , The stable derived category of a Noetherian scheme , Compos

    H. Krause , The stable derived category of a Noetherian scheme , Compos. Math. 141 (5) (2005), 1128--1162

    2005

  62. [70]

    Li , The injective Leavitt complex , Algebr

    H.H. Li , The injective Leavitt complex , Algebr. Represent. Theor. 21 (4) (2018), 833-858

    2018

  63. [71]

    Y. Liu, Z. Wang, and G. Zhou , The Batalin-Vilkovisky Structure on the Tate-Hochschild Cohomology Ring of a Group Algebra , arXiv:1901.03224. To appear in Inter. Math. Res. Not. IMRN

    1901 arXiv

  64. [72]

    Lopatkin , Derivations of Leavitt path algebra , arXiv: 1509.05075v19, 2017

    V. Lopatkin , Derivations of Leavitt path algebra , arXiv: 1509.05075v19, 2017

    2017 arXiv

  65. [73]

    Loday , Cyclic homology , Grundlehren Math

    J.-L. Loday , Cyclic homology , Grundlehren Math. Wiss. 301 . Springer-Verlag, Berlin, (1992)

    1992

  66. [74]

    Loday, and B

    J.-L. Loday, and B. Vallette , Algebraic operads , Grundlehren Math. Wiss. 346 . Springer, Heidelberg, (2012)

    2012

  67. [75]

    Lentner, S

    S. Lentner, S. Mierach C. Schweigert and Y. Sommerha\" u ser , Hochschild Cohomology and the Modular Group , J. Algebra. 507 (2018), 400-420

    2018

  68. [76]

    Loday and M

    J.-L. Loday and M. Ronco , On the strcutre of cofree Hopf algebras , J. Reine Angew. Math. 592 (2006), 123-155

    2006

  69. [77]

    Lowen , Hochschild cohomology, the characteristic morphism and derived deformations , Compositio Math

    W. Lowen , Hochschild cohomology, the characteristic morphism and derived deformations , Compositio Math. 144 (6) (2008), 1557--1580

    2008

  70. [78]

    Lowen, and M

    W. Lowen, and M. Van den Bergh , Hochschild cohomology of Abelian categories and ringed spaces , Advances in Math. 198 (1) (2005), 172--221

    2005

  71. [79]

    Lowen, and M

    W. Lowen, and M. Van den Bergh , The curvature problem for formal and infinitesimal deformations , arXiv:1505.03698

    arXiv

  72. [80]

    D.M. Lu, J.H. Palmieri, Q.S. Wu, and J.J. Zhang , A -infinity structure on -algebras , J. Pure Appl. Algebra 213 (11) (2009), 2017--2037

    2009

  73. [81]

    Lunts, and O.M

    V.A. Lunts, and O.M. Schn\"urer , New enhancements of derived categories of coherent sheaves and applications , J. Algebra 446 (2016), 203--274

    2016

  74. [82]

    Lurie , Higher algebra , available from Lurie's Home Page https://www.math.ias.edu/ lurie/papers/HA.pdf (2017)

    J. Lurie , Higher algebra , available from Lurie's Home Page https://www.math.ias.edu/ lurie/papers/HA.pdf (2017)

    2017

  75. [83]

    Lunts, and D

    V.A. Lunts, and D. Orlov , Uniqueness of enhancement for triangulated categories , J. Amer. Math. Soc., 23 (3) (2010), 853--908

    2010

  76. [84]

    MacLane , Homology ,

    S. MacLane , Homology ,

  77. [85]

    Mac Lane , Homology, Reprint of the 1975 Edition, Springer-Verlag, Berlin Heidelburg, 1995

    S. Mac Lane , Homology, Reprint of the 1975 Edition, Springer-Verlag, Berlin Heidelburg, 1995

    1975

  78. [86]

    Matsui and R

    H. Matsui and R. Takahashi , Thick tensor ideals of right bounded derived categories Algebra Number Theory, 11 (2017), 1677–1738

    2017

  79. [87]

    McClure, J.H

    J.E. McClure, J.H. Smith , A solution of Deligne's Hochschild cohomology conjecture , Recent progress in homotopy theory (Baltimore, MD, 2000), 153--193, Contemp. Math., 293 , Amer. Math. Soc., Providence. RI. 2002

    2000

  80. [88]

    Manetti , Lectures on deformations of complex manifolds (deformations from differential graded viewpoint) , Rend

    M. Manetti , Lectures on deformations of complex manifolds (deformations from differential graded viewpoint) , Rend. Mat. Appl. (7) 24 (2004),1--183

    2004

  81. [89]

    Merkulov , An L_ -algebra of an unobstructed deformation functor , Inter

    S. Merkulov , An L_ -algebra of an unobstructed deformation functor , Inter. Math. Res. Not. IMRN 2000 (3) (2000), 147-164

    2000

  82. [90]

    Murfet , Residues and duality for singularity categories of isolated Gorenstein singularities , Compos

    D. Murfet , Residues and duality for singularity categories of isolated Gorenstein singularities , Compos. Math. 149 (12) (2013), 2071--2100

    2013

  83. [91]

    Nakano, K.B

    D.K. Nakano, K.B. Vashaw, and M.T. Yakimov , Noncommutative tensor triangular geometry , Amer. J. Math. 144 (2022), 1681–1724

    2022

  84. [92]

    Nakano, K.B

    D.K. Nakano, K.B. Vashaw, and M.T. Yakimov , Noncommutative tensor triangular geometry and the tensor product property for support maps , Int. Math. Res. Not. 2022 (2022), 17766–17796

    2022

  85. [93]

    Neeman , The Grothendieck duality theorem via Bousfield's techniques and Brown representability , J

    A. Neeman , The Grothendieck duality theorem via Bousfield's techniques and Brown representability , J. Amer. Math. Soc. 9 (1996), 205--248

    1996

  86. [94]

    Orlov, Triangulated categories of singularities and D -branes in Landau-Ginzburg models , Proc

    D. Orlov, Triangulated categories of singularities and D -branes in Landau-Ginzburg models , Proc. Steklov Inst. Math. 246 (3) (2004), 227--248

    2004

  87. [95]

    Rivera, and Z

    M. Rivera, and Z. Wang , Singular Hochschild cohomology and algebraic string operations , J. Noncommut. Geom. 13 (2019), 297--361

    2019

  88. [96]

    Rizzardo, and M

    A. Rizzardo, and M. Van den Bergh , A note on non-unique enhancements , Proc. Amer. Math. Soc. 147 (2019), 451--453

    2019

  89. [97]

    Selinger , A survey of graphical languages for monoidal categories , in: New Structures for Physics, Lecture Notes in Phys

    P. Selinger , A survey of graphical languages for monoidal categories , in: New Structures for Physics, Lecture Notes in Phys. 813 , Springer, 2011, 289--355

    2011

  90. [98]

    Smith , A tour of support theory for triangulated categories through tensor triangular geometry , in Building bridges between algebra and topology

    L. Smith , A tour of support theory for triangulated categories through tensor triangular geometry , in Building bridges between algebra and topology. 129 (2018), 63--101

    2018

  91. [99]

    Smith , Homological algebra and the Eilenberg-Moore spectra sequence , Trans

    L. Smith , Homological algebra and the Eilenberg-Moore spectra sequence , Trans. Amer. Math. Soc. 129 (1967), 58--93

    1967

  92. [100]

    Smith , Category equivalences involving graded modules over path algebras of quivers, Adv

    S.P. Smith , Category equivalences involving graded modules over path algebras of quivers, Adv. Math. 230 (2012), 1780--1810

    2012

  93. [101]

    Schlichting , A note on K -theory and triangulated categories , Invent

    M. Schlichting , A note on K -theory and triangulated categories , Invent. Math. 150 (1) (2002), 111--116

    2002

  94. [102]

    Stasheff , Homotopy associativity of H -spaces, I , Trans

    J.D. Stasheff , Homotopy associativity of H -spaces, I , Trans. Amer. Math. Soc. 108 (1963), 275-292

    1963

  95. [103]

    Stasheff , Homotopy associativity of H -spaces, II , Trans

    J.D. Stasheff , Homotopy associativity of H -spaces, II , Trans. Amer. Math. Soc. 108 (1963), 293-312

    1963

  96. [104]

    Tamarkin , Formality of chain operad of small squares , arXiv:9809164

    D. Tamarkin , Formality of chain operad of small squares , arXiv:9809164

  97. [105]

    To\"en , The homotopy theory of dg-categories and derived Morita theory , Invent

    B. To\"en , The homotopy theory of dg-categories and derived Morita theory , Invent. Math. 167 (3) (2007), 615--667

    2007

  98. [106]

    To\"en , Lectures on dg-categories, Topics in algebraic and topological K -theory , Lecture Notes in Math., vol

    B. To\"en , Lectures on dg-categories, Topics in algebraic and topological K -theory , Lecture Notes in Math., vol. 2008, Springer, Berlin, 2011, 243--302

    2008

  99. [107]

    Tomforde, Uniqueness theorems and ideal structure for Leavitt path algebras, J

    M. Tomforde, Uniqueness theorems and ideal structure for Leavitt path algebras, J. Algebra 318 (2007), 270--299

    2007

  100. [108]

    Tradler , Infinity-inner-products on A -infinity-algebras , J

    T. Tradler , Infinity-inner-products on A -infinity-algebras , J. Homotopy Relat. Strcut. 3 (2008), 245-271

    2008

  101. [109]

    Turaev, and A

    V. Turaev, and A. Virelizier , Monoidal categories and topological field theory , Progr. Math. 322 (2017)

    2017

  102. [110]

    Vallette, Algebra+homotopy=operad

    B. Vallette, Algebra+homotopy=operad. Symplectic, Poisson, and noncommutative geometry, 229--290, Math. Sci. Res. Inst. Publ., 62 , Cambridge Univ. Press, New York, 2014

    2014

  103. [111]

    Voronov, Homotopy Gerstenhaber algebras, Conf\'erence Mosh\'e Flato 1999, Vol

    A. Voronov, Homotopy Gerstenhaber algebras, Conf\'erence Mosh\'e Flato 1999, Vol. II (Dijon), 307--331

    1999

  104. [112]

    Wang , Singular equivalence of Morita type with level , J

    Z. Wang , Singular equivalence of Morita type with level , J. Algebra 439 (2015), 245--269

    2015

  105. [113]

    Wang , Gerstenhaber algebra and Deligne's conjecture on Tate-Hochschild cohomology , Trans

    Z. Wang , Gerstenhaber algebra and Deligne's conjecture on Tate-Hochschild cohomology , Trans. Amer. Math. Soc. 374 (2021), 4537-4577

    2021

  106. [114]

    Wang , Invariance of the Gerstenhaber algebra structure on Tate-Hochschild cohomology , J

    Z.F. Wang , Invariance of the Gerstenhaber algebra structure on Tate-Hochschild cohomology , J. Inst. Math. Jussieu 136 (2019), DOI: 10.1017/S1474748019000367

    2019 doi

  107. [115]

    Weibel , An introduction to homological algebra , Cambridge University Press, Cambridge (1995)

    C.A. Weibel , An introduction to homological algebra , Cambridge University Press, Cambridge (1995)

    1995

  108. [116]

    Yalin , Maurer--Cartan spaces of filtered L_ -algebras , J

    S. Yalin , Maurer--Cartan spaces of filtered L_ -algebras , J. Homotopy Relat. Struct. 11 (2016), 375--407

    2016

  109. [117]

    Young , Brace bar-cobar duality , arXiv:1309.2820

    J. Young , Brace bar-cobar duality , arXiv:1309.2820

    arXiv

  110. [118]

    Yu and G

    J. Yu and G. Liu , Derived discrete Hopf algebras with the Chevalley property , arXiv: 2412.08093

    arXiv

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