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1. Maximal Tori in HH 1 and the Fundamental Group

   https://doi.org/10.1093/imrn/rnac026  

   Briggs, Benjamin; Rubio y Degrassi, Lleonard (March 2023, International Mathematics Research Notices)

   Abstract We investigate maximal tori in the Hochschild cohomology Lie algebra $${\operatorname {HH}}^1(A)$$ of a finite dimensional algebra $$A$$, and their connection with the fundamental groups associated to presentations of $$A$$. We prove that every maximal torus in $${\operatorname {HH}}^1(A)$$ arises as the dual of some fundamental group of $$A$$, extending the work by Farkas, Green, and Marcos; de la Peña and Saorín; and Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental group of $$A$$ is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Using this we prove that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras; hitherto this was known only for very restricted classes of monomial algebras. 

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2. On Tsirelson pairs of C*-algebras

   https://doi.org/10.1142/S0129055X23500162  

   Goldbring, Isaac; Hart, Bradd (September 2023, Reviews in Mathematical Physics)

   We introduce the notion of a Tsirelson pair of C*-algebras, which is a pair of C*-algebras for which the space of quantum strategies obtained by using states on the minimal tensor product of the pair is dense in the space of quantum strategies obtained by using states on the maximal tensor product. We exhibit a number of examples of such pairs that are “nontrivial” in the sense that the minimal tensor product and the maximal tensor product of the pair are not isomorphic. For example, we prove that any pair containing a C*-algebra with Kirchberg’s QWEP property is a Tsirelson pair. We then introduce the notion of a C*-algebra with the Tsirelson property (TP) and establish a number of closure properties for this class. We also show that the class of C*-algebras with the TP forms an elementary class (in the sense of model theory), but that this class does not admit an effective axiomatization. 

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3. On the Lie algebra structure of integrable derivations

   https://doi.org/10.1112/blms.12884  

   Briggs, Benjamin; Rubio_y_Degrassi, Lleonard (December 2023, Bulletin of the London Mathematical Society)

   Abstract Building on work of Gerstenhaber, we show that the space of integrable derivations on an Artin algebra forms a Lie algebra, and a restricted Lie algebra if contains a field of characteristic . We deduce that the space of integrable classes in forms a (restricted) Lie algebra that is invariant under derived equivalences, and under stable equivalences of Morita type between self‐injective algebras. We also provide negative answers to questions about integrable derivations posed by Linckelmann and by Farkas, Geiss and Marcos. Along the way, we compute the first Hochschild cohomology of the group algebra of any symmetric group. 

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4. Equivariant Witt complexes and twisted topological Hochschild homology

   https://doi.org/10.1016/j.topol.2025.109444  

   Bohmann, Anna Marie; Gerhardt, Teena; Krulewski, Cameron; Petersen, Sarah; Yang, Lucy (May 2025, Topology and its Applications)

   The topological Hochschild homology of a ring (or ring spectrum) R is an S1-spectrum, and the fixed points of THH(R) for subgroups C_n of S1 have been widely studied due to their use in algebraic K-theory computations. Hesselholt and Madsen proved that the fixed points of topological Hochschild homology are closely related to Witt vectors. Further, they defined the notion of a Witt complex, and showed that it captures the algebraic structure of the homotopy groups of the fixed points of THH. Recent work defines a theory of twisted topological Hochschild homology for equivariant rings (or ring spectra) that builds upon Hill, Hopkins and Ravenel's work on equivariant norms. In this paper, we study the algebraic structure of the equivariant homotopy groups of twisted THH. In particular, drawing on the definition of equivariant Witt vectors by Blumberg, Gerhardt, Hill and Lawson, we define an equivariant Witt complex and prove that the equivariant homotopy of twisted THH has this structure. Our definition of equivariant Witt complexes contributes to a growing body of research in the subject of equivariant algebra. 

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5. Noncommutative Tensor Triangular Geometry and the Tensor Product Property for Support Maps

   https://doi.org/10.1093/imrn/rnab221  

   Nakano, Daniel K; Vashaw, Kent B; Yakimov, Milen T (August 2021, International Mathematics Research Notices)

   Abstract The problem of whether the cohomological support map of a finite dimensional Hopf algebra has the tensor product property has attracted a lot of attention following the earlier developments on representations of finite group schemes. Many authors have focused on concrete situations where positive and negative results have been obtained by direct arguments. In this paper we demonstrate that it is natural to study questions involving the tensor product property in the broader setting of a monoidal triangulated category. We give an intrinsic characterization by proving that the tensor product property for the universal support datum is equivalent to complete primeness of the categorical spectrum. From these results one obtains information for other support data, including the cohomological one. Two theorems are proved giving compete primeness and non-complete primeness in certain general settings. As an illustration of the methods, we give a proof of a recent conjecture of Negron and Pevtsova on the tensor product property for the cohomological support maps for the small quantum Borel algebras for all complex simple Lie algebras. 

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