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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. Dirac Geometry I: Commutative Algebra

   https://doi.org/10.1007/s42543-023-00072-6  

   Hesselholt, Lars; Pstra̧gowski, Piotr (June 2023, Peking Mathematical Journal)

   Abstract The purpose of this paper and its sequel is to develop the geometry built from the commutative algebras that naturally appear as the homology of differential graded algebras and, more generally, as the homotopy of algebras in spectra. The commutative algebras in question are those in the symmetric monoidal category of graded abelian groups, and, being commutative, they form the affine building blocks of a geometry, as commutative rings form the affine building blocks of algebraic geometry. We name this geometry Dirac geometry, because the grading exhibits the hallmarks of spin in that it is a remnant of the internal structure encoded byanima, it distinguishes symmetric and anti-symmetric behavior, and the coherent cohomology of Dirac schemes and Dirac stacks, which we develop in the sequel, admits half-integer Serre twists. Thus, informally, Dirac geometry constitutes a “square root” of$$\mathbb {G}_m$$ $G_m$-equivariant algebraic geometry. 

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