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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. Polynomial Representations of the Witt Lie Algebra

   https://doi.org/10.1093/imrn/rnae139  

   Sam, Steven_V ; Snowden, Andrew ; Tosteson, Philip ( June 2024 , International Mathematics Research Notices)

   The Witt algebra  ${\mathfrak{W}}_{n}$ is the Lie algebra of all derivations of the $n$-variable polynomial ring $\textbf{V}_{n}=\textbf{C}[x_{1}, \ldots , x_{n}]$ (or of algebraic vector fields on $\textbf{A}^{n}$). A representation of ${\mathfrak{W}}_{n}$ is polynomial if it arises as a subquotient of a sum of tensor powers of $\textbf{V}_{n}$. Our main theorems assert that finitely generated polynomial representations of ${\mathfrak{W}}_{n}$ are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of $\textbf{Fin}^{\textrm{op}}$, where $\textbf{Fin}$ is the category of finite sets. We also show that polynomial representations of ${\mathfrak{W}}_{n}$ are equivalent to polynomial representations of the endomorphism monoid of $\textbf{A}^{n}$. These equivalences are a special case of an operadic version of Schur–Weyl duality, which we establish.

    

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3. Naturality and innerness for morphisms of compact groups and (restricted) Lie algebras

   https://doi.org/10.1090/bproc/164  

   Chirvasitu, Alexandru ( July 2024 , Proceedings of the American Mathematical Society, Series B)

   An extended derivation (endomorphism) of a (restricted) Lie algebra$L$is an assignment of a derivation (respectively) of$L’$for any (restricted) Lie morphism$f:L\to L’$, functorial in$f$in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of$L’$to every$f$; and (b) if$L$is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then$L$is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman.

   In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms.

    

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4. Hopf algebra structures and tensor products for group algebras

   Carlson, Jon F ; Iyengar, Srikanth B. ( March 2017 , New York journal of mathematics)

   The modular group algebra of an elementary abelian p-group is isomorphic to the restricted enveloping algebra of a commutative restricted Lie algebra. The different ways of regarding this algebra result in different Hopf algebra structures that determine cup products on cohomology of modules. However, it is proved in this paper that the products with elements of the polynomial subring of the cohomology ring generated by the Bocksteins of the degree one elements are independent of the choice of these coalgebra structures. 

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5. Contravariant forms on Whittaker modules

   https://doi.org/10.1090/proc/15205  

   Brown, Adam ; Romanov, Anna ( January 2021 , Proceedings of the American Mathematical Society)

   Let g \mathfrak {g} be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g \mathfrak {g} -modules Y ( χ , η ) Y(\chi , \eta ) introduced by Kostant. We prove that the set of all contravariant forms on Y ( χ , η ) Y(\chi , \eta ) forms a vector space whose dimension is given by the cardinality of the Weyl group of g \mathfrak {g} . We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M ( χ , η ) M(\chi , \eta ) introduced by McDowell. 

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