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1. Hodge decomposition of string topology

   https://doi.org/10.1017/fms.2021.26  

   Berest, Yuri; Ramadoss, Ajay C.; Zhang, Yining (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract Let X be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$ -equivariant homology $$ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $$ of the free loop space of X preserves the Hodge decomposition of $$ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $$ , making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture of [7]. 

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2. Monoidal categories, representation gap and cryptography

   https://doi.org/10.1090/btran/151  

   Khovanov, Mikhail; Sitaraman, Maithreya; Tubbenhauer, Daniel (January 2024, Transactions of the American Mathematical Society, Series B)

   The linear decomposition attack provides a serious obstacle to direct applications of noncommutative groups and monoids (or semigroups) in cryptography. To overcome this issue we propose to look at monoids with only big representations, in the sense made precise in the paper, and undertake a systematic study of such monoids. One of our main tools is Green’s theory of cells (Green’s relations). A large supply of monoids is delivered by monoidal categories. We consider simple examples of monoidal categories of diagrammatic origin, including the Temperley–Lieb, the Brauer and partition categories, and discuss lower bounds for their representations. 

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3. Stratified Noncommutative Geometry

   https://doi.org/10.1090/memo/1485  

   Ayala, David; Mazel-Gee, Aaron; Rozenblyum, Nick (May 2024, Memoirs of the American Mathematical Society)

   We introduce a theory of stratifications of noncommutative stacks (i.e., presentable stable $\mathrm{\infty <\#comment/>}$-categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as ${\mathbb{E}}_n$-monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction; this is closely related to Verdier duality and reflection functors, and gives a categorification of Möbius inversion. Our main application is to equivariant stable homotopy theory: for any compact Lie group $G$, we give a symmetric monoidal stratification of genuine $G$-spectra. In the case that $G$is finite, this expresses genuine $G$-spectra in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions). We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory. 

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4. The q-analog of Kostant’s partition function and the highest root of the simple Lie algebras

   Harris, Pamela; Insko, Erik; Omar, Mo (April 2018, The Australasian journal of combinatorics)

   For a given weight of a complex simple Lie algebra, the q-analog of Kostant’s partition function is a polynomial valued function in the variable q, where the coefficient of qk is the number of ways the weight can be written as a nonnegative integral sum of exactly k positive roots. In this paper we determine generating functions for the q-analog of Kostant’s partition function when the weight in question is the highest root of the classical Lie algebras of types B, C, and D, and the exceptional Lie algebras of type G2, F4, E6, E7, and E8. 

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5. Character Ratios for Exceptional Groups of Lie Type

   https://doi.org/10.1093/imrn/rnz206  

   Liebeck, Martin W.; Tiep, Pham Huu (October 2019, International Mathematics Research Notices)

   Abstract We prove character ratio bounds for finite exceptional groups $G(q)$ of Lie type. These take the form $$\dfrac{|\chi (g)|}{\chi (1)} \le \dfrac{c}{q^k}$$ for all nontrivial irreducible characters $$\chi$$ and nonidentity elements $$g$$, where $$c$$ is an absolute constant, and $$k$$ is a positive integer. Applications are given to bounding mixing times for random walks on these groups and also diameters of their McKay graphs. 

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