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1. A Plethora of Cluster Structures on 𝐺𝐿_{𝑛}

   https://doi.org/10.1090/memo/1486  

   Gekhtman, M; Shapiro, M; Vainshtein, A (May 2024, Memoirs of the American Mathematical Society)

   We continue the study of multiple cluster structures in the rings of regular functions on $G{L}_n$, $S{L}_n$and $Ma{t}_n$that are compatible with Poisson–Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin–Drinfeld classification of Poisson–Lie structures on a semisimple complex group $\mathcal{G}$corresponds to a cluster structure in $\mathcal{O}\left(\mathcal{G}\right)$. Here we prove this conjecture for a large subset of Belavin–Drinfeld (BD) data of $A_n$type, which includes all the previously known examples. Namely, we subdivide all possible $A_n$type BD data into oriented and non-oriented kinds. We further single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any oriented BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson–Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on $S{L}_n$compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of $S{L}_n$equipped with two different Poisson-Lie brackets. Similar results hold for aperiodic non-oriented BD data, but the analysis of the corresponding regular cluster structure is more involved and not given here. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address these situations in future publications. 

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2. Generalized cluster structures related to the Drinfeld double of 𝑮𝑳_𝒏

   Gekhtman, M.; Shapiro, M.; Vainshtein, A. (October 2021, Journal of the London Mathematical Society)

   We prove that the regular generalized cluster structure on the Drinfeld double of 𝐺𝐿𝑛 constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) is complete and compatible with the standard Poisson–Lie structure on the double. Moreover, we show that for 𝑛 = 4 this structure is distinct from a previously known regular generalized cluster structure on the Drinfeld double, even though they have the same compatible Poisson structure and the same collection of frozen variables. Further, we prove that the regular generalized cluster structure on band periodic matrices constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) possesses similar compatibility and completeness properties. 

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3. 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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4. Periodic Staircase Matrices and Generalized Cluster Structures

   https://doi.org/10.1093/imrn/rnaa148  

   Gekhtman, M.; Shapiro, M.; Vainshtein, A. (July 2022, International mathematics research notices)

   As is well known, cluster transformations in cluster structures of geometric type are often modeled on determinant identities, such as short Plücker relations, Desnanot– Jacobi identities, and their generalizations. We present a construction that plays a similar role in a description of generalized cluster transformations and discuss its applications to generalized cluster structures in GL_n compatible with a certain subclass of Belavin–Drinfeld Poisson–Lie brackets, in the Drinfeld double of GL_n, and in spaces of periodic difference operators. 

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5. Cyclic pairings and derived Poisson structures.

   Ramadoss, Ajay C; Zhang, Yining (April 2019, New York journal of mathematics)

   null (Ed.)

   By a fundamental theorem of D. Quillen, there is a natural duality - an instance of general Koszul duality - between differential graded (DG) Lie algebras and DG cocommutative coalgebras defined over a field k of characteristic 0. A cyclic pairing (i.e., an inner product satisfying a natural cyclicity condition) on the cocommutative coalgebra gives rise to an interesting structure on the universal enveloping algebra Ua of the Koszul dual Lie algebra a called the derived Poisson bracket. Interesting special cases of the derived Poisson bracket include the Chas-Sullivan bracket on string topology. We study the derived Poisson brackets on universal enveloping algebras Ua, and their relation to the classical Poisson brackets on the derived moduli spaces DRep_g(a) of representations of a in a finite dimensional reductive Lie algebra g. More specifically, we show that certain derived character maps of a intertwine the derived Poisson bracket with the classical Poisson structure on the representation homology HR(a, g) related to DRep_g(a). 

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