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1. Cluster algebra structures on Poisson nilpotent algebras

   https://doi.org/10.1090/memo/1445  

   Goodearl, K; Yakimov, M (October 2023, Memoirs of the American Mathematical Society)

   Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac–Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements. 

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2. PLETHORA OF CLUSTER STRUCTURES ON GL_n

   Gekhtman, M.; Shapiro, M.; Vainshtein, A. (January 2022, Memoirs of the American Mathematical Society)

   We continue the study of multiple cluster structures in the rings of regular functions on $$GL_n$$, $$SL_n$$ and $$\operatorname{Mat}_n$$ 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 $$\mathcal O(\mathcal G)$$. 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. In the oriented case, we single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any 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 $$SL_n$$ compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of $$SL_n$$ equipped with two different Poisson-Lie brackets. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address this situation in future publications. 

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3. 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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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. Poisson Trace Orders

   https://doi.org/10.1093/imrn/rnad086  

   Brown, Ken; Yakimov, Milen (May 2023, International Mathematics Research Notices)

   Abstract The two main approaches to the study of irreducible representations of orders (via traces and Poisson orders) have so far been applied in a completely independent fashion. We define and study a natural compatibility relation between the two approaches leading to the notion of Poisson trace orders. It is proved that all regular and reduced traces are always compatible with any Poisson order structure. The modified discriminant ideals of all Poisson trace orders are proved to be Poisson ideals and the zero loci of discriminant ideals are shown to be unions of symplectic cores, under natural assumptions (maximal orders and Cayley–Hamilton algebras). A base change theorem for Poisson trace orders is proved. A broad range of Poisson trace orders are constructed based on the proved theorems: quantized universal enveloping algebras, quantum Schubert cell algebras and quantum function algebras at roots of unity, symplectic reflection algebras, 3D and 4D Sklyanin algebras, Drinfeld doubles of pre-Nichols algebras of diagonal type, and root of unity quantum cluster algebras. 

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