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1. Root of unity quantum cluster algebras and discriminants

   https://doi.org/10.1112/jlms.70060  

   Nguyen, Bach; Trampel, Kurt; Yakimov, Milen (January 2025, Journal of the London Mathematical Society)

   Abstract We describe a connection between the subjects of cluster algebras, polynomial identity algebras, and discriminants. For this, we define the notion of root of unity quantum cluster algebras and prove that they are polynomial identity algebras. Inside each such algebra we construct a (large) canonical central subalgebra, which can be viewed as a far reaching generalization of the central subalgebras of big quantum groups constructed by De Concini, Kac, and Procesi and used in representation theory. Each such central subalgebra is proved to be isomorphic to the underlying classical cluster algebra of geometric type. When the root of unity quantum cluster algebra is free over its central subalgebra, we prove that the discriminant of the pair is a product of powers of the frozen variables times an integer. An extension of this result is also proved for the discriminants of all subalgebras generated by the cluster variables of nerves in the exchange graph. These results can be used for the effective computation of discriminants. As an application we obtain an explicit formula for the discriminant of the integral form over of each quantum unipotent cell of De Concini, Kac, and Procesi for arbitrary symmetrizable Kac–Moody algebras, where is a root of unity. 

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2. M$$M$$‐Ideals in real operator algebras

   https://doi.org/10.1002/mana.202400227  

   Blecher, David_P; Neal, Matthew; Peralta, Antonio_M; Su, Shanshan (March 2025, Mathematische Nachrichten)

   Abstract In a recent paper, we showed that a subspace of a real ‐triple is an ‐summand if and only if it is a ‐closed triple ideal. As a consequence, ‐ideals of real ‐triples, including real ‐algebras, real ‐algebras and real TROs, correspond to norm‐closed triple ideals. In this paper, we extend this result by identifying the ‐ideals in (possibly non‐self‐adjoint) real operator algebras and Jordan operator algebras. The argument for this is necessarily different. We also give simple characterizations of one‐sided ‐ideals in real operator algebras, and give some applications to that theory. 

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3. 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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4. Scattering diagrams, tight gradings, and generalized positivity

   https://doi.org/10.1073/pnas.2422893122  

   Burcroff, Amanda; Lee, Kyungyong; Mou, Lang (May 2025, Proceedings of the National Academy of Sciences)

   In 2013, Lee, Li, and Zelevinsky introduced combinatorial objects called compatible pairs to construct the greedy bases for rank-2 cluster algebras, consisting of indecomposable positive elements including the cluster monomials. Subsequently, Rupel extended this construction to the setting of generalized rank-2 cluster algebras by defining compatible gradings. We find a class of combinatorial objects which we call tight gradings. Using this, we give a directly computable, manifestly positive, and elementary but highly nontrivial formula describing rank-2 consistent scattering diagrams. This allows us to show that the coefficients of the wall-functions on a generalized cluster scattering diagram of any rank are positive, which implies the Laurent positivity for generalized cluster algebras and the strong positivity of their theta bases. 

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5. Affine cluster monomials are generalized minors

   https://doi.org/10.1112/S0010437X19007292  

   Rupel, Dylan; Stella, Salvatore; Williams, Harold (June 2019, Compositio Mathematica)

   We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with $$\mathbf{g}$$ -vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type $$A_{1}^{(1)}$$ , we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras. 

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