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1. 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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2. The minimal free resolution of a general principal symmetric ideal

   https://doi.org/10.1090/tran/9314  

   Harada, Megumi; Seceleanu, Alexandra; Şega, Liana (December 2024, Transactions of the American Mathematical Society)

   We introduce the class of principal symmetric ideals, which are ideals generated by the orbit of a single polynomial under the action of the symmetric group. Fixing the degree of the generating polynomial, this class of ideals is parametrized by points in a suitable projective space. We show that the minimal free resolution of a principal symmetric ideal is constant on a non-empty Zariski open subset of this projective space and we determine this resolution explicitly. Along the way, we study two classes of graded algebras which we term narrow and extremely narrow; both of which are instances of compressed artinian algebras. 

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3. The lowest discriminant ideal of a Cayley–Hamilton Hopf algebra

   https://doi.org/10.1090/tran/9354  

   Mi, Zhongkai; Wu, Quanshui; Yakimov, Milen (May 2025, Transactions of the American Mathematical Society)

   Discriminant ideals of noncommutative algebras $A$, which are module finite over a central subalgebra $C$, are key invariants that carry important information about $A$, such as the sum of the squares of the dimensions of its irreducible modules with a given central character. There has been substantial research on the computation of discriminants, but very little is known about the computation of discriminant ideals. In this paper we carry out a detailed investigation of the lowest discriminant ideals of Cayley–Hamilton Hopf algebras in the sense of De Concini, Reshetikhin, Rosso and Procesi, whose identity fiber algebras are basic. The lowest discriminant ideals are the most complicated ones, because they capture the most degenerate behaviour of the fibers in the exact opposite spectrum of the picture from the Azumaya locus. We provide a description of the zero sets of the lowest discriminant ideals of Cayley–Hamilton Hopf algebras in terms of maximally stable modules of Hopf algebras, irreducible modules that are stable under tensoring with the maximal possible number of irreducible modules with trivial central character. In important situations, this is shown to be governed by the actions of the winding automorphism groups. The results are illustrated with applications to the group algebras of central extensions of abelian groups, big quantum Borel subalgebras at roots of unity and quantum coordinate rings at roots of unity. 

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4. Gröbner bases, symmetric matrices, and type C Kazhdan–Lusztig varieties

   https://doi.org/10.1112/jlms.12856  

   Escobar, Laura; Fink, Alex; Rajchgot, Jenna; Woo, Alexander (February 2024, Journal of the London Mathematical Society)

   Abstract We study a class of combinatorially defined polynomial ideals that are generated by minors of a generic symmetric matrix. Included within this class are the symmetric determinantal ideals, the symmetric ladder determinantal ideals, and the symmetric Schubert determinantal ideals of A. Fink, J. Rajchgot, and S. Sullivant. Each ideal in our class is a type C analog of a Kazhdan–Lusztig ideal of A. Woo and A. Yong; that is, it is the scheme‐theoretic defining ideal of the intersection of a type C Schubert variety with a type C opposite Schubert cell, appropriately coordinatized. The Kazhdan–Lusztig ideals that arise are exactly those where the opposite cell is 123‐avoiding. Our main results include Gröbner bases for these ideals, prime decompositions of their initial ideals (which are Stanley–Reisner ideals of subword complexes), and combinatorial formulas for their multigraded Hilbert series in terms of pipe dreams. 

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5. Approximate ideal structures and K-theory

   Willett, Rufus (April 2021, New York journal of mathematics)

   null (Ed.)

   We introduce a notion of approximate ideal structure for a C*-algebra, and use it as a tool to study K-theory groups. The notion is motivated by the classical Mayer-Vietoris sequence, by the theory of nuclear dimension as introduced by Winter and Zacharias, and by the theory of dynamical complexity introduced by Guentner, Yu, and the author. A major inspiration for our methods comes from recent work of Oyono-Oyono and Yu in the setting of controlled K-theory of filtered C*-algebras; we do not, however, use that language in this paper. We give two main applications. The first is a vanishing result for K-theory that is relevant to the Baum-Connes conjecture. The second is a permanence result for the Kunneth formula in C*-algebra K-theory: roughly, this says that if A can be decomposed into a pair of subalgebras (C,D) such that C, D, and C∩ D all satisfy the Kunneth formula, then A itself satisfies the Kunneth formula. 

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