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1. The companion section for classical groups

   https://doi.org/10.1142/S0129167X24410106  

   Hameister, Thomas; Ngô, Bao Châu (August 2024, International Journal of Mathematics)

   We use the companion matrix construction for [Formula: see text] to build canonical sections of the Chevalley map [Formula: see text] for classical groups [Formula: see text] as well as the group [Formula: see text]. To do so, we construct canonical tensors on the associated spectral covers. As an application, we make explicit lattice descriptions of affine Springer fibers and Hitchin fibers for classical groups and [Formula: see text]. 

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2. A trace inequality of Ando, Hiai, and Okubo and a monotonicity property of the Golden–Thompson inequality

   https://doi.org/10.1063/5.0091111  

   Carlen, Eric A.; Lieb, Elliott H. (June 2022, Journal of Mathematical Physics)

   The Golden–Thompson trace inequality, which states that Tr  e H+ K ≤ Tr  e H e K , has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here, we make this G–T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, H = Δ or [Formula: see text] and K = potential, Tr  e H+(1− u) K e uK is a monotone increasing function of the parameter u for 0 ≤ u ≤ 1. Our proof utilizes an inequality of Ando, Hiai, and Okubo (AHO): Tr  X s Y t X 1− s Y 1− t ≤ Tr  XY for positive operators X, Y and for [Formula: see text], and [Formula: see text]. The obvious conjecture that this inequality should hold up to s + t ≤ 1 was proved false by Plevnik [Indian J. Pure Appl. Math. 47, 491–500 (2016)]. We give a different proof of AHO and also give more counterexamples in the [Formula: see text] range. More importantly, we show that the inequality conjectured in AHO does indeed hold in the full range if X, Y have a certain positivity property—one that does hold for quantum mechanical operators, thus enabling us to prove our G–T monotonicity theorem. 

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3. On the extension of the FKG inequality to n functions

   https://doi.org/10.1063/5.0065285  

   Lieb, Elliott H.; Sahi, Siddhartha (April 2022, Journal of Mathematical Physics)

   The 1971 Fortuin–Kasteleyn–Ginibre inequality for two monotone functions on a distributive lattice is well known and has seen many applications in statistical mechanics and other fields of mathematics. In 2008, one of us (Sahi) conjectured an extended version of this inequality for all n > 2 monotone functions on a distributive lattice. Here, we prove the conjecture for two special cases: for monotone functions on the unit square in [Formula: see text] whose upper level sets are k-dimensional rectangles and, more significantly, for arbitrary monotone functions on the unit square in [Formula: see text]. The general case for [Formula: see text], remains open. 

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4. Free fermions revisited

   https://doi.org/10.1142/S0219498826502695  

   Milas, Antun; Penn, Michael (June 2025, Journal of Algebra and Its Applications)

   Free fermion vertex superalgebras are discussed from the point of view of Urod vertex algebras [T. Arakawa, T. Creutzig and B. Feigin, Urod algebras and translation of [Formula: see text]-algebras, Forum Mathematics Sigma, Vol. 10 (Cambridge University Press, 2022) and M. Bershtein, B. Feigin and A. Litvinov, Coupling of two conformal field theories and Nakajima–Yoshioka blow-up equations, preprint (2013), arXiv:1310.7281]. We present all finite decompositions of the [Formula: see text]-fermion vertex algebra via Virasoro and [Formula: see text] superconformal vertex algebras. We also present decompositions of higher rank fermion algebras using affine [Formula: see text]-algebras, and a conjecture on the existence of the “square root” of the [Formula: see text] fermion algebra. 

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5. Bivariate continuous q-Hermite polynomials and deformed quantum Serre relations

   https://doi.org/10.1142/S0219498821400168  

   Riley Casper, W.; Kolb, Stefan; Yakimov, Milen (January 2021, Journal of Algebra and Its Applications)

   null (Ed.)

   To Nicolás Andruskiewitsch on his 60th birthday, with admiration We introduce bivariate versions of the continuous [Formula: see text]-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous [Formula: see text]-Hermite polynomials and the star product method of [S. Kolb and M. Yakimov, Symmetric pairs for Nichols algebras of diagonal type via star products, Adv. Math. 365 (2020), Article ID: 107042, 69 pp.] for quantum symmetric pairs to establish deformed quantum Serre relations for quasi-split quantum symmetric pairs of Kac–Moody type. We prove that these defining relations are obtained from the usual quantum Serre relations by replacing all monomials by multivariate orthogonal polynomials. 

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