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1. Computing weight q-multiplicities for the representations of the simple Lie algebras

   https://doi.org/10.1007/s00200-017-0346-7  

   Harris, Pamela E.; Insko, Erik; Simpson, Anthony (November 2017, Applicable Algebra in Engineering, Communication and Computing)

   The multiplicity of a weight μμ in an irreducible representation of a simple Lie algebra gg with highest weight λλ can be computed via the use of Kostant’s weight multiplicity formula. This formula is an alternating sum over the Weyl group and involves the computation of a partition function. In this paper we consider a q-analog of Kostant’s weight multiplicity and present a SageMath program to compute q-multiplicities for the simple Lie algebras. 

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2. 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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3. Semisimplification of contragredient Lie algebras

   https://doi.org/10.2140/om.2024.1.211  

   Angiono, Iván; Plavnik, Julia; Sanmarco, Guillermo (January 2025, Orbita Mathematicae)

   We describe the structure and different features of Lie algebras in the Verlinde category, obtained as semisimplification of contragredient Lie algebras in characteristic p with respect to the adjoint action of a Chevalley generator. In particular, we construct a root system for these algebras that arises as a parabolic restriction of the known root system for the classical Lie algebra. This gives a lattice grading with simple homogeneous components and a triangular decomposition for the semisimplified Lie algebra. We also obtain a non-degenerate invariant form that behaves well with the lattice grading. As an application, we exhibit concrete new examples of Lie algebras in the Verlinde category. 

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4. 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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5. Stability of the centers of group algebras of $$GL_n(q)$$

   https://doi.org/https://doi.org/10.1016/j.aim.2019.04.026  

   Wan, Jinkui; Wang, Weiqiang (April 2019, Advances in mathematics)

   The center $$Z_n(q)$$ of the integral group algebra of the general linear group $$GL_n(q)$$ over a finite field admits a filtration with respect to the reflection length. We show that the structure constants of the associated graded algebras $$\mathscr{G}_n(q)$$ are independent of $$n$$, and this stability leads to a universal stable center with positive integer structure constants which governs the algebras $$\mathscr{G}_n(q)$$ for all $$n$$. Various structure constants of the stable center are computed and several conjectures are formulated. Analogous stability properties for symmetric groups and wreath products were established earlier by Farahat-Higman and the second author. 

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