More Like this

1. 𝑆𝑝-equivariant modules over polynomial rings in infinitely many variables

   https://doi.org/10.1090/tran/8496  

   Sam, Steven; Snowden, Andrew (March 2022, Transactions of the American Mathematical Society)

   We study the category of S p \mathbf {Sp} -equivariant modules over the infinite variable polynomial ring, where S p \mathbf {Sp} denotes the infinite symplectic group. We establish a number of results about this category: for instance, we show that every finitely generated module M M fits into an exact triangle T → M → F → T \to M \to F \to where T T is a finite length complex of torsion modules and F F is a finite length complex of “free” modules; we determine the Grothendieck group; and we (partially) determine the structure of injective modules. We apply these results to show that the twisted commutative algebras Sym ⁡ ( C ∞ ⊕ ⋀ 2 C ∞ ) \operatorname {Sym}(\mathbf {C}^{\infty } \oplus \bigwedge ^2{\mathbf {C}^{\infty }}) and Sym ⁡ ( C ∞ ⊕ Sym 2 ⁡ C ∞ ) \operatorname {Sym}(\mathbf {C}^{\infty } \oplus \operatorname {Sym}^2{\mathbf {C}^{\infty }}) are noetherian, which are the strongest results to date of this kind. We also show that the free 2-step nilpotent twisted Lie algebra and Lie superalgebra are noetherian. 

   more » « less
2. M-ideals, yet again: the case of real JB*-triples

   https://doi.org/10.1007/s13398-024-01686-w  

   Blecher, David P; Neal, Matthew; Peralta, Antonio M; Su, Shanshan (January 2025, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas)

   We prove that a subspace of a real JBW∗-triple is an M-summand if and only if it is a weak∗-closed triple ideal. As a consequence, M-ideals of real JB∗-triples correspond to norm-closed triple ideals. As in the setting of complex JB∗-triples, a geometric property is characterized in purely algebraic terms. This is a newfangled treatment of the classical notion of M-ideal in the real setting, by a completely new approach necessitated by the unfeasibility of the known arguments from the setting of complex C∗-algebras and JB∗-triples. The results in this note also provide a full characterization of all M-ideals in real C∗-algebras, real JB∗-algebras and real TROs. 

   more » « less
3. Critical points of master functions and mKdV hierarchy of type $$A^{(2)}_{2n}$$, volume on Representations of Lie algebras, quantum groups and related topics, Contemp. Math., 713, Amer. Math. Soc., Providence, RI, 2018

   Alexander Varchenko, Tyler Woodruff (January 2018, Contemporary mathematics - American Mathematical Society)

   It is considered the population of critical points generated from the critical point of the master function with no variables, which is associated with the trivial representation of the twisted affine Lie algebra A(2)2n. The population is naturally partitioned into an infinite collection of complex cells C^m, where m are some positive integers. For each cell it is defined an injective rational map C^m→M(A^{(2)}_{2n}) of the cell to the space M(A^{(2)}_{2n}) of Miura opers of type A^{(2)}_{2n}. It is shown that the image of the map is invariant with respect to all mKdV flows on M(A^{(2)}_{2n}) and the image is point-wise fixed by all mKdV flows ∂/∂t_r with index r greater than 4m. 

   more » « less
4. Effects of Epitranscriptomic RNA Modifications on the Catalytic Activity of the SARS‐CoV‐2 Replication Complex**

   https://doi.org/10.1002/cbic.202300095  

   Apostle, Alexander; Yin, Yipeng; Chillar, Komal; Eriyagama, Adikari_M_D_N; Arneson, Reed; Burke, Emma; Fang, Shiyue; Yuan, Yinan (March 2023, ChemBioChem)

   Abstract SARS‐CoV‐2 causes individualized symptoms. Many reasons have been given. We propose that an individual's epitranscriptomic system could be responsible as well. The viral RNA genome can be subject to epitranscriptomic modifications, which can be different for different individuals, and thus epitranscriptomics can affect many events including RNA replication differently. In this context, we studied the effects of modifications including pseudouridine (Ψ), 5‐methylcytosine (m⁵C),N6‐methyladenosine (m⁶A),N1‐methyladenosine (m¹A) andN3‐methylcytosine (m³C) on the activity of SARS‐CoV‐2 replication complex (SC2RC). We found that Ψ, m⁵C, m⁶A and m³C had little effect, whereas m¹A inhibited the enzyme. Both m¹A and m³C disrupt canonical base pairing, but they had different effects. The fact that m¹A inhibits SC2RC implies that the modification can be difficult to detect. This fact also implies that individuals with upregulated m¹A including cancer, obesity and diabetes patients might have milder symptoms. However, this contradicts clinical observations. Relevant discussions are provided. 

   more » « less
5. Simple weight modules with finite weight multiplicities over the Lie algebra of polynomial vector fields

   https://doi.org/10.1515/crelle-2022-0053  

   Grantcharov, Dimitar; Serganova, Vera (September 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let 𝒲 n {{\mathcal{W}}_{n}} be the Lie algebra of polynomial vector fields.We classify simple weight 𝒲 n {{\mathcal{W}}_{n}} -modules M with finite weight multiplicities. We prove that every such nontrivial module M is either a tensor module or the unique simple submodule in a tensor module associatedwith the de Rham complex on ℂ n {\mathbb{C}^{n}} . 

   more » « less