Affine Type Research Articles

A generic classification of locally free representations of affine GLS algebras

Throughout, let K be an algebraically closed field of characteristic 0. We provide a generic classification of locally free representations of Geiß-Leclerc-Schröer's algebras HK(C,D,Ω) associated to affine Cartan matrices C with minimal symmetrizer D and acyclic orientation Ω. Affine GLS algebras are “smooth” degenerations of tame hereditary algebras and as such their representation theory is presumably still tractable. Indeed, we observe several “tame” phenomena of affine GLS algebras even though they are in general representation wild. For the GLS algebras of type BC˜1 we achieve a classification of all stable representations. For general GLS algebras of affine type, we construct a 1-parameter family of representations stable with respect to the defect. Our construction is based on a generalized one-point extension technique. This confirms in particular τ-tilted versions of the second Brauer-Thrall Conjecture recently raised by Mousavand and Schroll-Treffinger-Valdivieso for the class of GLS algebras. Finally, we show that generically every locally free H-module is isomorphic to a direct sum of τ-rigid modules and modules from our 1-parameter family. This generalizes Kac's canonical decomposition from the symmetric to the symmetrizable case in affine types and we obtain such a decomposition by folding the canonical decomposition of dimension vectors over path algebras. As a corollary we obtain that affine GLS algebras are E-tame in the sense of Derksen-Fei and Asai-Iyama.

Tuning Ferulic Acid Solubility in Choline-Chloride- and Betaine-Based Deep Eutectic Solvents: Experimental Determination and Machine Learning Modeling.

Deep eutectic solvents (DES) represent a promising class of green solvents, offering particular utility in the extraction and development of new formulations of natural compounds such as ferulic acid (FA). The experimental phase of the study undertook a systematic investigation of the solubility of FA in DES, comprising choline chloride or betaine as hydrogen bond acceptors and six different polyols as hydrogen bond donors. The results demonstrated that solvents based on choline chloride were more effective than those based on betaine. The optimal ratio of hydrogen bond acceptors to donors was found to be 1:2 molar. The addition of water to the DES resulted in a notable enhancement in the solubility of FA. Among the polyols tested, triethylene glycol was the most effective. Hence, DES composed of choline chloride and triethylene glycol (TEG) (1:2) with added water in a 0.3 molar ration is suggested as an efficient alternative to traditional organic solvents like DMSO. In the second part of this report, the affinities of FA in saturated solutions were computed for solute-solute and all solute-solvent pairs. It was found that self-association of FA leads to a cyclic structure of the C28 type, common among carboxylic acids, which is the strongest type of FA affinity. On the other hand, among all hetero-molecular bi-complexes, the most stable is the FA-TEG pair, which is an interesting congruency with the high solubility of FA in TEG containing liquids. Finally, this work combined COSMO-RS modeling with machine learning for the development of a model predicting ferulic acid solubility in a wide range of solvents, including not only DES but also classical neat and binary mixtures. A machine learning protocol developed a highly accurate model for predicting FA solubility, significantly outperforming the COSMO-RS approach. Based on the obtained results, it is recommended to use the support vector regressor (SVR) for screening new dissolution media as it is not only accurate but also has sound generalization to new systems.

Discovering novel targets of abscisic acid using computational approaches

Abscisic acid (ABA) is a crucial plant hormone that is naturally produced in various mammalian tissues and holds significant potential as a therapeutic molecule in humans. ABA is selected for this study due to its known roles in essential human metabolic processes, such as glucose homeostasis, immune responses, cardiovascular system, and inflammation regulation. Despite its known importance, the molecular mechanism underlying ABA's action remain largely unexplored. This study employed computational techniques to identify potential human ABA receptors. We screened 64 candidate molecules using online servers and performed molecular docking to assess binding affinity and interaction types with ABA. The stability and dynamics of the best complexes were investigated using molecular dynamics simulation over a 100 ns time period. Root mean square fluctuations (RMSF), root mean square deviation (RMSD), solvent-accessible surface area (SASA), radius of gyration (Rg), free energy landscape (FEL), and principal component analysis (PCA) were analyzed. Next, the molecular mechanics Poisson–Boltzmann surface area (MM-PBSA) method was employed to calculate the binding energies of the complexes based on the simulated data. Our study successfully pinpointed four key receptors responsible for ABA signaling (androgen receptor, glucocorticoid receptor, mineralocorticoid receptor, and retinoic acid receptor beta) that have a strong affinity for binding with ABA and remained structurally stable throughout the simulations. The simulations with Hydralazine as an unrelated ligand were conducted to validate the specificity of the identified receptors for ABA. The findings of this study can contribute to further experimental validation and a better understanding of how ABA functions in humans.

Mutation invariants of cluster algebras of rank 2

We consider the mutation invariants of cluster algebras of rank 2. We characterize the mutation invariants of finite type. Two examples are provided for the affine type and we prove the non-existence of Laurent mutation invariants of non-affine type. As an application, a class of Diophantine equations encoded with cluster algebras are studied.

Spreading primitive groups of diagonal type do not exist

The synchronization hierarchy of finite permutation groups consists of classes of groups lying between $2$ -transitive groups and primitive groups. This includes the class of spreading groups, which are defined in terms of sets and multisets of permuted points, and which are known to be primitive of almost simple, affine or diagonal type. In this paper, we prove that in fact no spreading group of diagonal type exists. As part of our proof, we show that all non-abelian finite simple groups, other than six sporadic groups, have a transitive action in which a proper normal subgroup of a point stabilizer is supplemented by all corresponding two-point stabilizers.

Infinite affine, hyperbolic and Lorentzian Weyl groups with their associated Calogero models

We propose generalizations of Calogero models that exhibit invariance with respect to the infinite Weyl groups of affine, hyperbolic, and Lorentzian types. Our approach involves deriving closed analytic formulas for the action of the associated Coxeter elements of infinite order acting on arbitrary roots within their respective root spaces. These formulas are then utilized in formulating the new type of Calogero models.

Experimental and density functional theory studies on some metal oxides and the derived nanoclusters: a comparative effects on human ferritin

A comprehensive investigation into the green synthesis of metal oxide nanoparticles (NPs) has garnered significant attention due to its commendable reliability, sustainability, and environmentally friendly attributes. Green synthesis methods play a crucial role in mitigating the adverse effects associated with conventional approaches employed for nanostructure preparation. This research endeavors to examine the impact of ginger plant extract-assisted green synthesis of metal oxides NPs on the serum ferritin levels of anemic diabetic patients in vitro, focusing specifically on α-Fe2O3 and ZnO NPs. Sixty diabetic volunteers with anemia (35–50 years) and thirty healthy volunteers were enrolled as controls. The assessment was conducted using the VIDAS Ferritin (FER) assay. Photoluminescence (PL) spectroscopy measurements were performed to elucidate the intrinsic and extrinsic transitions of these NPs, affirming the successful formation of α-structured iron oxide. Density functional theory (DFT) calculations were carried out at the B3LYP/6-311++G(d,2p) level of theory to investigate the geometry optimization and molecular electrostatic potential maps of the NPs. Furthermore, TD-DFT calculations were employed to explore their frontier molecular orbitals and various quantum chemical parameters. The binding affinity and interaction types of ZnO and α-Fe2O3 NPs to the active site of the human H-Chain Ferritin (PDB ID: 2FHA) target were determined with the help of molecular docking. Results unveiled the crystalline structure of ZnO and the α-structure of α-Fe2O3. Analysis of the frontier molecular orbitals and dipole moment values demonstrated that ZnO (total dipole moment (D) = 5.80 µ) exhibited superior chemical reactivity, biological activity, and stronger molecular interactions with diverse force fields compared to α-Fe2O3 (D = 2.65 µ). Molecular docking of the metal oxides NPs with human H-chain ferritin provided evidence of robust hydrogen bond interactions and metal-acceptor bonds between the metal oxides and the target protein. This finding could have a great impact on using metal oxides NPs-ferritin as a therapeutic protein, however, further studies on their toxicity are required.Graphical abstract

Affine Stresses, Inverse Systems, and Reconstruction Problems

Abstract A conjecture of Kalai asserts that for $d\geq 4$, the affine type of a prime simplicial $d$-polytope $P$ can be reconstructed from the space of affine $2$-stresses of $P$. We prove this conjecture for all $d\geq 5$. We also prove the following generalization: for all pairs $(i,d)$ with $2\leq i\leq \lceil \frac d 2\rceil -1$, the affine type of a simplicial $d$-polytope $P$ that has no missing faces of dimension $\geq d-i+1$ can be reconstructed from the space of affine $i$-stresses of $P$. A consequence of our proofs is a strengthening of the Generalized Lower Bound Theorem: it was proved by Nagel that for any simplicial $(d-1)$-sphere $\Delta $ and $1\leq k\leq \lceil \frac {d}{2}\rceil -1$, $g_{k}(\Delta )$ is at least as large as the number of missing $(d-k)$-faces of $\Delta $; here we show that, for $1\leq k\leq \lfloor \frac {d}{2}\rfloor -1$, equality holds if and only if $\Delta $ is $k$-stacked. Finally, we show that for $d\geq 4$, any simplicial $d$-polytope $P$ that has no missing faces of dimension $\geq d-1$ is redundantly rigid, that is, for each edge $e$ of $P$, there exists an affine $2$-stress on $P$ with a non-zero value on $e$.

Reduction for flag‐transitive point‐primitive 2‐(v, k, λ) designs with λ prime

AbstractIt is shown that the flag‐transitive, point‐primitive automorphism groups of 2‐ designs with prime must be of affine type or almost simple type.

Evaluation birepresentations of affine type A Soergel bimodules

In this paper, we use Soergel calculus to define a monoidal functor, called the evaluation functor, from extended affine type A Soergel bimodules to the homotopy category of bounded complexes in finite type A Soergel bimodules. This functor categorifies the well-known evaluation homomorphism from the extended affine type A Hecke algebra to the finite type A Hecke algebra. Through it, one can pull back the triangulated birepresentation induced by any finitary birepresentation of finite type A Soergel bimodules to obtain a triangulated birepresentation of extended affine type A Soergel bimodules. We show that if the initial finitary birepresentation in finite type A is a cell birepresentation, the evaluation birepresentation in extended affine type A has a finitary cover, which we illustrate by working out the case of cell birepresentations with subregular apex in detail.

Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras

Let g be a complex simple Lie algebra and UqLg the corresponding quantum affine algebra. We construct a functor Fθ between finite-dimensional modules over a quantum symmetric pair subalgebra of affine type Uqk⊂UqLg and an orientifold KLR algebra arising from a framed quiver with a contravariant involution, providing a boundary analogue of the Kang-Kashiwara-Kim-Oh generalized Schur-Weyl duality. With respect to their construction, our combinatorial model is further enriched with the poles of a trigonometric K-matrix intertwining the action of Uqk on finite-dimensional UqLg-modules. By construction, Fθ is naturally compatible with the Kang-Kashiwara-Kim-Oh functor in that, while the latter is a functor of monoidal categories, Fθ is a functor of module categories. Relying on a suitable isomorphism à la Brundan-Kleshchev-Rouquier, we prove that Fθ recovers the Schur-Weyl dualities due to Fan-Lai-Li-Luo-Wang-Watanabe in quasi-split type AIII.

Harder-Narasimhan filtrations and zigzag persistence

We introduce a sheaf-theoretic stability condition for finite acyclic quivers. Our main result establishes that for representations of affine type A˜ quivers, there is a precise relationship between the associated Harder-Narasimhan filtration and the barcode of the periodic zigzag persistence module obtained by unwinding the underlying quiver.

Antcin-B, a phytosterol-like compound from Taiwanofungus camphoratus inhibits SARS-CoV-2 3-chymotrypsin-like protease (3CLPro) activity in silico and in vitro

Despite the remarkable development of highly effective vaccines, including mRNA-based vaccines, within a limited timeframe, coronavirus disease 2019 (COVID-19) caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is not been entirely eradicated. Thus, it is crucial to identify new effective anti-3CLPro compounds, pivotal for the replication of SARS-CoV-2. Here, we identified an antcin-B phytosterol-like compound from Taiwanofungus camphoratus that targets 3CLPro activity. MTT assay and ADMET prediction are employed for assessing potential cytotoxicity. Computational molecular modeling was used to screen various antcins and non-antcins for binding affinity and interaction type with 3CLPro. Further, these compounds were subjected to study their inhibitory effects on 3CLPro activity in vitro. Our results indicate that antcin-B has the best binding affinity by contacting residues like Leu141, Asn142, Glu166, and His163 via hydrogen bond and salt bridge and significantly inhibits 3CLPro activity, surpassing the positive control compound (GC376). The 100 ns molecular dynamics simulation studies showed that antcin-B formed consistent, long-lasting water bridges with Glu166 for their inhibitory activity. In summary, antcin-B could be useful to develop therapeutically viable drugs to inhibit SARS-CoV-2 replication alone or in combination with medications specific to other SARS-CoV-2 viral targets.

Representation type of cyclotomic quiver Hecke algebras of type [formula omitted

We first investigate a connected quiver consisting of all dominant maximal weights for an integrable highest weight module in affine type A. This quiver provides an efficient method to obtain all dominant maximal weights. Then, we completely determine the representation type of cyclotomic Khovanov-Lauda-Rouquier algebras of arbitrary level in affine type A, by using the quiver we construct. This result gives a complete classification for the representation type of blocks of cyclotomic Hecke algebras since cyclotomic KLR algebras of type Aℓ(1) form a one-parameter family and cyclotomic Hecke algebras occur at a special parameter, i.e., t=−2 if ℓ=1 and t=(−1)ℓ+1 if ℓ≥2.

McKay correspondence, cohomological Hall algebras and categorification

Let π : Y → X \pi \colon Y\to X denote the canonical resolution of the two dimensional Kleinian singularity X X of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of ω \omega -semistable properly supported sheaves on Y Y with fixed slope μ \mu and ζ \zeta -semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on ζ \zeta depending on the polarization ω \omega and μ \mu . These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide a finer description of the cohomological, K-theoretical and categorified Hall algebra of ω \omega -semistable properly supported sheaves on Y Y with fixed slope μ \mu : for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams.

Cellularity and subdivision of KLR and weighted KLRW algebras

Weighted KLRW algebras are diagram algebras generalizing KLR algebras. This paper undertakes a systematic study of these algebras culminating in the construction of homogeneous affine cellular bases in affine types A and C, which immediately gives cellular bases for the cyclotomic quotients of these algebras. In addition, we construct subdivision homomorphisms that relate weighted KLRW algebras for different quivers. As an application we obtain new results about the (cyclotomic) KLR algebras of affine type, including (re)proving that the cyclotomic KLR algebras of type Ae(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^{(1)}_{e}$$\\end{document} and Ce(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{(1)}_{e}$$\\end{document} are graded cellular algebras.

The Cluster Multiplication Theorem for Acyclic Quantum Cluster Algebras

Abstract Let $Q$ be a finite acyclic valued quiver. We give the cluster multiplication formulas in the quantum cluster algebra of $Q$ with arbitrary coefficients, by applying certain quotients of derived Hall subalgebras of $Q$. These formulas can be viewed as the quantum version of the cluster multiplication theorem in the classical cluster algebra proved by Caldero–Keller for finite type, Hubery for affine type, and Xiao–Xu for acyclic quivers.

Property investment and the making of the ambivalent elective Polish diaspora in Israel

AbstractThe article explores investment decisions made by Israelis who purchased or intended to purchase a residential property in Poland. Specifically, it focuses on their set of motivations to invest there and the extent to which their ethno‐national or other types of affinity with the country played a role in their decision. Drawing on interviews with (Jewish) Israeli citizens, we argue that their Choice to invest in Poland was not only financial but influenced also by strong emotional connections to the country, a combination we term ‘sentrumental’ (instrumental and sentimental). We contend that the decision of Israeli Jews to buy property in Poland, against the historical backdrop of the traumatic experience of Jews there, is highly contentious. Analyzing the discursive strategies they use to explain, indeed justify, their unorthodox decision, we show how their emotional ties to Poland often conflict with its controversial history and their own personal identities. It is this conflict, we conclude, that makes Israeli Jews with various biographical ties to Poland an inherently ambivalent elective diaspora.

Reduction for flag-transitive symmetric designs with k > λ(λ − 2)

Let G be a flag-transitive automorphism group of a symmetric (v,k,λ) design D with k>λ(λ−2). O'Reilly Regueiro proved that if G is point-imprimitive, then D has parameters (v,k,λ)=(λ2(λ+2),λ(λ+1),λ). In the present paper, we consider the case that G is point-primitive. By applying the O'Nan-Scott Theorem, we prove that G must be of affine type or almost simple type.

Components of AR-quivers for string algebras of type [formula omitted] and a conjecture by Geiss-Leclerc-Schröer

We study modules of certain string algebras, which are referred to as of affine type C˜. We introduce minimal string modules and apply them to explicitly describe components of the Auslander-Reiten quivers of the string algebras and τ-locally free modules defined by Geiss-Lerclerc-Schröer. In particular, we show that an indecomposable module is τ-locally free if and only if it is preprojective, or preinjective or regular in a tube. As an application, we prove Geiss-Leclerc-Schröer's conjecture on the correspondence between positive roots of type C˜ and τ-locally free modules of the corresponding string algebras. Furthermore, given a positive root α, we show that if α is real, then there is a unique τ-locally free module M (up to isomorphism) with rank_M=α; otherwise there are families of τ-locally free modules with rank_M=α.