4d $$ \mathcal{N} $$ = 2 SCFTs and lisse W-algebras

We continue our studies of the correspondence between 4d $$ \mathcal{N} $$ N = 2 SCFTs and 2d W-algebras. The purpose of this paper is to study the relationship between 2d lisse W-algebras and their 4d SCFT partners. The lisse W-algebra is the W-algebra whose associated Zhu’s C2 algebra is finite dimensional. As the associated variety of Zhu’s C2 algebra is identified with the Higgs branch in the 4d/2d correspondence, the lisse condition is equivalent to the absence of the Higgs branch on the 4d side. We classify 4d $$ \mathcal{N} $$ N = 2 SCFTs which do not admit Higgs branch, then these theories would give lisse W-algebras through the 4d/2d correspondence. In particular, we predict the existence of a large class of new non-admissible lisse W-algebras, which have not been studied before. The 4d theories corresponding to lisse W-algebra can appear in the Higgs branches of generic 4d $$ \mathcal{N} $$ N = 2 SCFTs, therefore they are crucial to understand the Higgs branches of $$ \mathcal{N} $$ N = 2 SCFTs.

Topical application of Apremilast in the treatment of mild to moderate psoriasis

Mild to moderate psoriasis is highly prevalent in about 80% of the global psoriatic population. Current available treatment options for mild to moderate psoriasis are topical dosage forms and are associated variety of setbacks. To address these setbacks, Apremilast topical gels, 2% & 4%, w/w were developed, and a clinical proof of concept study (POC) was performed to establish efficacy and safety. A single centre randomized, double-blind, placebo-controlled study was conducted with apremilast topical gels 2% & 4% w/w in adult mild to moderate psoriatic patients for 12 weeks. The efficacy of the gels was evaluated by comparing the PASI scores before and after treatment of 12 weeks. Both gels exhibited a significant reduction in PASI values when compared with baseline PASI scores. An average percentage inhibition of PASI with test products, i.e. 2% and 4% w/w Apremilast topical gels, are about 46.8% and 34.6%, respectively, after 12 weeks of treatment. The results confirm that the apremilast topical gels are a good option for the treatment of mild to moderate psoriasis and have to be explored further.

The Miocene Tatatila–Las Minas IOCG skarn deposits (Veracruz) as a result of adakitic magmatism in the Trans-Mexican Volcanic Belt

The Cu- and Au-rich Tatatila–Las Minas IOCG skarn deposits in Veracruz (central-east Mexico) are circumscribed to the earliest stages of the Trans-Mexican Volcanic Belt (TMVB) and stand for a metallogenic province directly linked to its tectonomagmatic dynamics. This is the first well-documented case for such metallogenic province. These deposits were formed as skarns between rocks of the Mesozoic carbonate series and Miocene intermediate to acid hypabyssal rocks. New U-Pb zircon and 40Ar/39Ar ages provide evidence for four epochs of magmatic activity in the area: (1) early Permian (Artinskian), in association with the Paleozoic basement, (2) late Oligocene to early Miocene suite of pre-TMVB intrusive rocks, (3) middle to late Miocene suite of early TMVB-related intrusive rocks, and (4) Pliocene intrusive and extrusive rocks of the TMVB, possibly associated with the Los Humeros post-caldera stage. The obtained ages range between 24.60 ± 1.10 and 19.04 ± 0.69 Ma for stage 2, and between 16.34 ± 0.20 and 13.92 ± 0.22 Ma for stage 3. Stage 2 corresponds to a magmatic stage unheard of in the area, until this study. Only stage 3 rocks are associated with the IOCG skarn mineralization, with retrograde stages dated at 12.44 ± 0.09 (chromian muscovite, phyllic association) and 12.18 ± 0.21 Ma (zircon, potassic association). Therefore, the ages of stage-3 intrusive rocks are interpreted to date the formation of the prograde skarn associations (mostly ~15.4 to <14 Ma). The petrogenetic affinity of stage-2 and stage-3 rocks is about the same—the main difference has to do with higher Y and Yb contents in stage-3 rocks (although no affinity with within-plate granites was found), which is suggestive of an interaction of their parental magmas with alkaline magmas that most likely belong to the conterminous and contemporaneous Eastern Mexico Alkaline Province. Petrological indicators (elemental and isotopic) in Cenozoic rocks consistently point to intermediate to acid, metaluminous, I- and S-type rocks that were emplaced in a subduction-related continental arc, within the medium- to high-potassium calc-alkaline series, with high-silica adakitic signatures due associated to deep-sourced magmas that underwent crustal contamination to some degree. The various possible sources for the magmas with adakitic signature in this context can be narrowed down to two of them that are not mutually exclusive: adakitic derived from subducted slab melting and melting-assimilation-storage-homogenization (MASH)-derived adakites. Both sources are, in principle, capable of generating magmas that would eventually produce magmatic-hydrothermal mineralizing systems with an associated variety of ore deposit types, including IOCG. Also, both possible sources for adakites are compatible with the renewed steepening of the subducted slab after a period of flat subduction, for the earliest stage in the evolution of the TMVB.

Shakespearean Pedagogy and Copious Paradox: How Might we Queer Shakespeare’s Work?

Despite opposition to ‘queering the text’ by established Shakespeare critics, this essay sets out to ponder the possibility of teaching Shakespeare in a queer context. The essay begins by examining the social and sexual conditions of the early modern period, making the observation that there were a wide variety of sexualities, moral attitudes, and sexual practices at that time, and that often the early modern moral codes of conduct contradicted each other. It then traces the significance of the rhetorical device of paradox through Lyly and Castiglione, and examines the method of Elizabethan rhetoric, which involved ‘copia’ (the elaboration of one idea into a variety of ideas through language) and its associated variety of meanings. The essay then turns to the text of Shakespeare’s Venus and Adonis, observing that its subject matter—the love of an older woman for a younger man—may have reflected the early modern heterosexual fear of women’s sexuality—and continues to baffle male critics today. Finally, turning to the poem itself—analyzing it in terms of ‘copia’ and paradox—the conclusion suggests that perhaps there was a relationship between Shakespeare’s use of these rhetorical techniques and the sexuality of his time.

Limited diversity—business models of German cooperative banks

Zusammenfassung: Die strukturelle Vielfalt auf dem deutschen Bankenmarkt und die damit verbundene Vielfalt von Geschäftsmodellen dürfte in den vergangenen Jahrzehnten ein wichtiger Stabilitätsfaktor am deutschen Bankenmarkt gewesen sein. Diese Vielfalt zeigt sich zum einen in der 3-Säulen-Struktur der deutschen Kreditwirtschaft, zum anderen aber auch innerhalb der Säulen selbst. Für die deutschen Genossenschaftsbanken dokumentiert eine empirische Analyse eine begrenzte Vielfalt der Geschäftsmodelle. Fast alle Genossenschaftsbanken konzentrieren sich auf das Kredit- und Einlagengeschäft mit Kunden. Innerhalb des traditionellen Geschäftsmodells lassen sich zwei unterschiedliche Geschäftstypen identifizieren, von denen der eine stärker aktiv- und der andere stärker passivlastig ausfällt. Summary: The structural diversity of the German banking market and the associated variety of business models is thought to have been an important factor in the stability of the German banking market in recent decades. This diversity is manifested both in the three-pillar structure of the German banking industry and within the pillars themselves. However, empirical analysis reveals that there is limited variety within the business models used by the German cooperative banks. Almost all cooperative banks concentrate on the lending and deposit-taking business with customers. Within the traditional business model, two different types of business can be identified, one of which is more focused on lending and the other on deposit-taking.

Tropical ideals

We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals strictly includes the tropicalizations of classical ideals, and allows us to define subschemes of tropical toric varieties, generalizing Giansiracusa and Giansiracusa [Equations of tropical varieties, Duke Math. J. 165 (2016), 3379–3433]. We investigate some of the basic structure of tropical ideals, and show that they satisfy many desirable properties that mimic the classical setup. In particular, every tropical ideal has an associated variety, which we prove is always a finite polyhedral complex. In addition we show that tropical ideals satisfy the ascending chain condition, even though they are typically not finitely generated, and also the weak Nullstellensatz.

Gelfand–Kirillov Dimensions of Highest Weight Harish-Chandra Modules for $\boldsymbol{{SU}(p,q)}$

Let $(G,K)$ be an irreducible Hermitian symmetric pair of non-compact type with $G={SU}(p,q)$, and let $\lambda$ be an integral weight such that the simple highest weight module $L(\lambda)$ is a Harish-Chandra $({\mathfrak{g}},K)$-module. We give a combinatorial algorithm for the Gelfand–Kirillov (GK) dimension of $L(\lambda)$. This enables us to prove that the GK dimension of $L(\lambda)$ decreases as the integer $\langle{\lambda+\rho},{\beta}^{\vee} \rangle$ increases, where $\rho$ is the half sum of positive roots and $\beta$ is the maximal non-compact root. Finally by the combinatorial algorithm, we obtain a description of the associated variety of $L(\lambda)$.

A superdimension formula for 𝔤𝔩(m|n)-modules

We give a formula for the superdimension of a finite-dimensional simple [Formula: see text]-module using the Su–Zhang character formula. This formula coincides with the superdimension formulas proven by Weissauer and Heidersdorf–Weissauer. As a corollary, we obtain a simple algebraic proof of a conjecture of Kac–Wakimoto for [Formula: see text], namely, a simple module has nonzero superdimension if and only if it has maximal degree of atypicality. This conjecture was proven originally by Serganova using the Duflo–Serganova associated variety.

Classification of finite-dimensional irreducible modules over -algebras

Finite $W$-algebras are certain associative algebras arising in Lie theory. Each $W$-algebra is constructed from a pair of a semisimple Lie algebra ${\mathfrak{g}}$ (our base field is algebraically closed and of characteristic 0) and its nilpotent element $e$. In this paper we classify finite-dimensional irreducible modules with integral central character over $W$-algebras. In more detail, in a previous paper the first author proved that the component group $A(e)$ of the centralizer of the nilpotent element under consideration acts on the set of finite-dimensional irreducible modules over the $W$-algebra and the quotient set is naturally identified with the set of primitive ideals in $U({\mathfrak{g}})$ whose associated variety is the closure of the adjoint orbit of $e$. In this paper, for a given primitive ideal with integral central character, we compute the corresponding $A(e)$-orbit. The answer is that the stabilizer of that orbit is basically a subgroup of $A(e)$ introduced by G. Lusztig. In the proof we use a variety of different ingredients: the structure theory of primitive ideals and Harish-Chandra bimodules for semisimple Lie algebras, the representation theory of $W$-algebras, the structure theory of cells and Springer representations, and multi-fusion monoidal categories.

On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type

We consider the finiteW-algebraU(𝔤,e) associated to a nilpotent elemente∈𝔤 in a simple complex Lie algebra 𝔤 of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem forU(𝔤,e), we verify a conjecture of Premet, thatU(𝔤,e) always has a 1-dimensional representation when 𝔤 is of typeG2,F4,E6orE7. Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal inU(𝔤) whose associated variety is the coadjoint orbit corresponding to e.