Abstract Let and be the Witt Lie algebras. Clearly, is a proper subalegbra of . Surprisingly, we prove that simple smooth modules over are exactly the simple modules over studied by Rudakov (no need to take completion). Then, we find an easy and elementary way to classify all simple smooth modules over . When the height or , any nontrivial simple smooth ‐module is isomorphic to an induced module from a simple smooth ‐module . When and , any such module is the unique simple quotient of the tensor module for some simple ‐module , where is a particular simple module over the Weyl algebra . We further show that a simple ‐module is a smooth module if and only if the action of each of particular vectors in is locally finite on .

Abstract Let F be a non-Archimedean local field. For any irreducible smooth representation $$\pi $$ π of $$\textrm{GL}_n(F)$$ GL n ( F ) and a multisegment $${\mathfrak {m}}$$ m , we have an operation $$D_{{\mathfrak {m}}}(\pi )$$ D m ( π ) to construct a simple quotient $$\tau $$ τ of a Bernstein-Zelevinsky derivative of $$\pi $$ π . This article continues the previous one to study the following poset $$\begin{aligned} {\mathcal {S}}(\pi , \tau ) {:}{=}\left\{ {\mathfrak {n}} : D_{{\mathfrak {n}}}(\pi )\cong \tau \right\} , \end{aligned}$$ S ( π , τ ) : = n : D n ( π ) ≅ τ , where $${\mathfrak {n}}$$ n runs for all the multisegments. Here the partial ordering on $${\mathcal {S}}(\pi , \tau )$$ S ( π , τ ) comes from the Zelevinsky ordering. We show that the poset has a unique minimal multisegment. Along the way, we introduce two new ingredients: fine chain orderings and local minimizability.

We study stable homotopy through unstable methods applied to its representing infinite loop space, as pioneered by Curtis and Wellington. Using cohomology instead of homology, we find a width filtration whose subquotients are simple quotients of Dickson algebras. We make initial calculations and determine towers in the resulting width spectral sequence. We also make calculations related to the image of $J$ and conjecture that it is captured exactly by the lowest filtration in the width spectral sequence.

Let F F be a local non-Archimedean field. A sequence of derivatives of generalized Steinberg representations can be used to construct simple quotients of Bernstein–Zelevinsky derivatives of irreducible representations of G L n ( F ) \mathrm {GL}_n(F) . In the first of a series of articles, we introduce a notion of a highest derivative multisegment, which in turn gives a combinatorial approach to study problems about those simple quotients. We also prove a double derivative result along the way.

In this paper, we describe how to explicitly construct infinitely many finite simple groups as characteristic quotients of the rank 2 free group F_{2} . This shows that a “baby” version of the Wiegold conjecture [in: Geometry, Rigidity, and Group Actions (2011), 609–643] fails for F_{2} and provides counterexamples to two conjectures in the theory of noncongruence subgroups of \mathrm{SL}_{2}(\mathbb{Z}) by Chen [Math. Ann. 371 (2018), 41–126]. Our main result explicitly produces, for every prime power q\ge 7 , the groups \mathrm{SL}_{3}(\mathbb{F}_{q}) and \mathrm{SU}_{3}(\mathbb{F}_{q}) as characteristic quotients of F_{2} . Our strategy is to study specializations of the Burau representation for the braid group B_{4} , exploiting an exceptional relationship between F_{2} and B_{4} first observed by Dyer, Formanek, and Grossman [Arch. Math. (Basel) 38 (1982), 404–409]. Weisfeiler’s strong approximation theorem guarantees that our specializations are surjective for infinitely many primes, but they are not effective. To make our result effective, we give another proof of surjectivity via a careful analysis of the maximal subgroup structures of \mathrm{SL}_{3}(\mathbb{F}_{q}) and \mathrm{SU}_{3}(\mathbb{F}_{q}) . These examples are minimal in the sense that no finite simple group of the form \mathrm{PSL}_{2}(\mathbb{F}_{q}) appears as a characteristic quotient of F_{2} .

Calcification is vital to marine organisms that produce calcium carbonate shells and skeletons. However, how calcification is impacted by ongoing environmental changes, including ocean acidification, remains incompletely understood due to complex relationships among the carbonate system variables hypothesized to drive calcification. Here, we experimentally decouple these drivers in an exploration of shell formation in adult marine mussels, Mytilus californianus. In contrast to models that focus on single parameters like calcium carbonate saturation state, our results implicate two independent factors, bicarbonate concentration and seawater pH, in governing calcification. While qualitatively similar to ideas embodied in the related substrate-inhibitor ratio (bicarbonate divided by hydrogen ion concentration), our data highlight that merging bicarbonate ion and hydrogen ion concentrations into a simple quotient obscures important features of calcification. Considering a dual-parameter framework improves mechanistic understanding of how calcifiers interact with complex and changing chemical conditions.

We prove that any unitary highest weight module over a universal minimal quantum affine W-algebra at non-critical level descends to its simple quotient. We find the defining relations of the unitary simple minimal quantum affine W-algebras and the list of all their irreducible positive energy modules. We also classify all irreducible highest weight modules for the simple affine vertex algebras in the cases when the associated simple minimal W-algebra is unitary.

We investigate the source algebra class of a p-block with cyclic defect groups of the group algebra of a finite group. By the work of Linckelmann this class is parametrized by the Brauer tree of the block together with a sign function on its vertices and an endo-permutation module of a defect group. We prove that this endo-permutation module can be read off from the character table of the group. We also prove that this module is trivial for all cyclic p-blocks of quasisimple groups with a simple quotient which is a sporadic group, an alternating group, a group of Lie type in defining characteristic, or a group of Lie type in cross-characteristic for which the prime p is large enough in a certain sense.

We construct several series of explicit presentations of infinite hyperbolic groups enjoying Kazhdan's property (T). Some of them are significantly shorter than the previously known shortest examples. Moreover, we show that some of those hyperbolic Kazhdan groups possess finite simple quotient groups of arbitrarily large rank; they constitute the first-known specimens combining those properties. All the hyperbolic groups we consider are non-positively curved k $k$ -fold generalized triangle groups, that is, groups that possess a simplicial action on a CAT(0) triangle complex, which is sharply transitive on the set of triangles, and such that edge-stabilizers are cyclic of order k $k$ . Appendices A, B and C are provided separately as supplementary material with the published article.

The Bershadsky–Polyakov algebras are the original examples of nonregular W-algebras, obtained from the affine vertex operator algebras associated with mathfrak {sl}_3 by quantum Hamiltonian reduction. In Fehily et al. (Comm Math Phys 385:859–904, 2021), we explored the representation theories of the simple quotients of these algebras when the level mathsf {k} is nondegenerate-admissible. Here, we combine these explorations with Adamović’s inverse quantum Hamiltonian reduction functors to study the modular properties of Bershadsky–Polyakov characters and deduce the associated Grothendieck fusion rules. The results are not dissimilar to those already known for the affine vertex operator algebras associated with mathfrak {sl}_2, except that the role of the Virasoro minimal models in the latter is here played by the minimal models of Zamolodchikov’s mathsf {W}_3 algebras.

We study disemisimple Lie algebras, i.e., Lie algebras which can be written as a vector space sum of two semisimple subalgebras. We show that a Lie algebra g is disemisimple if and only if its solvable radical coincides with its nilradical and is a prehomogeneous s-module for a Levi subalgebra s of g. We use the classification of prehomogeneous s-modules for simple Lie algebras s given by Vinberg to show that the solvable radical of a disemisimple Lie algebra with simple Levi subalgebra is abelian. We extend this result to disemisimple Lie algebras having no simple quotients of type A.

Let [Formula: see text] be the Poisson enveloping algebra of the Lie algebra [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic zero. The quotient algebras [Formula: see text] [Formula: see text], where [Formula: see text] is the standard Casimir element of [Formula: see text] in [Formula: see text] and [Formula: see text], are proven to be simple in [U. Umirbaev and V. Zhelyabin, A Dixmier theorem for Poisson enveloping algebras, J. Algebra 568 (2021) 576–600]. Using a result by Makar–Limanov [22], we describe generators of the automorphism group of [Formula: see text] and represent this group as an amalgamated product of its subgroups. Moreover, using similar results by Dixmier [Quotients simples de l’algebre enveloppante de [Formula: see text], J. Algebra 24 (1973) 551–564] and O. Fleury [Sur les sous-groupes finis de [Formula: see text] et [Formula: see text], J. Algebra 200 (1998) 404–427] for the quotient algebras [Formula: see text], where [Formula: see text] is the standard Casimir element of [Formula: see text] in the universal enveloping algebra [Formula: see text], we prove that the automorphism groups of [Formula: see text] and [Formula: see text] are isomorphic.

The basis argument of this study departures from the spatial differences between north and south region of Central Java. The north-south corridor classification derives from the rapid growth in north coastal region and the prosperous inland principalities in south-central region during Java colonial. The rivalry between The Dutch and inland principalities to attain absolute control over north-east region indicates that the north corridor plays the primary region. This geographical interpretation of north-(interior)-south region has been embedded in mind. This study explores the spatial pattern of socio-economic distribution in Central Java and Yogyakarta based on regional specialization. The result of the simple Location Quotient (??) indicator from 1990-2010 shows that structural change from the agriculture to the non-agriculture sectors was mostly found in the north corridor. On the contrary, the primary sector in south corridor remained agriculture.

We prove the conjecture of Gaiotto and Rap\v{c}\'ak that the $Y$-algebras $Y_{L,M,N}[\psi]$ with one of the parameters $L,M,N$ zero, are simple one-parameter quotients of the universal two-parameter $\mathcal{W}_{1+\infty}$-algebra, and satisfy a symmetry known as triality. These $Y$-algebras are defined as the cosets of certain non-principal $\mathcal{W}$-algebras and $\mathcal{W}$-superalgebras by their affine vertex subalgebras, and triality is an isomorphism between three such algebras. Special cases of our result provide new and unified proofs of many theorems and open conjectures in the literature on $\mathcal{W}$-algebras of type $A$. This includes (1) Feigin-Frenkel duality, (2) the coset realization of principal $\mathcal{W}$-algebras due to Arakawa and us, (3) Feigin and Semikhatov's conjectured triality between subregular $\mathcal{W}$-algebras, principal $\mathcal{W}$-superalgebras, and affine vertex superalgebras, (4) the rationality of subregular $\mathcal{W}$-algebras due to Arakawa and van Ekeren, (5) the identification of Heisenberg cosets of subregular $\mathcal{W}$-algebras with principal rational $\mathcal{W}$-algebras that was conjectured in the physics literature over 25 years ago. Finally, we prove the conjectures of Proch\'azka and Rap\v{c}\'ak on the explicit truncation curves realizing the simple $Y$-algebras as $\mathcal{W}_{1+\infty}$-quotients, and on their minimal strong generating types.

We study the representation theory of non-admissible simple affine vertex algebra [Formula: see text]. We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra [Formula: see text], and show that it generates the maximal ideal in [Formula: see text]. We classify irreducible [Formula: see text]-modules in the category [Formula: see text], and determine the fusion rules between irreducible modules in the category of ordinary modules [Formula: see text]. It turns out that this fusion algebra is isomorphic to the fusion algebra of [Formula: see text]. We also prove that [Formula: see text] is a semi-simple, rigid braided tensor category. In our proofs, we use the notion of collapsing level for the affine [Formula: see text]-algebra, and the properties of conformal embedding [Formula: see text] at level [Formula: see text] from D. Adamovic et al. [Finite vs infinite decompositions in conformal embeddings, Comm. Math. Phys. 348 (2016) 445–473.]. We show that [Formula: see text] is a collapsing level with respect to the subregular nilpotent element [Formula: see text], meaning that the simple quotient of the affine [Formula: see text]-algebra [Formula: see text] is isomorphic to the Heisenberg vertex algebra [Formula: see text]. We prove certain results on vanishing and non-vanishing of cohomology for the quantum Hamiltonian reduction functor [Formula: see text]. It turns out that the properties of [Formula: see text] are more subtle than in the case of minimal reduction.

Duality of subregular [formula omitted]-algebras and principal [formula omitted]-superalgebras

The Bershadsky-Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to $\mathfrak{sl}_3$ and their simple quotients have a long history of applications in conformal field theory and string theory. Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications [arxiv:1005.0185, arxiv:1910.13781]. In particular, we prove that the simple Bershadsky-Polyakov algebras with admissible nonintegral $\mathsf{k}$ are always rational in category $\mathscr{O}$, whilst they always admit nonsemisimple relaxed highest-weight modules unless $\mathsf{k}+\frac{3}{2} \in \mathbb{Z}_{\ge0}$.

Simple weight modules with finite-dimensional weight spaces over Witt superalgebras

In this note, we prove that the universal affine vertex algebra associated with a simple Lie algebra 𝔤 is simple if and only if the associated variety of its unique simple quotient is equal to 𝔤 * . We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra.

Permutation orbifolds of Virasoro vertex algebras and W-algebras