Given an algebra and a finite group acting on it via automorphisms, a natural object of study is the associated skew group algebra. In this article, we study the relationship between quasi-hereditary structures on the original algebra and on the corresponding skew group algebra. Assuming a natural compatibility condition on the partial order, we show that the skew group algebra is quasi-hereditary if and only if the original algebra is. Moreover, we show that in this setting an exact Borel subalgebra of the original algebra which is invariant as a set under the group action gives rise to an exact Borel subalgebra of the skew group algebra and that under this construction, properties such as normality and regularity of the exact Borel subalgebra are preserved.

We obtain a new interpretation of the cohomological Hall algebra $\mathcal {H}_Q$ of a symmetric quiver Q in the context of the theory of vertex algebras. Namely, we show that the graded dual of $\mathcal {H}_Q$ is naturally identified with the underlying vector space of the principal free vertex algebra associated to the Euler form of Q. Properties of that vertex algebra are shown to account for the key results about $\mathcal {H}_Q$. In particular, it has a natural structure of a vertex bialgebra, leading to a new interpretation of the product of $\mathcal {H}_Q$. Moreover, it is isomorphic to the universal enveloping vertex algebra of a certain vertex Lie algebra, which leads to a new interpretation of Donaldson–Thomas invariants of Q (and, in particular, re-proves their positivity). Finally, it is possible to use that vertex algebra to give a new interpretation of CoHA modules made of cohomologies of non-commutative Hilbert schemes.

In this paper, we investigate extensions between graded Verma modules in the Bernstein–Gelfand–Gelfand category $\mathcal{O}$. In particular, we determine exactly which information about extensions between graded Verma modules is given by the coefficients of the R-polynomials. We also give some upper bounds for the dimensions of graded extensions between Verma modules in terms of Kazhdan–Lusztig combinatorics. We completely determine all extensions between Verma module in the regular block of category $\mathcal{O}$ for $\mathfrak{sl}_4$ and construct various “unexpected” higher extensions between Verma modules.

We introduce the notions of quasi-Laurent and Laurent families of simple modules over quiver Hecke algebras of arbitrary symmetrizable types. We prove that such a family plays a similar role of a cluster in quantum cluster algebra theory and exhibits a quantum Laurent positivity phenomenon similar to the basis of the quantum unipotent coordinate ring $\mathcal {A}_q(\mathfrak {n}(w))$, coming from the categorification. Then we show that the families of simple modules categorifying Geiß–Leclerc–Schröer (GLS) clusters are Laurent families by using the Poincaré–Birkhoff–Witt (PBW) decomposition vector of a simple module $X$ and categorical interpretation of (co)degree of $[X]$. As applications of such $\mathbb {Z}\mspace {1mu}$-vectors, we define several skew-symmetric pairings on arbitrary pairs of simple modules, and investigate the relationships among the pairings and $\Lambda$-invariants of $R$-matrices in the quiver Hecke algebra theory.

The main objective of the present paper is to present a version of the Tannaka–Krein type reconstruction theorems: if $F:{\mathcal B}\to {\mathcal C}$ is an exact faithful monoidal functor of tensor categories, one would like to realize ${\mathcal B}$ as category of representations of a braided Hopf algebra $H(F)$ in ${\mathcal C}$. We prove that this is the case iff ${\mathcal B}$ has the additional structure of a monoidal ${\mathcal C}$-module category compatible with F, which equivalently means that F admits a monoidal section. For Hopf algebras, this reduces to a version of the Radford projection theorem. The Hopf algebra is constructed through the relative coend for module categories. We expect this basic result to have a wide range of applications, in particular in the absence of fiber functors, and we give some applications. One particular motivation was the logarithmic Kazhdan–Lusztig conjecture.

In this paper, we give Pieri rules for skew dual immaculate functions and their recently discovered row-strict counterparts. We establish our rules using a right-action analogue of the skew Littlewood–Richardson rule for Hopf algebras of Lam–Lauve–Sottile. We also obtain Pieri rules for row-strict (dual) immaculate functions.

We generalize the influential $C^*$-algebraic results of Kawamura–Tomiyama and Archbold–Spielberg for crossed products of discrete groups actions to the realm of Banach algebras and twisted actions. We prove that topological freeness is equivalent to the intersection property for all reduced twisted Banach algebra crossed products coming from subgroups, and in the untwisted case to a generalized intersection property for a full $L^p$-operator algebra crossed product for any $p\in [1,\,\infty ]$. This gives efficient simplicity criteria for various Banach algebra crossed products. We also use it to identify the prime ideal space of some crossed products as the quasi-orbit space of the action. For amenable actions we prove that the full and reduced twisted $L^p$-operator algebras coincide.

Our goal of the paper is to investigate the Waring problem for upper triangular matrix algebras, which gives a complete solution of a conjecture proposed by Panja and Prasad in 2023.

Many connections and dualities in representation theory and Lie theory can be explained using quasi-hereditary covers in the sense of Rouquier. Recent work by the first-named author shows that relative dominant (and codominant) dimensions are natural tools to classify and distinguish distinct quasi-hereditary covers of a finite-dimensional algebra. In this paper, we prove that the relative dominant dimension of a quasi-hereditary algebra, possessing a simple preserving duality, with respect to a direct summand of the characteristic tilting module is always an even number or infinite and that this homological invariant controls the quality of quasi-hereditary covers that possess a simple preserving duality. To resolve the Temperley–Lieb algebras, we apply this result to the class of Schur algebras $S(2, d)$ and their $q$-analogues. Our second main result completely determines the relative dominant dimension of $S(2, d)$ with respect to $Q=V^{\otimes d}$, the $d$-th tensor power of the natural two-dimensional module. As a byproduct, we deduce that Ringel duals of $q$-Schur algebras $S(2,d)$ give rise to quasi-hereditary covers of Temperley–Lieb algebras. Further, we obtain precisely when the Temperley–Lieb algebra is Morita equivalent to the Ringel dual of the $q$-Schur algebra $S(2, d)$ and precisely how far these two algebras are from being Morita equivalent, when they are not. These results are compatible with the integral setup, and we use them to deduce that the Ringel dual of a $q$-Schur algebra over the ring of Laurent polynomials over the integers together with some projective module is the best quasi-hereditary cover of the integral Temperley–Lieb algebra.

For an action of a finite group on a finite EI quiver, we construct its ‘orbifold’ quotient EI quiver. The free EI category associated to the quotient EI quiver is equivalent to the skew group category with respect to the given group action. Specializing the result to a finite group action on a finite acyclic quiver, we prove that, under reasonable conditions, the skew group category of the path category is equivalent to a finite EI category of Cartan type. If the ground field is of characteristic $p$ and the acting group is a cyclic $p$-group, we prove that the skew group algebra of the path algebra is Morita equivalent to the algebra associated to a Cartan matrix, defined in [C. Geiss, B. Leclerc, and J. Schröer, Quivers with relations for symmetrizable Cartan matrices I: Foundations, Invent. Math. 209 (2017), 61–158]. We apply the Morita equivalence to construct a categorification of the folding projection between the root lattices with respect to a graph automorphism. In the Dynkin cases, the restriction of the categorification to indecomposable modules corresponds to the folding of positive roots.

We explore when the silting-discreteness is inherited. As a result, one obtains that taking idempotent truncations and homological epimorphisms of algebras transmit the silting-discreteness. We also study classification of silting-discrete simply-connected tensor algebras and silting-indiscrete self-injective Nakayama algebras. This paper contains two appendices; one states that every derived-discrete algebra is silting-discrete, and the other is about triangulated categories whose silting objects are tilting.

We describe the one-generator braces A satisfying the condition $A^3 = \langle 0 \rangle$.

The aim of this paper is to study supersoluble skew braces, a class of skew braces that encompasses all finite skew braces of square-free order. It turns out that finite supersoluble skew braces have Sylow towers and that in an arbitrary supersoluble skew brace B many relevant skew brace-theoretical properties are easier to identify: For example, a centrally nilpotent ideal of B is B-centrally nilpotent, a fact that simplifies the computational search for the Fitting ideal; also, B has finite multipermutational level if and only if $(B,+)$ is nilpotent.

Given a finite presentation of the structure skew brace $G(X,r)$ associated with a finite nondegenerate solution of the Yang–Baxter equation (YBE), there is an algorithm that decides if $G(X,r)$ is supersoluble or not. Moreover, supersoluble skew braces are examples of almost polycyclic skew braces, so they give rise to solutions of the YBE on which one can algorithmically work on.

The main theme of this paper is to study $\tau $-tilting subcategories in an abelian category $\mathscr {A}$ with enough projective objects. We introduce the notion of $\tau $-cotorsion torsion triples and investigate a bijection between the collection of $\tau $-cotorsion torsion triples in $\mathscr {A}$ and the collection of support $\tau $-tilting subcategories of $\mathscr {A}$, generalizing the bijection by Bauer, Botnan, Oppermann, and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr {A}$. General definitions and results are exemplified using persistent modules. If $\mathscr {A}=\mathrm{Mod}\mbox {-}R$, where R is a unitary associative ring, we characterize all support $\tau $-tilting (resp. all support $\tau ^-$-tilting) subcategories of $\mathrm{Mod}\mbox {-}R$ in terms of finendo quasitilting (resp. quasicotilting) modules. As a result, it will be shown that every silting module (resp. every cosilting module) induces a support $\tau $-tilting (resp. support $\tau ^{-}$-tilting) subcategory of $\mathrm{Mod}\mbox {-}R$. We also study the theory in $\mathrm {Rep}(Q, \mathscr {A})$, where Q is a finite and acyclic quiver. In particular, we give an algorithm to construct support $\tau $-tilting subcategories in $\mathrm {Rep}(Q, \mathscr {A})$ from certain support $\tau $-tilting subcategories of $\mathscr {A}$.

The minimum number of idempotent generators is calculated for an incidence algebra of a finite poset over a commutative ring. This quantity equals either $\lceil \log _2 n\rceil $ or $\lceil \log _2 n\rceil +1$, where n is the cardinality of the poset. The two cases are separated in terms of the embedding of the Hasse diagram of the poset into the complement of the hypercube graph.

We prove an equality, predicted in the physical literature, between the Jeffrey–Kirwan residues of certain explicit meromorphic forms attached to a quiver without loops or oriented cycles and its Donaldson–Thomas type invariants.

In the special case of complete bipartite quivers we also show independently, using scattering diagrams and theta functions, that the same Jeffrey–Kirwan residues are determined by the the Gross–Hacking–Keel mirror family to a log Calabi–Yau surface.

Carlsen [‘$\ast $-isomorphism of Leavitt path algebras over $\Bbb Z$’, Adv. Math. 324 (2018), 326–335] showed that any $\ast $-homomorphism between Leavitt path algebras over $\mathbb Z$ is automatically diagonal preserving and hence induces an isomorphism of boundary path groupoids. His result works over conjugation-closed subrings of $\mathbb C$ enjoying certain properties. In this paper, we characterise the rings considered by Carlsen as precisely those rings for which every $\ast $-homomorphism of algebras of Hausdorff ample groupoids is automatically diagonal preserving. Moreover, the more general groupoid result has a simpler proof.

In this article, we explore the problem of determining isomorphisms between the twisted complex group algebras of finite $p$-groups. This problem bears similarity to the classical group algebra isomorphism problem and has been recently examined by Margolis-Schnabel. Our focus lies on a specific invariant, referred to as the generalized corank, which relates to the twisted complex group algebra isomorphism problem. We provide a solution for non-abelian $p$-groups with generalized corank at most three.

If E is a graph and K is a field, we consider an ideal I of the Leavitt path algebra $L_K(E)$ of E over K. We describe the admissible pair corresponding to the smallest graded ideal which contains I where the grading in question is the natural grading of $L_K(E)$ by ${\mathbb {Z}}$. Using this description, we show that the right and the left annihilators of I are equal (which may be somewhat surprising given that I may not be self-adjoint). In particular, we establish that both annihilators correspond to the same admissible pair and its description produces the characterisation from the title. Then, we turn to the property that the right (equivalently left) annihilator of any ideal is a direct summand and recall that a unital ring with this property is said to be quasi-Baer. We exhibit a condition on E which is equivalent to unital $L_K(E)$ having this property.

We prove new results concerning the additive Galois module structure of wildly ramified non-abelian extensions $K/\mathbb{Q}$ with Galois group isomorphic to $A_4$, $S_4$, $A_5$, and dihedral groups of order $2p^n$ for certain prime powers $p^n$. In particular, when $K/\mathbb{Q}$ is a Galois extension with Galois group $G$ isomorphic to $A_4$, $S_4$ or $A_5$, we give necessary and sufficient conditions for the ring of integers $\mathcal{O}_{K}$ to be free over its associated order in the rational group algebra $\mathbb{Q}[G]$.