Abstract We show that, for a finite spectrum $X$ , Spanier–Whitehead duality induces an isomorphism between the cohomological and homological Atiyah–Hirzebruch spectral sequences. As an application, it follows that Poincaré duality for a Poincaré duality complex that is oriented over a ring spectrum $\mathcal{R}$ induces an isomorphism between the two spectral sequences.

Abstract We give a new definition of a Frobenius structure on an algebra object in a monoidal category, generalising Frobenius algebras in the category of vector spaces. Our definition allows Frobenius forms valued in objects other than the unit object and can be seen as a categorical version of Frobenius extensions of the second kind. When the monoidal category is pivotal, we define a Nakayama morphism for the Frobenius structure and explain what it means for this morphism to have finite order. Our main example is a well-studied algebra object in the (additive and idempotent completion of the) Temperley–Lieb category at a root of unity. We show that this algebra has a Frobenius structure and that its Nakayama morphism has order 2. As a consequence, we obtain information about Nakayama morphisms of preprojective algebras of Dynkin type, considered as algebras over the semisimple algebras on their vertices.

Abstract Let $p$ be an odd prime, and let $E_1$ and $E_2$ be two elliptic curves defined over a number field $K$ , with good ordinary reduction at $p$ . We compare the $\Lambda$ -ranks and (generalized) Iwasawa invariants of the Pontryagin duals of the Selmer groups of $E_1$ and $E_2$ over ${\mathbb{Z}}_p^d$ -extensions $\mathbb{L}_\infty$ of $K$ for general $d \ge 1$ under the hypothesis that $E_1[p^i] \cong E_2[p^i]$ as Galois modules for a sufficiently large $i$ . This generalizes and complements previous work over ${\mathbb{Z}}_p$ -extensions. The comparison of generalized Iwasawa invariants is related via an up-down approach to the comparison of the variation of classical Iwasawa invariants over the ${\mathbb{Z}}_p$ -extensions of $K$ which are contained in $\mathbb{L}_\infty$ .

Abstract Let $\Bbbk$ be a field, $H$ a Hopf algebra over $\Bbbk$ , and $R = (_iM_j)_{1 \leq i,j \leq n}$ a generalized matrix algebra. In this work, we establish necessary and sufficient conditions for $H$ to act partially on $R$ . To achieve this, we introduce the concept of an opposite covariant pair and demonstrate that it satisfies a universal property. In the special case where $H = \Bbbk G$ is the group algebra of a group $G$ , we recover the conditions given in [7] for the existence of a unital partial action of $G$ on $R$ .

Abstract We derive faithful inclusions of C*-algebras from a coend-type construction in unitary tensor categories. This gives rise to different potential notions of discreteness for an inclusion in the non-irreducible case and provides a unified framework that encloses the theory of compact quantum group actions. We also provide examples coming from semi-circular systems and from factorization homology. In the irreducible case, we establish conditions under which the C*-discrete and W*-discrete conditions are equivalent.

Abstract Fulton’s matrix Schubert varieties are affine varieties that arise in the study of Schubert calculus in the complete flag variety. Weigandt showed that arbitrary intersections of matrix Schubert varieties, now called ASM varieties, are indexed by alternating sign matrices (ASMs), objects with a long history in enumerative combinatorics. It is very difficult to assess Cohen–Macaulayness of ASM varieties or to compute their codimension, though these properties are well understood for matrix Schubert varieties due to work of Fulton. In this paper, we study these properties of ASM varieties with a focus on the relationship between a pair of ASMs and their direct sum. We also consider ASM pattern avoidance from an algebro-geometric perspective.

Abstract Given a finite abelian group $G$ and $t\in \mathbb{N}$ , there are two natural types of subsets of the Cartesian power $G^t$ ; namely, Cartesian powers $S^t$ where $S$ is a subset of $G$ and (cosets of) subgroups $H$ of $G^t$ . A basic question is whether two such sets intersect. In this paper, we show that this decision problem is NP-complete. Furthermore, for fixed $G$ and $S$ , we give a complete classification: we determine conditions for when the problem is NP-complete and show that in all other cases the problem is solvable in polynomial time. These theorems play a key role in the classification of algebraic decision problems in finitely generated rings developed in later work of the author.

Abstract In this paper, we first describe the cohomology theory of Lie supertriple systems by using the cohomology theory of the associated Leibniz superalgebras. Then we focus on Lie supertriple systems with superderivations, called LSTSDer pairs. We introduce the notion of representations of LSTSDer pairs and investigate their corresponding cohomology theory. We also construct a differential graded Lie algebra whose Maurer–Cartan elements are LSTSDer pairs. Moreover, we consider the relationship between a LSTSDer pair and the associated LeibSDer pair. Furthermore, we develop the 1-parameter formal deformation theory of LSTSDer pairs and prove that it is governed by the cohomology groups. At last, we study abelian extensions of LSTSDer pairs and show that equivalent abelian extensions of LSTSDer pairs are classified by the third cohomology groups.

Abstract This paper focuses on quadratic Hom–Leibniz algebras, defined as (left or right) Hom–Leibniz algebras equipped with symmetric, non-degenerate, and invariant bilinear forms. In particular, we demonstrate that every quadratic regular Hom–Leibniz algebra is symmetric, meaning that it is simultaneously a left and a right Hom–Leibniz algebra. We provide characterizations of symmetric (resp. quadratic) Hom–Leibniz algebras. We also investigate the $\mathrm{T}^*$ -extensions of Hom–Leibniz algebras, establishing their compatibility with solvability and nilpotency. We study the equivalence of such extensions and provide the necessary and sufficient conditions for a nilpotent quadratic Hom–Leibniz algebra to be isometric to a $\mathrm{T}^*$ -extension. Furthermore, through the procedure of double extension, which is a central extension followed by a generalized semi-direct product, we get an inductive description of all quadratic regular Hom–Leibniz algebras, allowing us to reduce their study to that of quadratic regular Hom–Lie algebras. Finally, we construct several non-trivial examples of symmetric (resp. quadratic) Hom–Leibniz algebras.

Abstract Let $\Gamma$ be a Schottky subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and let $X=\Gamma \backslash {\mathbb{H}}^2$ be the associated hyperbolic surface. We consider the family of Hecke congruence coverings of $X$ , which we denote as usual by $ X_0(q) = \Gamma _0(q)\backslash {\mathbb{H}}^2$ . Conditional on the Lindelöf Hypothesis for quadratic L-functions, we establish a uniform and explicit spectral gap for the Laplacian on $ X_0(q)$ for “almost” all prime levels $q$ . Assuming the generalized Riemann hypothesis for quadratic $L$ -functions, we obtain an even larger spectral gap.