We prove that the essential dimension of central simple algebras of degree $p^{\ell m}$ and exponent $p^{m}$ over fields $F$ containing a base-field $k$ of characteristic $p$ is at least $\ell +1$ when $k$ is perfect. We do this by observing that the $p$-rank of $F$ bounds the symbol length in $\text{Br}_{p^{m}}(F)$ and that there exist indecomposable $p$-algebras of degree $p^{\ell m}$ and exponent $p^{m}$. We also prove that the symbol length of the Kato-Milne cohomology group $\text{H}_{p^{m}}^{n+1}(F)$ is bounded from above by $\binom{r}{n}$ where $r$ is the $p$-rank of the field, and provide upper and lower bounds for the essential dimension of Brauer classes of a given symbol length.

We give a complete description of a basis of the extension spaces between indecomposable string and quasi-simple band modules in the module category of a gentle algebra.

We study the derived category of a complete intersection $X$ of bilinear divisors in the orbifold $\operatorname{Sym}^{2}\mathbb{P}(V)$. Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between $\operatorname{Sym}^{2}\mathbb{P}(V)$ and a category of modules over a sheaf of Clifford algebras on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$. The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating $D^{b}(X)$ into a derived category of factorisations on a Landau–Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi–Yau 3-folds have equivalent derived categories.

Let Φ be a finite-dimensional algebra over a field k. Kleiner described the Auslander–Reiten sequences in a precovering extension closed subcategory ${\rm {\cal X}}\subseteq {\rm mod }\,\Phi $. If $X\in \mathcal {X}$ is an indecomposable such that ${\rm Ext}_\Phi ^1 (X,{\rm {\cal X}})\ne 0$ and $\zeta X$ is the unique indecomposable direct summand of the $\mathcal {X}$-cover $g:Y\to D\,{\rm Tr}\,X$ such that ${\rm Ext}_\Phi ^1 (X,\zeta X)\ne 0$, then there is an Auslander–Reiten sequence in $\mathcal {X}$ of the form

$${\rm \epsilon }:0\to \zeta X\to {X}^{\prime}\to X\to 0.$$

${\rm En}{\rm d}_\Phi (X)$

$$\delta :0\to Y\to {Y}^{\prime}\buildrel \eta \over \longrightarrow X\to 0$$

$\mathcal {X}$

${\rm mod}\,\Phi $

In this paper, we give higher-dimensional generalizations of this. Let $d\geq 1$ be an integer. A d-cluster tilting subcategory ${\rm {\cal F}}\subseteq {\rm mod}\,\Phi $ plays the role of a higher ${\rm mod}\,\Phi $. Such an $\mathcal {F}$ is a d-abelian category, where kernels and cokernels are replaced by complexes of d objects and short exact sequences by complexes of d + 2 objects. We give higher versions of the above results for an additive ‘d-extension closed’ subcategory $\mathcal {X}$ of $\mathcal {F}$.

The main result of this note implies that any function from the product of several vector spaces to a vector space can be uniquely decomposed into the sum of mutually orthogonal functions that are odd in some of the arguments and even in the other arguments. Probabilistic notions and facts are employed to simplify statements and proofs.

Let $\mathfrak{o}$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ and $\mathfrak{X}_{0}$ a smooth formal $\mathfrak{o}$-scheme. Let $\mathfrak{X}\rightarrow \mathfrak{X}_{0}$ be an admissible blow-up. In the first part, we introduce sheaves of differential operators $\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ on $\mathfrak{X}$, for every sufficiently large positive integer $k$, generalizing Berthelot’s arithmetic differential operators on the smooth formal scheme $\mathfrak{X}_{0}$. The coherence of these sheaves and several other basic properties are proven. In the second part, we study the projective limit sheaf $\mathscr{D}_{\mathfrak{X},\infty }=\mathop{\varprojlim }\nolimits_{k}\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ and introduce its abelian category of coadmissible modules. The inductive limit of the sheaves $\mathscr{D}_{\mathfrak{X},\infty }$, over all admissible blow-ups $\mathfrak{X}$, is a sheaf $\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$ on the Zariski–Riemann space of $\mathfrak{X}_{0}$, which gives rise to an abelian category of coadmissible modules. Analogues of Theorems A and B are shown to hold in each of these settings, that is, for $\mathscr{D}_{\mathfrak{X},k}^{\dagger }$, $\mathscr{D}_{\mathfrak{X},\infty }$, and $\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$.

Let $R$ be a ring and $T$ be a good Wakamatsu-tilting module with $S=\text{End}(T_{R})^{op}$. We prove that $T$ induces an equivalence between stable repetitive categories of $R$ and $S$ (i.e., stable module categories of repetitive algebras $\hat{R}$ and ${\hat{S}}$). This shows that good Wakamatsu-tilting modules seem to behave in Morita theory of stable repetitive categories as that tilting modules of finite projective dimension behave in Morita theory of derived categories.

The first fundamental theorem of invariant theory for the orthosymplectic supergroup scheme $\text{OSp}(m|2n)$ states that there is a full functor from the Brauer category with parameter $m-2n$ to the category of tensor representations of $\text{OSp}(m|2n)$. This has recently been proved using algebraic supergeometry to relate the problem to the invariant theory of the general linear supergroup. In this work, we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. Specifically, we give a linear description of the kernel of the surjective homomorphism from the Brauer algebra to endomorphisms of tensor space, which commute with the orthosymplectic supergroup. The main result has a clear and succinct formulation in terms of Brauer diagrams. Our proof includes, as special cases, new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues, which are independent of the Capelli identities. The results of this paper have led to the result that the map from the Brauer algebra ${\mathcal{B}}_{r}(m-2n)$ to endomorphisms of $V^{\otimes r}$ is an isomorphism if and only if $r<(m+1)(n+1)$.

Let D be a unital associative division ring and D[t, σ, δ] be a skew polynomial ring, where σ is an endomorphism of D and δ a left σ-derivation. For each f ϵ D[t, σ, δ] of degree m > 1 with a unit as leading coefficient, there exists a unital nonassociative algebra whose behaviour reflects the properties of f. These algebras yield canonical examples of right division algebras when f is irreducible. The structure of their right nucleus depends on the choice of f. In the classical literature, this nucleus appears as the eigenspace of f and is used to investigate the irreducible factors of f. We give necessary and sufficient criteria for skew polynomials of low degree to be irreducible. These yield examples of new division algebras Sf.

A new homological dimension, called rigidity dimension, is introduced to measure the quality of resolutions of finite dimensional algebras (especially of infinite global dimension) by algebras of finite global dimension and big dominant dimension. Upper bounds of the dimension are established in terms of extensions and of Hochschild cohomology, and finiteness in general is derived from homological conjectures. In particular, the rigidity dimension of a non-semisimple group algebra is finite and bounded by the order of the group. Then invariance under stable equivalences is shown to hold, with some exceptions when there are nodes in case of additive equivalences, and without exceptions in case of triangulated equivalences. Stable equivalences of Morita type and derived equivalences, both between self-injective algebras, are shown to preserve rigidity dimension as well.

The cup product in the cohomology of algebras over quadratic operads has been studied in the general setting of Koszul duality for operads. We study the cup product on the cohomology of n-ary totally associative algebras with an operation of even (homological) degree. This cup product endows the cohomology with the structure of an n-ary partially associative algebra with an operation of even or odd degree depending on the parity of n. In the cases n=3 and n=4, we provide an explicit definition of this cup product and prove its basic properties.

Let R be a semiprime ring with the extended centroid C and Q the maximal right ring of quotients of R. Set [y, x]1 = [y, x] = yx − xy for x, y ∈ Q and inductively [y, x]k = [[y, x]k−1, x] for k > 1. Suppose that f : R → Q is an additive map satisfying [f(x), x]n = 0 for all x ∈ R, where n is a fixed positive integer. Then it can be shown that there exist λ ∈ C and an additive map μ : R → C such that f(x) = λx + μ(x) for all x ∈ R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996.

We study exact sequences of finite tensor categories of the form Rep G → 𝒞 → 𝒟, where G is a finite group. We show that, under suitable assumptions, there exists a group Γ and mutual actions by permutations ⊳ : Γ × G → G and ⊲ : Γ × G→ Γ that make (G, Γ) into matched pair of groups endowed with a natural crossed action on 𝒟 such that 𝒞 is equivalent to a certain associated crossed extension 𝒟(G,Γ) of 𝒟. Dually, we show that an exact sequence of finite tensor categories VecG → 𝒞 → 𝒟 induces an Aut(G)-grading on 𝒞 whose neutral homogeneous component is a (Z(G), Γ)-crossed extension of a tensor subcategory of 𝒟. As an application we prove that such extensions 𝒞 of 𝒟 are weakly group-theoretical fusion categories if and only if 𝒟 is a weakly group-theoretical fusion category. In particular, we conclude that every semisolvable semisimple Hopf algebra is weakly group-theoretical.

We answer a question of Skalski and Sołan (2016) about inner faithfulness of the Curran’s map of extending a quantum increasing sequence to a quantum permutation. Roughly speaking, we find a inductive setting in which the inner faithfulness of Curran’s map can be boiled down to inner faithfulness of similar map for smaller algebras and then rely on inductive generation result for quantum permutation groups of Brannan, Chirvasitu and Freslon (2018).

We show that a directed graph $E$ is a finite graph with no sinks if and only if, for each commutative unital ring $R$, the Leavitt path algebra $L_{R}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if the $C^{\ast }$-algebra $C^{\ast }(E)$ is unital and $\text{rank}(K_{0}(C^{\ast }(E)))=\text{rank}(K_{1}(C^{\ast }(E)))$. Let $k$ be a field and $k^{\times }$ be the group of units of $k$. When $\text{rank}(k^{\times })<\infty$, we show that the Leavitt path algebra $L_{k}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if $L_{k}(E)$ is unital and $\text{rank}(K_{1}(L_{k}(E)))=(\text{rank}(k^{\times })+1)\text{rank}(K_{0}(L_{k}(E)))$. We also show that any unital $k$-algebra which is Morita equivalent or stably isomorphic to an algebraic Cuntz–Krieger algebra, is isomorphic to an algebraic Cuntz–Krieger algebra. As a consequence, corners of algebraic Cuntz–Krieger algebras are algebraic Cuntz–Krieger algebras.

We classify tame block algebras of Hecke algebras of classical type over an algebraically closed field of characteristic not equal to two.

In this paper, we study some connections between the polynomial ring $R[y]$ and the differential polynomial ring $R[x;D]$. In particular, we answer a question posed by Smoktunowicz, which asks whether $R[y]$ is nil when $R[x;D]$ is nil, provided that $R$ is an algebra over a field of positive characteristic and $D$ is a locally nilpotent derivation.

We prove that $A_{R}(G)\otimes _{R}A_{R}(H)\cong A_{R}(G\times H)$ if $G$ and $H$ are Hausdorff ample groupoids. As part of the proof, we give a new universal property of Steinberg algebras. We then consider the isomorphism problem for tensor products of Leavitt algebras, and show that no diagonal-preserving isomorphism exists between $L_{2,R}\otimes L_{3,R}$ and $L_{2,R}\otimes L_{2,R}$. In fact, there are no unexpected diagonal-preserving isomorphisms between tensor products of finitely many Leavitt algebras. We give an easy proof that every $\ast$-isomorphism of Steinberg algebras over the integers preserves the diagonal, and it follows that $L_{2,\mathbb{Z}}\otimes L_{3,\mathbb{Z}}\not \cong L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ (as $\ast$-rings).

The Dixmier Conjecture says that every endomorphism of the (first) Weyl algebra $A_{1}$ (over a field of characteristic zero) is an automorphism, i.e., if $PQ-QP=1$ for some $P,Q\in A_{1}$, then $A_{1}=K\langle P,Q\rangle$. The Weyl algebra $A_{1}$ is a $\mathbb{Z}$-graded algebra. We prove that the Dixmier Conjecture holds if the elements $P$ and $Q$ are sums of no more than two homogeneous elements of $A_{1}$ (there is no restriction on the total degrees of $P$ and $Q$).

Kaplansky introduced the notions of CCR and GCR $C^{\ast }$-algebras, because they have a tractable representation theory. Many years later, he introduced the notions of CCR and GCR rings. In this paper we characterize when the algebra of an ample groupoid over a field is CCR and GCR. The results turn out to be exact analogues of the corresponding characterization of locally compact groupoids with CCR and GCR $C^{\ast }$-algebras. As a consequence, we classify the CCR and GCR Leavitt path algebras.