For two bimodules NBA and MAB with M⊗AN=0=N⊗BM, the monomorphism category M(A,M,N,B) and its dual, the epimorphism E(A,M,N,B), are introduced and studied. By definition, M(A,M,N,B) is the subcategory of Δ-mod consisting of (X,Y,f,g) such that f:M⊗AX→Y is a monic B-map and g:N⊗BY→X is a monic A-map, where Δ=(ANMB) is a Morita ring. This monomorphism category is a resolving subcategory of Δ-mod if and

We describe a general technique to classify blocks of finite groups, and we apply it to determine Morita equivalence classes of blocks with elementary abelian defect groups of order 32 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two. As a consequence we verify that a conjecture of Harada holds on these blocks.

In this article we explore an approach to the problem of Albert described by U. Umirbaev. We characterize the space of invariant bilinear forms of the polarization algebra of a polynomial endomorphism F:Kn→Kn, where K is a field with characteristic different from two. We obtain some conjectures expressed in the language of polynomial endomorphisms, which are equivalent to the existence of invariant

In this paper, we introduce a class of exact structures in terms of finiteness conditions of modules, which are called n-pure exact structures. We investigate the properties of n-pure derived categories of module categories using n-pure exact structures, and show that n-pure derived categories share many nice properties of classical derived categories. In particular, we show that bounded n-pure derived

In [6] and [7], Joyce and Matveev showed that for given a group G and an automorphism ϕ, there is a quandle structure on the underlying set of G. When the automorphism is an inner-automorphism by ζ, we denote this quandle structure as (G,◃ζ). In this paper, we show a relationship between group extensions of a group G and quandle extensions of the quandle (G,◃ζ). In fact, there exists a group homomorphism

We prove that δ-derivations of a simple finite-dimensional Lie algebra over a field of characteristic zero, with values in a finite-dimensional module, are either inner derivations, or, in the case of adjoint module, multiplications by a scalar, or some exceptional cases related to sl(2). This can be viewed as an extension of the classical first Whitehead Lemma.

We study a simple, self-dual, rational, and C2-cofinite vertex operator algebra of CFT-type whose simple current modules are graded by Z2k. Based on those simple current modules, a vertex operator algebra associated with a Z2k-code is constructed. The classification of irreducible modules for such a vertex operator algebra is established. Furthermore, all the irreducible modules are realized in a module

In 2007, Dubouloz introduced Danielewski varieties. Such varieties generalize Danielewski surfaces and provide counterexamples to generalized Zariski cancellation problem in arbitrary dimension. In the present work we describe the automorphism group of a Danielewski variety. This result is a generalization of a description of automorphisms of Danielewski surfaces due to Makar-Limanov.

One proves a far-reaching upper bound for the degree of a generically finite rational map between projective varieties over a base field of arbitrary characteristic. The bound is expressed as a product of certain degrees that appear naturally by considering the Rees algebra (blowup) of the base ideal defining the map. Several special cases are obtained as consequences, some of which cover and extend

In this paper we study the Galois group of the Galois cover of the composition of a q-cyclic étale cover and a cyclic p-gonal cover for any odd prime p. Furthermore, we give properties of isogenous decompositions of certain Prym and Jacobian varieties associated to intermediate subcovers given by subgroups.

We provide a classification of developable cubic hypersurfaces in P4. Using the correspondence between forms of degree 3 on P4 and Artinian Gorenstein K-algebras, given by Macaulay-Matlis duality, we describe the locus in GOR(1,5,5,1) corresponding to those algebras which satisfy the Strong Lefschetz property.

We construct a uniform model for highest weight crystals and B(∞) for generalized Kac–Moody algebras using rigged configurations. We also show an explicit description of the ⁎-involution on rigged configurations for B(∞): that the ⁎-involution interchanges the rigging and the corigging. We do this by giving a recognition theorem for B(∞) using the ⁎-involution. As a consequence, we also characterize

We study the problem when the product of two non-linear Galois conjugate characters of a finite group are irreducible. We also prove new results on irreducible tensor products of cross-characteristic Brauer characters of quasisimple groups of Lie type.

Let U1,U2 be connected commutative unipotent algebraic groups defined over an algebraically closed field k of characteristic p>0 and let L be a bimultiplicative Q‾ℓ-local system on U1×U2. In this paper we will study the Q‾ℓ-cohomology Hc⁎(U1×U2,L), which turns out to be supported in only one degree. We will construct a finite Heisenberg group Γ which naturally acts on Hc⁎(U1×U2,L) as an irreducible

Let X be a weighted projective line and CX the associated cluster category. It is known that CX can be realized as a generalized cluster category of quiver with potential. In this note, under the assumption that X has at most three weights or is of tubular type, we prove that if the generalized cluster category C(Q,W) of a Jacobi-finite non-degenerate quiver with potential (Q,W) shares a 2-CY tilted

The objects of study in this paper are Hopf algebras H which are finitely generated algebras over an algebraically closed field and are extensions of a commutative Hopf algebra A by a finite dimensional Hopf algebra H‾:=H/A+H, where A+ is the augmentation ideal of A. Basic structural and homological properties are recalled and classes of examples are listed. A structure theorem is proved when (⁎) H‾

In this paper we study two VOAs: first we study the VOA generated by two Ising vectors whose inner product is 128 with |τeτf|=3, then we study the VOA generated by two Ising vectors whose inner product is 329. We also determine they both have unique VOA structures.

Let α be a coprime automorphism of a group G of prime order and let P be an α-invariant Sylow p-subgroup of G. Assume that p∉π(CG(α)). Firstly, we prove that G is p-nilpotent if and only if CNG(P)(α) centralizes P. In the case that G is Sz(2r) and PSL(2,2r)-free where r=|α|, we show that G is p-closed if and only if CG(α) normalizes P. As a consequence of these two results, we obtain that G≅P×H for

Let k be a field of characteristic not two or three, let g be a finite-dimensional colour Lie algebra and let V be a finite-dimensional representation of g. In this article we give various ways of constructing a colour Lie algebra g˜ whose bracket in some sense extends both the bracket of g and the action of g on V. Colour Lie algebras, originally introduced by R. Ree ([18]), generalise both Lie algebras

Suppose that A is a positively filtered algebra such that the associated graded algebra gr A is commutative Calabi-Yau. Then gr A has a canonical Poisson structure with a modular derivation. In general, A is skew Calabi-Yau by a result of Van den Bergh, so A has an invariant, called Nakayama automorphism. A connection between the Nakayama automorphism of A and the modular derivation of gr A is described

We describe a generalization of Hashimoto and Kurano's Cauchy filtration for divided powers algebras. This filtration is then used to provide a cellular structure for generalized Schur algebras associated to an arbitrary cellular algebra. Applications to the cellularity of wreath product algebras A≀Sd are also considered.

We give infinite triangularization and strict triangularization results for algebras of operators on infinite-dimensional vector spaces. We introduce a class of algebras we call Ore-solvable algebras: these are similar to iterated Ore extensions but need not be free as modules over the intermediate subrings. Ore-solvable algebras include many examples as particular cases, such as group algebras of

Let G be a graph on [n] and JG be the binomial edge ideal of G in the polynomial ring S=K[x1,…,xn,y1,…,yn]. In this paper we investigate some topological properties of a poset associated to the minimal primary decomposition of JG. We show that this poset admits some specific subposets which are contractible. This in turn, provides some interesting algebraic consequences. In particular, we characterize

We examine the subgroup D(G) of a transitive permutation group G which is generated by the derangements in G. Our main results bound the index of this subgroup: we conjecture that, if G has degree n and is not a Frobenius group, then |G:D(G)|⩽n−1; we prove this except when G is a primitive affine group. For affine groups, we translate our conjecture into an equivalent form regarding |H:R(H)|, where

We investigate the structure constants of the center Hn of the group algebra Z[Spn(q)] over a finite field. The reflection length on the group GL2n(q) induces a filtration on the algebras Hn. We prove that the structure constants of the associated graded algebra Sn are independent of n. As tool in the proof we consider the embedding Spm↪Spn(q) and determine the behavior of the centralizers and intersection

This paper generalizes former works of Derksen, Weyman and Zelevinsky about quivers with potentials. We consider semisimple finite-dimensional algebras E over a field F, such that E⊗FEop is semisimple. We assume that E contains a certain type of F-basis which is a generalization of a multiplicative basis. We study potentials belonging to the algebra of formal power series, with coefficients in the

For a partial Galois extension of commutative rings we give a seven term exact sequence which generalize the Chase-Harrison-Rosenberg sequence.

Given an embedding of a smooth projective curve X of genus g≥1 into PN, we study the locus of linear subspaces of PN of codimension 2 such that projection from said subspace, composed with the embedding, gives a Galois morphism X→P1. For genus g≥2 we prove that this locus is a smooth projective variety with components isomorphic to projective spaces. If g=1 and the embedding is given by a complete

Let R be a standard graded commutative algebra over a field k, let K be its Koszul complex viewed as a differential graded k-algebra, and let H be the homology algebra of K. This paper studies the interplay between homological properties of the three algebras R, K, and H. In particular, we introduce two definitions of Koszulness that extend the familiar property originally introduced by Priddy: one

In this paper we introduce the definition of the (k,l)-universal transversal property for permutation groups, which is a refinement of the definition of k-universal transversal property, which in turn is a refinement of the classical definition of k-homogeneity for permutation groups. In particular, a group possesses the (2,n)-universal transversal property if and only if it is primitive; it possesses

We prove a broad generalization of a theorem of W. Burnside about the existence of real characters of finite groups to permutation characters. If G is a finite group, under the necessary hypothesis of O2′(G)=G, we can also give some control on the parity of multiplicities of the constituents of permutation characters (a result that needs the Classification of Finite Simple Groups). Along the way, we

We prove the p-version of a conjecture raised by Isaacs et al in Proc. AMS [8] on the degree of a primitive character that divides the size of some conjugacy class of a finite solvable group, thus generalizing Marchi's results [10] in J. Algebra 2020.

Let G be a finite group. Roughly speaking, a subgroup A of G is called a local partial covering subgroup if AG=AB for a suitable maximal G-invariant subgroup B of AG. Our main result is as follows: Let pd be a prime power with p≤pd≤|G|p, assume that all subgroups of pd, and all cyclic subgroups of order 4 when pd=2 and a Sylow 2-subgroup of G is nonabelian, of G are local partial covering subgroups

It is proved that every unital right alternative bimodule over a Cayley algebra (over an algebraically closed field of characteristic not 2) is alternative. Using this result, a coordinatization theorem is proved for unital right alternative algebras containing a Cayley subalgebra with the same unit. In particular, any such an algebra is alternative.

The structure of 3C-algebra and 5A-algebra constructed by Lam-Yamada-Yamauchi is studied and the uniqueness of the vertex operator algebra structure of these two algebras is established. We also give the fusion rules for these two algebras.

In this paper we study rings of invariants arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form K[U]Γ where Γ is a product of general linear groups over a field K of characteristic zero, and U is a finite dimensional rational representation of Γ. We will calculate the Hilbert series of such rings using the representation theory of the

Let E be an n-dimensional vector space. Then the symmetric group Sym(n) acts on E by permuting the elements of a basis and hence on the r-fold tensor product E⊗r. Bowman, Doty and Martin ask, in [1], whether the endomorphism algebra EndSym(n)(E⊗r) is cellular. The module E⊗r is the permutation module for a certain Young Sym(n)-set. We shall show that the endomorphism algebra of the permutation module

The aim of this paper is to compare the Cartier and covariant Dieudonné modules of connected p-divisible groups over perfect fields of positive characteristic p. In fact, we construct an explicit, canonical and functorial isomorphism and its inverse from the Dieudonné module to the Cartier module. The fact that the Cartier and (covariant) Dieudonné modules of a p-divisible formal Lie group over a perfect

In 1986, Van de Leur introduced and classified affine Lie superalgebras. An affine Lie superalgebra is defined as the quotient of certain Lie superalgebra G defined by generators and relations, corresponding to a symmetrizable generalized Cartan matrix, over the so-called radical of G. Because of the interesting applications of affine Lie (super)algebras in combinatorics, number theory and physics

We study the eigenspace decomposition of a basic classical Lie superalgebra under the adjoint action of a toral subalgebra, thus extending results of Kostant. In recognition of Kostant's contribution we refer to the eigenfunctions appearing in the decomposition as Kostant roots. We then prove that Kostant root systems inherit the main properties of classical root systems. Our approach is combinatorial

This is the first of two papers determining the saturated 2-fusion systems in which the centralizer of some fully centralized involution contains a component that is the 2-fusion system of a large group of Lie type over a field of odd order.

We prove the algorithmic undecidability of the problem: whether there exists a solution of an equationw(x1,…,xn)=[a,b] in a free group F2 of rank 2 with free generators, a and b, with constraints of the type x1∈F2(2), where w(x1,…,xn) is a word in the alphabet of unknowns, {x1,…,xn}; [a,b] is the commutator of the free generators, a and b; F2(2) is its second derived subgroup. We also build a polynomial

The (q,Q)-current algebra associated with the general linear Lie algebra was introduced by the second author in the study of representation theory of cyclotomic q-Schur algebras. In this paper, we study the (q,Q)-current algebra Uq(sln〈Q〉[x]) associated with the special linear Lie algebra sln. In particular, we classify finite dimensional simple Uq(sln〈Q〉[x])-modules.

For each positive integer n we introduce the notion of n-exangulated categories as higher dimensional analogues of extriangulated categories defined by Nakaoka–Palu. We characterize which n-exangulated categories are n-exact in the sense of Jasso and which are (n+2)-angulated in the sense of Geiss–Keller–Oppermann.

Let R be a local domain, v a valuation of its quotient field centred in R at its maximal ideal. We investigate the relationship between Rh, the henselisation of R as local ring, and v˜, the henselisation of the valuation v, by focussing on the recent result by de Felipe and Teissier referred to in the title. We give a new proof that simplifies the original one by using purely algebraic arguments. This

We classify Jet modules for the Lie (super)algebras L=W⋉(g⊗C[t,t−1]), where W is the Witt algebra and g is a Lie superalgebra with an even diagonlizable derivation. Then we give a conceptional method to classify all simple Harish-Chandra modules for L and the map superalgebras, which are of the form L⊗R, where R is a Noetherian unital supercommutative associative superalgebra.

We study permutation orbifolds of the 2-fold and 3-fold tensor product for the Virasoro vertex algebra Vc of central charge c. In particular, we show that for all but finitely many central charges (Vc⊗3)Z3 is a W-algebra of type (2,4,5,63,7,83,93,102). We also study orbifolds of their simple quotients and obtain new realizations of certain rational affine W-algebras associated to a principal nilpotent

We introduce a type B analogue of the nil Temperley-Lieb algebra in terms of generators and relations, that we call the (extended) nil-blob algebra. We show that this algebra is isomorphic to the endomorphism algebra of a Bott-Samelson bimodule in type A˜1. We also prove that it is isomorphic to an idempotent truncation of the classical blob algebra.

We calculate the structure of H3(SL2(Q),Z[12]). Let H3(SL2(Q),Z)0 denote the kernel of the (split) surjective homomorphism H3(SL2(Q),Z)→K3ind(Q). Each prime number p determines an operator 〈p〉 on H3(SL2(Q),Z) with square the identity. We prove that H3(SL2(Q),Z[12])0 is the direct sum of the (−1)-eigenspaces of these operators. The (−1)-eigenspace of 〈p〉 is the scissors congruence group, over Z[12]

We study locally graded groups whose non-modular subgroups are soluble and satisfy some rank condition. In particular, in order to characterize locally graded groups whose subgroups are either modular or polycyclic, we describe (generalized) soluble groups whose non-modular subgroups are finitely generated.

We investigate geometric properties of algebraic sets defined by matrix pairs with special commutators. Specifically, we prove that the set consisting of the n×n matrix pairs whose commutators are diagonal matrices is a complete intersection with two components, one of which is the so called commuting variety that consists of commuting matrix pairs, and that the algebraic set as a scheme is reduced

We classify irreducible representations of finite W-algebra for the queer Lie superalgebra Q(n) associated with the principal nilpotent coadjoint orbits. We use this classification and our previous results to obtain a classification of irreducible finite-dimensional representations of the super Yangian YQ(1).

A locally compact contraction group is a pair (G,α), where G is a locally compact group and α:G→G an automorphism such that αn(x)→e pointwise as n→∞. We show that every surjective, continuous, equivariant homomorphism between locally compact contraction groups admits an equivariant continuous global section. As a consequence, extensions of locally compact contraction groups with abelian kernel can

In this paper we show that the center of the partition algebra A2k(δ), in the semisimple case, is given by the subalgebra of supersymmetric polynomials in the normalised Jucys-Murphy elements. For the non-semisimple case, such a subalgebra is shown to be central, and in particular it is large enough to recognise the block structure of A2k(δ). This allows one to give an alternative description for when

We show that the geodesic growth function of any finitely generated virtually abelian group is either polynomial or exponential; and that the geodesic growth series is holonomic, and rational in the polynomial growth case. In addition, we show that the language of geodesics is blind multicounter.

Building on the classification of modules for algebraic groups with finitely many orbits on subspaces [9], we determine all faithful irreducible modules V for a connected simple algebraic group H, such that H≤SO(V) and H has finitely many orbits on singular 1-spaces of V. We do the same for H connected semisimple, and maximal among connected semisimple subgroups. This question is naturally connected

We prove Clifford theoretic results which only hold in characteristic 2. Let G be a finite group, let N be a normal subgroup of G and let φ be an irreducible 2-Brauer character of N. We show that φ occurs with odd multiplicity in the restriction of some self-dual irreducible Brauer character θ of G if and only if φ is G-conjugate to its dual. Moreover, if φ is self-dual then θ is unique and the multiplicity

We apply minimal weakly generating sets to study the existence of Add(UR)-covers for a uniserial module UR. If UR is a uniserial right module over a ring R, then S:=End(UR) has at most two maximal (right, left, two-sided) ideals: one is the set I of all endomorphisms that are not injective, and the other is the set K of all endomorphisms of UR that are not surjective. We prove that if UR is either

In this paper, we study (G,χϕ)-equivariant ϕ-coordinated quasi modules for nonlocal vertex algebras. Among the main results, we establish several conceptual results, including a generalized commutator formula and a general construction of weak quantum vertex algebras and their (G,χϕ)-equivariant ϕ-coordinated quasi modules. As an application, we also construct (equivariant) ϕ-coordinated quasi modules

We develop a notion of degree for functions between two abelian groups that allows us to generalize the Chevalley-Warning Theorems from fields to noncommutative rings or abelian groups of prime power order.