Given a representation of a 3-Lie algebra, we construct a Lie 3-algebra, whose Maurer-Cartan elements are relative Rota-Baxter operators on the 3-Lie algebra. We define the cohomology of relative Rota-Baxter operators on 3-Lie algebras, by which we study deformations of relative Rota-Baxter operators. We show that if two formal deformations of a relative Rota-Baxter operator on a 3-Lie algebra are

A finite p-group S is said to be of supersoluble type if every fusion system over S is supersoluble. The main aim of this paper is to characterise the finite p-groups of supersoluble type. Abelian and metacyclic p-groups of supersoluble type are completely described. Furthermore, we show that the Sylow p-subgroups of supersoluble type of a finite simple group must be cyclic.

In this paper we discuss finite presentability of the universal central extensions of Lie algebras sln(R), where n≥3 and R is a unital associative k-algebra. We show that a universal central extension is finitely presented if and only if the algebra R is finitely presented.

The objective of this paper is to study topological properties of the prime spectrum of a unitary semimodule over a semiring with zero and nonzero identity. We define a top semimodule over a semiring, show that the prime spectrum is a T0-space, and prove that each irreducible closed subset of the prime spectrum has a generic point and that the prime spectrum is a compact space if the top semimodule

By restricting to a class of localic open groupoids G which, similarly to Lie groupoids, possess appropriate covers Gˆ→G by étale groupoids, we extend results about groupoid actions and quantales that were previously proved for étale groupoids but do not seem to work for arbitrary open groupoids. In particular we obtain a characterization of the category of G-actions as a category of quantale modules

To study s-homogeneous algebras, we introduce the category of quivers with s-homogeneous corelations and the category of s-homogeneous triples. We show that both of these categories are equivalent to the category of s-homogeneous algebras. We prove some properties of the elements of s-homogeneous triples and give some consequences for s-Koszul algebras. Then we discuss the relations between the s-Koszulity

The classification of the nilpotent Jacobians with some structure has been an object of study because of its relationship with the Jacobian conjecture. In this paper we classify the polynomial maps in dimension n of the form H=(u(x,y),u2(x,y,x3),…,un−1(x,y,xn),h(x,y)) with JH nilpotent. In addition we prove that the maps X+H are invertible, which shows that for this kind of maps the Jacobian conjecture

In this paper, we focus on Hall's criterion for nilpotence in semi-abelian categories, and we improve the bound of the main theorem of [2, Theorem 3.4] (see Main Theorem). And this bound is the best possible.

The generalized quantum group U(ϵ) of type A is an affine analogue of quantum group associated to a general linear Lie superalgebra glM|N. We prove that there exists a unique R matrix on the tensor product of fundamental type representations of U(ϵ) for arbitrary parameter sequence ϵ corresponding to a non-conjugate Borel subalgebra of glM|N. We give an explicit description of its spectral decomposition

Let B=A〈X|dX=t〉 be an extended DG algebra by the adjunction of a variable of positive even degree n, and let N be a semi-free DG B-module that is assumed to be bounded below as a graded module. We prove in this paper that N is liftable to A if ExtBn+1(N,N)=0. Furthermore such a lifting is unique up to DG isomorphisms if ExtBn(N,N)=0.

We introduce the notion of Hitchin variety over C. Let L be a holomorphic line bundle over a Hitchin variety X. We investigate the space of all global sections of sheaf of differential operators Dk(L) and symmetric powers of sheaf of first order differential operators Sk(D1(L)) over X and show that for a projective Hithcin variety both the spaces are one dimensional. As an application, we show that

Let C be an irreducible plane curve of PG(2,K) where K is an algebraically closed field of characteristic p≥0. A point Q∈C is an inner Galois point for C if the projection πQ from Q is Galois. Assume that C has two different inner Galois points Q1 and Q2, both simple. Let G1 and G2 be the respective Galois groups. Under the assumption that Gi fixes Qi, for i=1,2, we provide a complete classification

If H is a subgroup of an Abelian p-group G, we say G is H-fully transitive if using the height valuation from G, for every x∈H, every valuated (i.e., non-height decreasing) homomorphism 〈x〉→G extends to a valuated homomorphism H→G. This notion is a generalization of the classical definition of fully transitive groups due to Kaplansky. A number of interesting properties of this idea are established

We show that the Steinberg group St(C2,Z) associated with the Lie type C2 and with integer coefficients can be realized as a quotient of the braid group B6 by one relation. As an application we give a new braid-like presentation of the symplectic modular group Sp4(Z).

Let Fq be a finite field of characteristic p and let W2(Fq) be the ring of Witt vectors of length two over Fq. We prove that for any integer n such that p divides n, the groups SLn(Fq[t]/t2) and SLn(W2(Fq)) have the same number of irreducible representations of dimension d, for each d.

We consider the notion of mixed multiplicities for multigraded modules by using Hilbert series, and this is later applied to study the projective degrees of rational maps. We use a general framework to determine the projective degrees of a rational map via a computation of the multiplicity of the saturated special fiber ring. As specific applications, we provide explicit formulas for all the projective

We consider the affine vertex algebra at the critical level associated with the centralizer of a nilpotent element in the Lie algebra glN. Due to a recent result of Arakawa and Premet, the center of this vertex algebra is an algebra of polynomials. We construct a family of free generators of the center in an explicit form. As a corollary, we obtain generators of the corresponding quantum shift of argument

We define the degenerate two boundary affine Hecke-Clifford algebra Hd, and show it admits a well-defined q(n)-linear action on the tensor space M⊗N⊗V⊗d, where V is the natural module for q(n), and M,N are arbitrary modules for q(n), the Lie superalgebra of Type Q. When M and N are irreducible highest weight modules parameterized by a staircase partition and a single row, respectively, this action

We give an explicit formula showing how the double Poisson algebra introduced in [17] appears as a particular part of a pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the Maurer-Cartan equation on A⊕A⁎. Specific part of this solution is described, which is in one-to-one correspondence with the double Poisson algebra structures. The result holds

Let q be a prime power, and let Fq be the finite field with q elements. In connection with Galois theory and algebraic curves, this paper investigates rational functions h(x)=f(x)/g(x)∈Fq(x) for which the value sets Vh={h(α)|α∈Fq∪{∞}} are relatively small. In particular, under certain circumstances, it proves that h(x) having a small value set is equivalent to the field extension Fq(x)/Fq(h(x)) being

Gröbner basis methods are used to solve systems of polynomial equations over finite fields, but their complexity is poorly understood. In this work an upper bound on the time complexity of constructing a Gröbner basis according to a total degree monomial ordering and finding a solution of a system is proved. A key parameter in this estimate is the degree of regularity of the leading forms of the polynomials

We use the gluing construction introduced by Jia Huang to explore the rings of invariants for a range of modular representations. We construct generating sets for the rings of invariants of the maximal parabolic subgroups of a finite symplectic group and their common Sylow p-subgroup. We also investigate the invariants of singular finite classical groups. We introduce parabolic gluing and use this

In this paper, we first consider the relationship between a polynomial ring B over a Noetherian domain R and the ring of invariants A of a Ga-action on B, when A occurs as a retract of B. Next, we study retracts of a polynomial ring in general and address the questions of D. L. Costa raised in [5]. Finally, we examine the behaviour of ideals and certain properties of rings under retractions.

Let b be a block with normal abelian defect group and abelian inertial quotient. We prove that every Morita auto-equivalence of b has linear source. We note that this improves upon results of Zhou and also Boltje, Kessar and Linckelmann.

By a result of Cocke and Venkataraman we know that if G is a group with at most m elements of maximal order, then |G| is m-bounded. In this paper we consider the following related setting. Suppose G is a group with at most m commutators whose order is maximal among all commutators. What can we say about the structure of the group G?

We completely determine the structure constants between real root vectors in a rank 2 Kac–Moody algebra g. Our description is computationally efficient, even in the rank 2 hyperbolic case where the coefficients of roots on the root lattice grow exponentially with height. Our approach is to extend Carter's method of finding structure constants from those on extraspecial pairs to the rank 2 Kac–Moody

For any exact category A with splitting idempotents, a maximal exact category T(A) containing A as a biresolving subcategory, is constructed. Important types of exact categories, including n-tilting torsion classes, categories of Cohen-Macaulay modules over a Cohen-Macaulay order, or categories of Gorenstein projectives, are shown to be of the form T(A). The quotient category T(A)/A in the sense of

We study Hironaka's idealistic exponents over Spec(Z). We give an idealistic interpretation of the tangent cone, the directrix, and the ridge. The main purpose is to introduce the notion of characteristic polyhedra of idealistic exponents and deduce from them intrinsic data on the idealistic exponent.

Suppose that F is a field such that the value groups of the R-places on F, i.e., places from F into the real numbers R, are all trivial or countable. The path-connected components of the space M(F(x1,x2,⋯,xn)) of R-places on F(x1,x2,⋯,xn) are shown then to correspond bijectively to those of M(F). For example, the space M(R(x1,x2,⋯,xn)) of R-places on the rational function field R(x1,x2,⋯,xn) is path-connected

We show that a symmetry property that we call the up-down symmetry implies that the Kazhdan–Lusztig Rx-polynomials of a pircon P are a P-kernel, and we show that this property holds in the classical cases. Then, we enhance and extend to this context a duality of Deodhar in parabolic Kazhdan–Lusztig theory.

Happel constructed a fully faithful functor H:Db(modΛ)→mod_ZT(Λ) for a finite dimensional algebra Λ. He also showed that this functor H gives an equivalence precisely when gldimΛ<∞. Thus if H gives an equivalence, then it provides a canonical tilting object H(Λ) of modZT(Λ). In this paper we generalize the Happel functor H in the case where T(Λ) is replaced with a finitely graded IG-algebra A. We study

Let f(X1,…,Xn) be a nonzero multilinear noncommutative polynomial. If A is a unital algebra with a surjective inner derivation, then every element in A can be written as f(a1,…,an) for some ai∈A.

We determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding skew left braces. Besides we determine the separable field extensions of degree twice an odd prime square allowing a cyclic Hopf Galois structure and the number of these structures.

Axial algebras are non-associative algebras generated by semisimple idempotents, known as axes, whose definition is motivated by examples of algebras related to group theory and theoretical physics. In this paper, we construct an axial algebra of Monster type over the ring R[t]. By specialising the parameter t to a value in R, we construct an infinite family of pairwise non-isomorphic axial algebras

We introduce a generalization of the Euclidean algorithm for rings equipped with an involution, and completely enumerate all isomorphism classes of orders over definite, rational quaternion algebras equipped with an orthogonal involution that admit such an algorithm. We give two applications: first, any order that admits such an algorithm has class number 1; second, we show how the existence of such

This paper studies, for p a prime, rank one isolated p-minimal subgroups P in a finite group G. Such subgroups share many of the features of the minimal parabolic subgroups in groups of Lie type. The structure of Y, the normal closure in G of Op(P) is determined where Op(P) is the smallest normal subgroup of P such that P/Op(P) is a p-group. We find that if Y≠Op(P) and Op(G)=1, then either Y/Z(Y) is

Let R be a ring such that (GP,GP⊥) forms a cotorsion pair cogenerated by a set, where GP denotes the category of all Gorenstein projective R-modules. Recently, the first author defined for any complex X the relative cohomology functors ExtGP⁎(X,−) as H−⁎(Hom(G,−)) in which G is a special Gorenstein projective precover of X. In the present paper, we introduce the dimension of X related to Gorenstein

We present examples of algebras A having dominant dimension n, for every n≥1, such that the algebra B=EndA(I0⊕Ω−n(A)) has dominant dimension different from n, where I0 is the injective hull of A. This gives a counterexample to a recent conjecture of Chen and Xi. While the conjecture is false in general, we show that a large class of algebras containing higher Auslander algebras satisfies the conjecture

Let g be a complex semisimple Lie algebra, let b be a Borel subalgebra of g, let n be the nilradical of b, and let U(n) be the universal enveloping algebra of n. We study primitive ideals of U(n). Almost all primitive ideals are centrally generated, i.e., are generated by their intersections with the center Z(n) of U(n). We present an explicit characterization of the centrally generated primitive ideals

Let X be a projective K3 surface over C. We prove that its Cox ring has a generating set whose degrees are either classes of smooth rational curves, sums of at most three elements of the Hilbert basis of the nef cone, or of the form 2(f+f′), where f,f′ are classes of smooth elliptic curves with f⋅f′=2. This result and techniques using Koszul's type exact sequences are then applied to determine a generating

Exceptional cycles in a triangulated category T with Serre duality, introduced by N. Broomhead, D. Pauksztello, and D. Ploog, have a notable impact on the global structure of T. In this paper we show that if T is homotopy-like, then any exceptional 1-cycle is indecomposable and at the mouth (i.e., the middle term of the Auslander-Reiten triangle ending at it is indecomposable); and any object in an

We develop the theory of geometric degenerations for finite-dimensional algebras over algebraically closed fields and study the orbit closures of weighted surface algebras, investigated recently by Erdmann and Skowroński [11], [12], [13], [14], [15].

We reformulate the problem of bounding the total rank of the homology of perfect chain complexes over the group ring Fp[G] of an elementary abelian p-group G in terms of commutative algebra. This extends results of Carlsson for p=2 to all primes. As an intermediate step, we construct an embedding of the derived category of perfect chain complexes over Fp[G] into the derived category of p-DG modules

We introduce Morita and Rickard equivalences over a group graded G-algebra between block extensions. A consequence of such equivalences is that Späth's central order relation holds between two corresponding character triples.

Wolfgang Rump showed that there is a one-to-one correspondence between nondegenerate involutive set-theoretic solutions of the Yang-Baxter equation and binary algebras in which all left translations Lx are bijections, the squaring map is a bijection, and the identity (xy)(xz)=(yx)(yz) holds. We call these algebras rumples in analogy with quandles, another class of binary algebras giving solutions of

We explicitly construct modular forms on a 4-dimensional bounded symmetric domain of type IV based on the variation of the Hodge structures of K3 surfaces. We study the ring of our modular forms. Because of the Kneser conditions of the transcendental lattice of our family of K3 surfaces, our modular group has a good arithmetic property. Also, our results can be regarded as natural extensions of the

Given a finite group G and an abelian variety A acted on by G, to any subgroup H of G, we associate an abelian subvariety AH on which the associated Hecke algebra HH for H in G acts. Any irreducible rational representation W˜ of HH induces an abelian subvariety of AH in a natural way. In this paper we give equations for this abelian subvariety. In a special case these equations become much easier.

We say that an irreducible character ξ of a subgroup H of G is fully extendible to G if every irreducible constituent of the induced character ξG is actually an extension of ξ. Now let N◃G, where G/N is π-separable for some set π of primes, and let H/N be a Hall π-subgroup of G/N. If ξ∈Irr(H) is fully extendible to G, we show that a Hall π′-subgroup K/N of G/N must be abelian. Also, if λ is an irreducible

More than four decades ago, Eisenbud, Khimšiašvili, and Levine introduced an analogue in the algebro-geometric setting of the notion of local degree from differential topology. Their notion of degree, which we call the EKL-degree, can be thought of as a refinement of the usual notion of local degree in algebraic geometry that works over non-algebraically closed base fields, taking values in the Grothendieck-Witt

For any vertex operator algebra V and any finite automorphism g of V, we prove that the associative algebra Ag,n(V) is isomorphic to some (sub)quotient of the universal enveloping algebra of V with respect to g.

In this note we correct two oversights in Çanakçi et al. (2019) [6] which only occur when a band complex is involved. As a consequence we see that the mapping cone of a morphism between two band complexes can decompose into arbitrarily many indecomposable direct summands.

We prove that in q-Weyl generators the multi-parameter quantizations of type An+ and Bn+ are quadratic-linear Koszul algebras.

We discuss relations between some category-theoretical notions for a finite tensor category and cointegrals on a quasi-Hopf algebra. Specifically, for a finite-dimensional quasi-Hopf algebra H, we give an explicit description of categorical cointegrals of the category MH of left H-modules in terms of cointegrals on H. Provided that H is unimodular, we also express the Frobenius structure of the ‘adjoint

Young tableaux carry an associative product, described by the Schensted algorithm. They thus form a monoid Pl, called plactic. It is central in numerous combinatorial and algebraic applications. In this paper, the tableaux product is shown to be completely determined by an idempotent braiding σ on the (much simpler!) set of columns Col. Here a braiding is a set-theoretic solution to the Yang–Baxter

We describe explicitly the vertex algebra of (twisted) chiral differential operators on certain nilmanifolds and construct their logarithmic modules. This is achieved by generalizing the construction of vertex operators in terms of exponentiated scalar fields to Jacobi theta functions naturally appearing in these nilmanifolds. This construction furnishes non-trivial examples of logarithmic vertex algebra

This work provides the first step toward the classification of irreducible finite weight modules over twisted affine Lie superalgebras. We divide the class of such modules into two subclasses called hybrid and tight. We reduce the classification of hybrid irreducible finite weight modules to the classification of cuspidal modules of finite dimensional cuspidal Lie superalgebras which is discussed in

A flat pseudo-Euclidean Lie algebra is a left-symmetric algebra endowed with a non-degenerate symmetric bilinear form such that left multiplications are skew-symmetric. We observe that, in many classes of flat pseudo-Euclidean Lie algebras, the left-symmetric product is actually of Novikov. This leads us to study pseudo-Euclidean Novikov algebras (A,〈,〉), that is, Novikov algebra A endowed with a non-degenerate

The injective spectrum is a topological space associated to a ring R, which agrees with the Zariski spectrum when R is commutative noetherian. We consider injective spectra of right noetherian rings (and locally noetherian Grothendieck categories) and establish some basic topological results and a functoriality result, as well as links between the topology and the Krull dimension of the ring (in the

Peirce-evanescent baric identities are polynomial identities verified by baric algebras such that their Peirce polynomials are the null polynomial. In this paper procedures for constructing such homogeneous and non homogeneous identities are given. For this we define an algebraic system structure on the free commutative nonassociative algebra generated by a set T which provides for classes of baric

Let R denote a commutative Noetherian ring and I an ideal of R. The concept of quintessential sequences over zero ideal was introduced by McAdam and Ratliff (1985) [8]. They showed that these sequences enjoy many of the basic properties of asymptotic sequences over zero ideal. It was shown, that quintessential sequences over ideals I≠(0)R are not a good analogue of asymptotic sequences over I≠(0)R