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We deal with Perazzo hypersurfaces X = V (f) in P n + 2 defined by a homogeneous polynomial f (x 0 , x 1 , ... , x n , u , v) = p 0 (u , v) x 0 + p 1 (u , v) x 1 + ⋯ + p n (u , v) x n + g (u , v) , where p 0 , p 1 , ... , p n are algebraically dependent but linearly independent forms of degree d − 1 in K [ u , v ] and g is a form in K [ u , v ] of degree d. Perazzo hypersurfaces have vanishing hessian and, hence, the associated graded artinian Gorenstein algebra A f fails the strong Lefschetz property. In this paper, we first determine the maximum and minimum Hilbert function of A f , we prove that the Hilbert function of A f is always unimodal and we determine when A f satisfies the weak Lefschetz property. We illustrate our results with many examples and we show that our results do not generalize to Perazzo hypersurfaces X = V (f) in P n + 3 defined by a homogeneous polynomial f (x 0 , x 1 , ... , x n , u , v , w) = p 0 (u , v , w) x 0 + p 1 (u , v , w) x 1 + ⋯ + p n (u , v , w) x n + g (u , v , w) , where p 0 , p 1 , ... , p n are algebraically dependent but linearly independent forms of degree d − 1 in K [ u , v , w ] and g is a form in K [ u , v , w ] of degree d. [ABSTRACT FROM AUTHOR]

This paper is a commutative algebra introduction to the homological theory of quasi-coherent sheaves and contraherent cosheaves over quasi-compact semi-separated schemes. Antilocality is an alternative way in which global properties are locally controlled in a finite affine open covering. For example, injectivity of modules over non-Noetherian commutative rings is not preserved by localizations, while homotopy injectivity of complexes of modules is not preserved by localizations even for Noetherian rings. The latter also applies to the contraadjustedness and cotorsion properties. All the mentioned properties of modules or complexes over commutative rings are actually antilocal. They are also colocal, if one presumes contraadjustedness. Generally, if the left class in a (hereditary complete) cotorsion theory for modules or complexes of modules over commutative rings is local and preserved by direct images with respect to affine open immersions, then the right class is antilocal. If the right class in a cotorsion theory for contraadjusted modules or complexes of contraadjusted modules is colocal and preserved by such direct images, then the left class is antilocal. As further examples, the class of flat contraadjusted modules is antilocal, and so are the classes of acyclic, Becker-coacyclic, or Becker-contraacyclic complexes of contraadjusted modules. The same applies to the classes of homotopy flat complexes of flat contraadjusted modules and acyclic complexes of flat contraadjusted modules with flat modules of cocycles. [ABSTRACT FROM AUTHOR]

The duality relation is a basic family of linear relations for multiple zeta values. The extended double shuffle relation (EDSR) is one of the families of relations expected to generate all linear relations among multiple zeta values, but it remains unclear as to whether all duality relations can be deduced from the EDSR. In the present paper, regarding the family generated by the duality relation and the family generated by the derivation relation, an explicit characterization of their intersection is obtained. Here, the derivation relation is a specialization of the EDSR. [ABSTRACT FROM AUTHOR]

The extension dimensions of an Artin algebra give a reasonable way of measuring how far an algebra is from being representation-finite. In this paper we mainly study the behavior of the extensions dimensions of algebras under different equivalences. We show that the difference of the extension dimensions of two derived equivalent algebras is bounded above by the length of the tilting complex associated with the derived equivalence, and that the extension dimension is an invariant under the stable equivalence. In addition, we provide two sufficient conditions such that the extension dimension is an invariant under particular derived equivalences. [ABSTRACT FROM AUTHOR]

Let A be a finite-dimensional algebra, and M be a d -cluster tilting subcategory of mod A. From the viewpoint of higher homological algebra, a natural question to ask is when M induces a d -cluster tilting subcategory in Mod A. In this paper, we investigate this question in a more general form. We consider M as an essentially small d -abelian category, known to be equivalent to a d -cluster tilting subcategory of an abelian category A. The completion of M , denoted by Ind (M) , is defined as the universal completion of M with respect to filtered colimits. We explore Ind (M) and demonstrate its equivalence to the full subcategory L d (M) of Mod M , comprising left d -exact functors. Notably, Ind (M) as a subcategory of Mod M Eff (M) falls short of being a d -cluster tilting subcategory since it satisfies all properties of a d -cluster tilting subcategory except d -rigidity. For a d -cluster tilting subcategory M of mod A , M → consists of all filtered colimits of objects from M , is a generating-cogenerating, functorially finite subcategory of Mod A. The question of whether M → is a d -rigid subcategory remains unanswered. However, if it is indeed d -rigid, it qualifies as a d -cluster tilting subcategory. In the case d = 2 , employing cotorsion theory, we establish that M → is a 2-cluster tilting subcategory if and only if M is of finite type. Thus, the question regarding whether M → is a d -cluster tilting subcategory of Mod A appears to be equivalent to Iyama's question about the finiteness of M. Furthermore, for general d , we address the problem and present several equivalent conditions for Iyama's question. [ABSTRACT FROM AUTHOR]

A difference Lie group is a Lie group equipped with a difference operator, equivalently a crossed homomorphism with respect to the adjoint action. In this paper, first we introduce the notion of a representation of a difference Lie group, and establish the relation between representations of difference Lie groups and representations of difference Lie algebras via differentiation and integration. Then we introduce a cohomology theory for difference Lie groups and justify it via the van Est theorem. Finally, we classify abelian extensions of difference Lie groups using the second cohomology group as applications. [ABSTRACT FROM AUTHOR]

Let R d be the space of stable sheaves F which satisfy the Hilbert polynomial χ (F (m)) = d m + 1 and are supported on rational curves in the projective plane P 2. Then R 1 (resp. R 2) is isomorphic to P 2 (resp. P 5). In addition, R 3 is well-known to be a P 6 -bundle over P 2. In particular, R d is smooth for d ≤ 3. However, for d ≥ 4 , in general, the space R d is no longer smooth because of the complexity of boundary curves. In this paper, we obtained an SL (3 , C) -equivariant smooth resolution of R 4 for d = 4 , which is a P 5 -bundle over a blow-up of a Kronecker modules space. [ABSTRACT FROM AUTHOR]

A pseudo-Euclidean non-associative algebra (g , •) is a finite dimensional algebra over a field K that has a metric, i.e., a bilinear, symmetric, and non-degenerate form 〈 , 〉. The metric is considered L-invariant (resp. R-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if 〈 u • v , w 〉 = 〈 u , v • w 〉 for all u , v , w ∈ g. These three notions coincide when g is a Lie algebra and in this case g endowed with the metric is known as a quadratic Lie algebra. This paper provides a complete description of L-invariant, R-invariant, or associative pseudo-Euclidean metrics on left Leibniz algebras over a commutative field of characteristic zero. It shows that a left Leibniz algebra with an associative metric is also right Leibniz and can be obtained easily from its underlying Lie algebra, which is a quadratic Lie algebra. Additionally, it shows that at the core of a left Leibniz algebra endowed with a L-invariant or R-invariant metric, there are two Lie algebras with one quadratic and the left Leibniz algebra can be built from these Lie algebras. We derive many important results from this complete description. Finally, the paper provides a list of left Leibniz algebras with an associative metric up to dimension 6, as well as a list of left Leibniz algebras with an L-invariant metric, up to dimension 4, and R-invariant metric up to dimension 5. [ABSTRACT FROM AUTHOR]

In recent years, averaging operators on Lie algebras (also called embedding tensors in the physics literature) and associated tensor hierarchies have formed an efficient tool for constructing supergravity and higher gauge theories. A Lie algebra with an averaging operator is called an averaging Lie algebra. In the present paper, we introduce 2-term averaging L ∞ -algebras and give characterizations of some particular classes of such homotopy algebras. Next, we study non-abelian extensions of an averaging Lie algebra by another averaging Lie algebra. We define the second non-abelian cohomology group to classify the equivalence classes of such non-abelian extensions. Next, given a non-abelian extension of averaging Lie algebras, we show that the obstruction for a pair of averaging Lie algebra automorphisms to be inducible can be seen as the image of a suitable Wells map. Finally, we discuss the Wells short exact sequence in the above context. [ABSTRACT FROM AUTHOR]

In this paper we begin the study of set-theoretic type solution of the braid equation. Our theory includes set-theoretical solutions as basic examples. More precisely, the linear solution associated to a set-theoretic solution on a set X can be regarded as coming from the coalgebra kX , where k is a field and the elements of X are grouplike. We introduce and study a broader class of linear solutions associated in a similar way to more general coalgebras. We show that the relationships between set-theoretical solutions, q -cycle sets, q -braces, skew-braces, matched pairs of groups and invertible 1-cocycles remain valid in our setting. [ABSTRACT FROM AUTHOR]

BiHom-associative algebras are generalized associative algebras with two multiplicative linear maps. In this paper, we give the Hochschild homology and cyclic homology structure of BiHom-associative algebras. Then, generalize the dual bimodule action to define the cyclic cohomology. Finally, we introduce the crossed modules of BiHom-associative algebras and show that the Hochschild cohomology of BiHom-associative algebra classifies crossed modules. [ABSTRACT FROM AUTHOR]

Given a finite group G and an irreducible complex character χ of G , the codegree of χ is defined as the integer cod (χ) = | G : ker (χ) | / χ (1). It was conjectured by G. Qian in [16] that, for every element g of G , there exists an irreducible character χ of G such that cod (χ) is a multiple of the order of g ; the conjecture has been verified under the assumption that G is solvable ([16]) or almost-simple ([13]). In this paper, we prove that Qian's conjecture is true for every finite group whose Fitting subgroup is trivial, and we show that the analysis of the full conjecture can be reduced to groups having a solvable socle. [ABSTRACT FROM AUTHOR]

In the present paper, we give some sufficient conditions for Cl (k [ P ]) to be torsionfree, where Cl (k [ P ]) denote the divisor class group of the toric ring k [ P ] of an integral polytope P. We prove that Cl (k [ P ]) is torsionfree if P is compressed, and Cl (k [ P ]) is torsionfree if P is a (0 , 1) -polytope which has at most dim ⁡ P + 2 facets. Moreover, we characterize the toric rings of (0 , 1) -polytopes in the case Cl (k [ P ]) ≅ Z. [ABSTRACT FROM AUTHOR]

This paper is about the orbifold theory of affine vertex operator superalgebras. Among the main results, we classify the irreducible modules and determine the fusion rules for the orbifold of the simple affine vertex operator superalgebra L o s p (1 | 2) ˆ (k , 0). [ABSTRACT FROM AUTHOR]

Let g be a complex simple Lie algebra and Z (g) be the center of the universal enveloping algebra U (g). Denote by V λ the finite-dimensional irreducible g -module with highest weight λ. Lehrer and Zhang defined the notion of strongly multiplicity free representations for simple Lie algebras motivated by studying the structure of the endomorphism algebra End U (g) (V λ ⊗ r) in terms of the quotients of the Kohno's infinitesimal braid algebra. Kostant introduced the g -invariant endomorphism algebras R λ (g) = (End V λ ⊗ U (g)) g and R λ , π (g) = (End V λ ⊗ π (U (g))) g. In this paper, we give some other criteria for a multiplicity free representation to be strongly multiplicity free by classifying the pairs (g , V λ) , which are multiplicity free and for such pairs, R λ (g) and R λ , π (g) are generated by generalizations of the quadratic Casimir elements of Z (g). [ABSTRACT FROM AUTHOR]

We study the category of representations of sl m + 2 n over a field of characteristic p with p ≫ 0 , whose p -character is a nilpotent whose Jordan type is the two-row partition (m + n , n). In a previous paper with Anno, we used Bezrukavnikov-Mirkovic-Rumynin's theory of positive characteristic localization and exotic t-structures to give a geometric parametrization of the simples using annular crossingless matchings. Building on this, here we give combinatorial dimension formulae for the simple objects, and compute the Jordan-Hölder multiplicities of the simples inside the baby Vermas. We use Cautis-Kamnitzer's geometric categorification of the tangle calculus to study the images of the simple objects under the BMR equivalence. Our results generalize Jantzen's formulae in the subregular nilpotent case (i.e. when n = 1), and may be viewed as a positive characteristic analogue of the combinatorial description for Kazhdan-Lusztig polynomials of Grassmannian permutations due to Lascoux and Schutzenberger. [ABSTRACT FROM AUTHOR]

In recent years, the notion of characteristic polynomials of representations of Lie algebras has been widely studied. This paper provides more properties of these characteristic polynomials. For simple Lie algebras, we characterize the linearization of characteristic polynomials. Additionally, we characterize nilpotent Lie algebras via characteristic polynomials of the adjoint representation. [ABSTRACT FROM AUTHOR]

Let A be a finite nilpotent group acting fixed point freely on the finite (solvable) group G by automorphisms. It is conjectured that the nilpotent length of G is bounded above by ℓ (A) , the number of primes dividing the order of A counted with multiplicities. In the present paper we consider the case A is cyclic and obtain that the nilpotent length of G is at most 2 ℓ (A) if | G | is odd. More generally we prove that the nilpotent length of G is at most 2 ℓ (A) + c (G ; A) when G is of odd order and A normalizes a Sylow system of G where c (G ; A) denotes the number of trivial A -modules appearing in an A -composition series of G. [ABSTRACT FROM AUTHOR]

A finitely generated residually finite group G is an O E ˆ -group if any action of its profinite completion G ˆ on a profinite tree with finite edge stabilizers admits a global fixed point. In this paper, we study the profinite genus of free products G 1 ⁎ H G 2 of O E ˆ -groups G 1 , G 2 with finite amalgamation H. Given such G 1 , G 2 , H we give precise formulas for the number of isomorphism classes of G 1 ⁎ H G 2 and of its profinite completion. We compute the genus of G 1 ⁎ H G 2 and list various situations when the formula for the genus simplifies. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic 0, let S be a complete local ring with coefficient field K , let K [ [ x 1 , ... , x n ] ] be the ring of formal power series in variables x 1 , ... , x n with coefficients from K , let K [ [ x 1 , ... , x n ] ] → S be a K -algebra surjection and let E • • , • be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of S. Nicholas Switala [12] proved that this spectral sequence is independent of the surjection beginning with the E 2 page, and the groups E 2 p , q are all finite-dimensional over K. In this paper we extend this result to affine varieties. Namely, let Y be an affine variety over K , let X be a non-singular affine variety over K , let Y ⊂ X be an embedding over K and let E • • , • be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of Y. Then this spectral sequence is independent of the embedding beginning with the E 2 page, and the groups E 2 p , q are all finite-dimensional over K. [ABSTRACT FROM AUTHOR]

Hopf monads generalise Hopf algebras. They clarify several aspects of the theory of Hopf algebras and capture several related structures such as weak Hopf algebras and Hopf algebroids. However, important parts of Hopf algebra theory are not reached by Hopf monads, most noticeably Drinfeld's quasi-Hopf algebras. In this paper we introduce a generalisation of Hopf monads, that we call slack Hopf monads. This generalisation retains a clean theory and is flexible enough to encompass quasi-Hopf algebras as examples. A slack Hopf monad is a colax magma monad T on a magma category C such that the forgetful functor U T : C T → C 'slackly' preserves internal Homs. We give a number of different descriptions of slack Hopf monads, and study special cases such as slack Hopf monads on cartesian categories and k -linear exact slack Hopf monads on Vect k , that is comagma algebras such that a modified fusion operator is invertible. In particular, we characterise quasi-Hopf algebras in terms of slackness. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove a conjecture made by T. Fukaya in 2008. This conjecture concerns the exact value of the Z 2 -cup-length of the Grassmann manifold G ˜ n , 3 of oriented 3-planes in R n. Along the way, we calculate the heights of the Stiefel–Whitney classes of the canonical vector bundle over G ˜ n , 3. [ABSTRACT FROM AUTHOR]

A finite group G is said to be rational if every character of G is rational-valued. The Gruenberg-Kegel graph of a finite group G is the undirected graph whose vertices are the primes dividing the order of G and an edge connects a pair of different vertices p and q whenever G has an element of order pq. In this paper, we complete the classification of the Gruenberg-Kegel graphs of finite solvable rational groups initiated in [1]. [ABSTRACT FROM AUTHOR]

We introduce a new class of quiver varieties, called saw (quiver) varieties. These can be seen as a generalization of the three existing theories: the quiver gauge theory of monopoles, Kac's representation theory of Dynkin types, and Donaldson's theory on the based maps of P 1. E.g., in our terminology, the handsaw (quiver) varieties and chainsaw varieties are finite and affine A type saw varieties respectively with the maximal symmetry breaking. These varieties are the monopole spaces and the caloron spaces in 3d gauge theory, respectively. We expand the theory over to the DE type quiver gauge theory. The saw varieties are constructed via a Hamiltonian formalism, so that the moment maps defining these varieties naturally arise. The main assertion of the paper is that the moment map μ defining a saw variety is a flat morphism for arbitrary dimension vector, if and only if the underlying graph is of finite ADE types. Furthermore under the assumption of maximal symmetry breaking or all the higher frame dimension vectors, the affine ADE types also attain the flatness property of μ. We give an algebro-geometric description of the momentum-zero scheme μ − 1 (0) and the saw variety defined as the Hamiltonian reduction. Various partition functions given rise to by the above mentioned theories are the byproducts of the geometric structure of the corresponding saw varieties. [ABSTRACT FROM AUTHOR]

In a categorification of skew-symmetric cluster algebras, each cluster variable corresponds with an indecomposable module over the associated Jacobian algebra. Buan, Marsh and Reiten studied when the denominator vector of each cluster variable in an acyclic cluster algebra coincides with the dimension vector of the corresponding module. In this paper, we give analogues of their results for cluster algebras from triangulated surfaces by comparing two kinds of intersection numbers of tagged arcs. [ABSTRACT FROM AUTHOR]

In this paper, we define a G -graded Hopf module associative pseudoalgebra A for any grading group G and Hopf algebra H , and construct a smash product A # H of A with H , which is also a G -graded associative pseudoalgebra. Then we establish a Mortia context between A # H and A H , the fixed sub-pseudoalgebra under the action of H. In addition, we prove a similarity of the Schur-Weyl type duality between the group algebra k [ S e (n) ] of Sergeev group S e (n) and a super pseudoalgebra. [ABSTRACT FROM AUTHOR]

Dlab–Ringel's standardization method gives a realization of a standardly stratified algebra. In this paper, we construct mixed stratified algebras, which are a generalization of standardly stratified algebras, following Dlab–Ringel's standardization method. Moreover, we study a Ringel duality of mixed stratified algebras from the viewpoint of stratifying systems. [ABSTRACT FROM AUTHOR]

A subgroup A of a finite group G is called Φ-isolator of G if A covers the Frattini chief factors of G and avoids the supplemented ones. Let H be a Φ-isolator of a soluble finite group G. In the present paper, we prove that there exist elements x , y ∈ G such that the equality H ∩ H x ∩ H y = Φ (G) holds. This result is an answer to Question 19.38 in "The Kourovka notebook". [ABSTRACT FROM AUTHOR]

Lie-Yamaguti algebras generalize both the notions of Lie algebras and Lie triple systems. In this paper, we consider the inducibility problem for automorphisms of Lie-Yamaguti algebra extensions. More precisely, given an abelian extension [Display omitted] of a Lie-Yamaguti algebra L , we are interested in finding the pairs (ϕ , ψ) ∈ Aut (V) × Aut (L) , which are inducible by an automorphism in Aut (L ˜). We connect the inducibility problem to the (2 , 3) -cohomology of Lie-Yamaguti algebra. In particular, we show that the obstruction for a pair of automorphisms in Aut (V) × Aut (L) to be inducible lies in the (2 , 3) -cohomology group H (2 , 3) (L , V). We develop the Wells exact sequence for Lie-Yamaguti algebra extensions, which relates the space of derivations, automorphism groups, and (2 , 3) -cohomology groups of Lie-Yamaguti algebras. As an application, we describe certain automorphism groups of semi-direct product Lie-Yamaguti algebras. In a sequel, we apply our results to discuss inducibility problem for nilpotent Lie-Yamaguti algebras of index 2. We give examples of infinite families of such nilpotent Lie-Yamaguti algebras and characterize the inducible pairs of automorphisms for extensions arising from these examples. Finally, we write an algorithm to find out all the inducible pairs of automorphisms for extensions arising from nilpotent Lie-Yamaguti algebras of index 2. [ABSTRACT FROM AUTHOR]

An order is a commutative ring that as an abelian group is finitely generated and free. A commutative ring is reduced if it has no non-zero nilpotent elements. In this paper we use a new tool, namely, the fact that every reduced order has a universal grading, to answer questions about realizing orders as group rings. In particular, we address the Isomorphism Problem for group rings in the case where the ring is a reduced order. We prove that any non-zero reduced order R can be written as a group ring in a unique "maximal" way, up to isomorphism. More precisely, there exist a ring A and a finite abelian group G , both uniquely determined up to isomorphism, such that R ≅ A [ G ] as rings, and such that if B is a ring and H is a group, then R ≅ B [ H ] as rings if and only if there is a finite abelian group J such that B ≅ A [ J ] as rings and J × H ≅ G as groups. Computing A and G for given R can be done by means of an algorithm that is not quite polynomial-time. We also give a description of the automorphism group of R in terms of A and G. [ABSTRACT FROM AUTHOR]

Previously we showed that the tensor product of a weight module over a generalized Weyl algebra (GWA) with a weight module over another GWA is a weight module over a third GWA. In this paper we compute tensor products of simple and indecomposable weight modules over generalized Weyl algebras supported on a finite orbit. This allows us to give a complete presentation by generators and relations of the Grothendieck ring of the categories of weight modules over a tower of generalized Weyl algebras in this setting. We also obtain partial results about the split Grothendieck ring. We described the case of infinite orbits in previous work. [ABSTRACT FROM AUTHOR]

We define parabolic quasi-Coxeter elements in well generated complex reflection groups. We characterize them in multiple natural ways, and we study two combinatorial objects associated with them: the collections Red W (g) of reduced reflection factorizations of g and RGS (W , g) of relative generating sets of g. We compute the cardinalities of these sets for large families of parabolic quasi-Coxeter elements and, in particular, we relate the size # Red W (g) with geometric invariants of Frobenius manifolds. This paper is second in a series of three; we will rely on many of its results in part III to prove uniform formulas that enumerate full reflection factorizations of parabolic quasi-Coxeter elements, generalizing the genus-0 Hurwitz numbers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the projectively coresolved Gorenstein flat dimension of a group G over a commutative ring R and we prove that this dimension enjoys all the properties of the cohomological and the Gorenstein cohomological dimension. We also provide good estimations for the Gorenstein global dimension of RG in terms of this dimension and the Gorenstein global dimension of R. Moreover, we study special cases of groups, such as ▪-groups, and show that for such a group every Gorenstein projective RG -module is Gorenstein flat when the global dimension of R is finite. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let p be a prime. In this paper, we study the structure of finite groups with a large number of p -regular conjugacy classes or, equivalently, a large number of irreducible p -modular representations. We prove sharp lower bounds for this number in terms of p and the p ′ -part of the order of G which ensure that G is p -solvable. A bound for the p -length is obtained which is sharp for odd primes p. We also prove a new best possible criterion for the existence of a normal Sylow p -subgroup in terms of these quantities. [ABSTRACT FROM AUTHOR]

Primitive axial algebras of Monster type are a class of non-associative algebras with a strong link to finite (especially simple) groups. The motivating example is the Griess algebra, with the Monster as its automorphism group. A crucial step towards the understanding of such algebras is the explicit description of the 2-generated symmetric objects. Recent work of Yabe, and Franchi and Mainardis shows that any such algebra is either explicitly known, or is a quotient of the infinite-dimensional Highwater algebra H , or its characteristic 5 cover H ˆ. In this paper, we complete the classification of symmetric axial algebras of Monster type by determining the quotients of H and H ˆ. We proceed in a unified way, by defining a cover of H in all characteristics. This cover has a previously unseen fusion law and provides an insight into why the Highwater algebra has a cover which is of Monster type only in characteristic 5. [ABSTRACT FROM AUTHOR]

In this paper we identify different classes of free group extension using core graphs, by further developing machinery from [3]. We show that every free group extension H ≤ K ≤ F has a basis B such that the associated pointed graph morphism Γ B (H) → Γ B (H) is onto. But if we examine graphs without base points, there is an extension 〈 b 〉 ≤ 〈 b , a b a − 1 〉 < F { a , b } such that for every basis of F { a , b } the associated graph morphisms are injective. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of surjective radical parametrization and we prove sufficient conditions for a radical curve parametrization to be surjective. [ABSTRACT FROM AUTHOR]

For a prime p , the Local Structure Theorem [15] studies finite groups G with the property that a Sylow p -subgroup S of G is contained in at least two maximal p -local subgroups. Under the additional assumptions that G contains a so called large p -subgroup Q ≤ S , and that composition factors of the normalizers of non-trivial p -subgroups are from the list of the known simple groups, [15] partially describes the p -local subgroups of G containing S , which are not contained in N G (Q). In the Global Structure Theorem, we extend the work of [15] and describe N G (Q) and, in almost all cases, the isomorphism type of the almost simple subgroup H generated by the p -local over-groups of S in G. Furthermore, for p = 2 , the isomorphism type of G is determined. In this paper, we provide a reduction framework for the proof of the Global Structure Theorem and also prove the Global Structure Theorem when Q is abelian. [ABSTRACT FROM AUTHOR]

In this paper, we associate the TKK algebra G ˆ (J) with vertex algebras through twisted modules. Firstly, we prove that for any complex number ℓ , the category of restricted G ˆ (J) -modules of level ℓ is canonically isomorphic to the category of σ -twisted V C g ˆ (ℓ , 0) -modules, where V C g ˆ (ℓ , 0) is a vertex algebra arising from the toroidal Lie algebra of type C 2 and σ is an isomorphism of V C g ˆ (ℓ , 0) induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer ℓ , the integrable restricted G ˆ (J) -modules of level ℓ are exactly the σ -twisted modules for the quotient vertex algebra L C g ˆ (ℓ , 0) of V C g ˆ (ℓ , 0). Finally, we classify the irreducible 1 2 N -graded σ -twisted L C g ˆ (ℓ , 0) -modules. [ABSTRACT FROM AUTHOR]

We give a complete answer to the rationality problem (up to stable k -equivalence) for norm one tori R K / k (1) (G m) of K / k whose Galois closures L / k are dihedral extensions with the aid of Endo and Miyata [17, Theorem 1.5, Theorem 2.3] and Endo [15, Theorem 2.1]. By using a similar technique, we give refinements of the proof of stably rational cases of Endo and Miyata's theorems as an appendix of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we examine the class of K-absolutely pure complexes. These are the complexes which are right orthogonal in the homotopy category K (R) to the acyclic complexes of pure-projective modules. The class K -abspure of these complexes is preenveloping in K (R) ; in fact, a Bousfield localization exists for the embedding K -abspure ⊆ K (R) and the quotient K (R) / K -abspure is equivalent to the homotopy category of acyclic complexes of pure-projective modules. We examine the role of K-absolutely pure complexes in the pure derived category D pure (R) and show that K -abspure is the isomorphic closure of the class of K-injective complexes therein. We explore the relevance of strongly fp-injective modules in the study of K-absolutely pure complexes and characterize the K-absolutely pure complexes of strongly fp-injective modules. Finally, we show that a K-absolutely pure complex of strongly fp-injective modules admits a K-injective complex of injective modules as a K (PInj) -preenvelope, in the case where the ring is left coherent. The notion of K-absolute purity is dual to the notion of K-flatness in the homotopy category, in a way analogous to the duality between (strongly) fp-injectivity and flatness in the module category. [ABSTRACT FROM AUTHOR]

In this paper, we construct and study various dual pairs acting on the oscillator modules of the symplectic toroidal Lie algebras coordinated by irrational quantum tori. This extends the classical Howe dual pairs to the toroidal setup. [ABSTRACT FROM AUTHOR]

A cover of an associative (not necessarily commutative nor unital) ring R is a collection of proper subrings of R whose set-theoretic union equals R. If such a cover exists, then the covering number σ (R) of R is the cardinality of a minimal cover, and a ring R is called σ -elementary if σ (R) < σ (R / I) for every nonzero two-sided ideal I of R. If R is a ring with unity, then we define the unital covering number σ u (R) to be the size of a minimal cover of R by subrings that contain 1 R (if such a cover exists), and R is σ u -elementary if σ u (R) < σ u (R / I) for every nonzero two-sided ideal of R. In this paper, we classify all σ -elementary unital rings and determine their covering numbers. Building on this classification, we are further able to classify all σ u -elementary rings and prove σ u (R) = σ (R) for every σ u -elementary ring R. We also prove that, if R is a ring without unity with a finite cover, then there exists a unital ring R ′ such that σ (R) = σ u (R ′) , which in turn provides a complete list of all integers that are the covering number of a ring. Moreover, if E (N) : = { m : m ⩽ N , σ (R) = m for some ring R } , then we show that | E (N) | = Θ (N / log ⁡ N) , which proves that almost all integers are not covering numbers of a ring. [ABSTRACT FROM AUTHOR]

The bricks over preprojective algebras of type A are known to be in bijection with certain combinatorial objects called "arcs". In this paper, we show how one can use arcs to compute bases for the Hom-spaces and first extension spaces between bricks. We then use this description to classify the "weak exceptional sequences" over these algebras. Finally, we explain how our result relates to a similar combinatorial model for the exceptional sequences over hereditary algebras of type A. [ABSTRACT FROM AUTHOR]

In this paper, we classify irreducible modules for map extended Witt algebras with finite dimensional weight spaces. They turn out to be either modules with uniformly bounded weight spaces or highest weight modules. We further prove that all these modules are single point evaluation modules (n ≥ 2). So they are actually irreducible modules for extended Witt algebras. [ABSTRACT FROM AUTHOR]

Let F be an infinite field and M 1 , 1 (F) be the algebra of 2 × 2 matrices over F endowed with non-trivial Z 2 -grading. We consider the involutions ⁎ defined on M 1 , 1 (F) which preserve the homogeneous components of the grading. In this paper, we deal with the ⁎-superalgebra (M 1 , 1 (F) , ⁎) and determine the generators of its ideal of (Z 2 , ⁎) -identities, considering that F has characteristic zero and also, we explicitly construct the generators of its space of central (Z 2 , ⁎) -polynomials, when the characteristic of F is different from 2. [ABSTRACT FROM AUTHOR]

In this paper, we mainly answer a Lampe's question [3] about the solutions of a Diophantine equation, that is, we give a criterion to determine which solutions of the Diophantine equation are in the orbit of the initial solution (ϵ , ϵ , ϵ , ϵ , ϵ) under the actions of the group G which is defined by mutations of a cluster algebra. In order to do this, using a rational map φ , we transform the Diophantine equation to a related equation whose all positive integral solutions form the orbit of an initial solution φ (ϵ , ϵ , ϵ , ϵ , ϵ) = (3 , 4 , 4) under the actions of the group G ˜ , and the set S (3 , 4 , 4) is shown to be the orbit of (ϵ , ϵ , ϵ , ϵ , ϵ) under the actions of a subgroup of G. Then the criterion is proved as the main conclusion. [ABSTRACT FROM AUTHOR]

Let U q − be the negative half of the quantum group associated to the Kac-Moody algebra g , and U _ q − the quantum group obtained by a folding of g. Let A = Z [ q , q − 1 ]. McNamara showed that U _ q − is categorified over a certain extension ring A ˜ of A , by using the folding theory of KLR algebras. He posed a question whether A ˜ coincides with A or not. In this paper, we give an affirmative answer for this problem. [ABSTRACT FROM AUTHOR]

Let K be a finite field of characteristic 2, and U T 2 : = U T 2 (K) be the Lie algebra of 2 × 2 upper triangular matrices over K with the multiplication x ∘ y = x y + y x = x y − y x. In this paper, we exhibit a finite basis of graded identities for the variety of Lie algebras generated by U T 2 for any grading and show that it has the Specht property. It is important to highlight that the technique used in order to solve the Specht problem is independent of the characteristic of the field and also of its cardinality. [ABSTRACT FROM AUTHOR]

In this paper, we begin the study of highest weight representations of the quantized enveloping superalgebra U q p n of type P. We introduce a Drinfeld-Jimbo representation and establish a triangular-decomposition of U q p n. We explain how to relate modules over U q p n to modules over p n , the Lie superalgebra of type P , and we prove that the category of tensor modules over U q p n is not semisimple. [ABSTRACT FROM AUTHOR]