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We use Regev's double centralizer theorem from [7] to study invariants of matrices with group actions. This theory then has applications to trace identities and Poincaré series. [ABSTRACT FROM AUTHOR]

In their paper of 1993, Meyer and Neutsch established the existence of a 48-dimensional associative subalgebra in the Griess algebra G . By exhibiting an explicit counter example, the present paper shows a gap in the proof one of the key results in Meyer and Neutsch's paper, which states that an idempotent a in the Griess algebra is indecomposable if and only its Peirce 1-eigenspace (i.e. the 1-eigenspace of the linear transformation L a : x ↦ a x ) is one-dimensional. The present paper fixes this gap, and shows a more general result: let V be a real commutative nonassociative algebra with an associative inner product, and let c be a nonzero idempotent of V such that its Peirce 1-eigenspace is a subalgebra; then, c is indecomposable if and only if its Peirce 1-eigenspace is one-dimensional. The proof of this result is based on a general variational argument for real commutative metrised algebras with inner product. [ABSTRACT FROM AUTHOR]

Abstract: This appendix collects some material on the cohomology of discrete subgroups of Lie groups and the theory of their representation varieties to provide background for the results of Sin and Thompson (2010, 2013) in [22,23]. [Copyright &y& Elsevier]

Abstract: We associate to a good cell decomposition of a manifold M a quadratic algebra and show that the Koszulity of the algebra implies a restriction on the Euler characteristic of M. For a two-dimensional manifold M the algebra is Koszul if and only if the Euler characteristic of M is two. [Copyright &y& Elsevier]

Abstract: In 1.9 of the paper [A. Hahn, A Hermitian Morita theorem for algebras with anti-structure, J. Algebra 93 (1985) 215–235], should be replaced by . This leads to minor changes in the rest of the paper where the ring should be replaced by its opposite and vice versa. [Copyright &y& Elsevier]

In this paper we investigate the image of the center Z of the distribution algebra D i s t (G L (m | n)) of the general linear supergroup over a ground field of positive characteristic under the Harish-Chandra morphism h : Z → D i s t (T) obtained by the restriction of the natural map D i s t (G L (m | n)) → D i s t (T). We define supersymmetric elements in D i s t (T) and show that each image h (c) for c ∈ Z is supersymmetric. The central part of the paper is devoted to a description of a minimal set of generators of the algebra of supersymmetric elements over Frobenius kernels T r. [ABSTRACT FROM AUTHOR]

Let F q 2 be the finite field with q 2 elements. Most of the known F q 2 -maximal curves arise as quotient curves of the F q 2 -maximal Hermitian curve H q. After a seminal paper by Garcia, Stichtenoth and Xing, many papers have provided genera of quotients of H q , but their complete determination is a challenging open problem. In this paper we determine completely the spectrum of genera of quotients of H q for any q ≡ 1 (mod 4). [ABSTRACT FROM AUTHOR]

We show that depth (I) ≤ 1 for each ideal I of C (X). This gives a positive answer to a conjecture in Azarpanah et al. (2019) [3]. The present article is in fact an attempt to complete the aforementioned paper. Using the above fact, the depth of some ideals of C (X) such as principal ideals, the ideals O p , p ∈ β X , and prime ideals are determined in the sense that when they are 0 and when they are 1. We have generalized Proposition 2.9 in that paper and we have shown that depth (C (X) (f)) is at most 1, for each principal ideal (f) in C (X) , and it is exactly 1 if and only if int X Z (f) contains at least one non-almost P -point. Also, we prove that for each non-essential principal ideal (f), depth (C (X) (f)) = 1 or equivalently depth (Ann (f)) = 1 if and only if the set of non-almost P -points of X is dense in X. Finally, it has been shown that depth (O p) = 1 , for each p ∈ β X , if and only if X contains at least two non-almost P -points, and topological spaces X are characterized for which depth (I) = 1 , for each essential ideal of C (X). [ABSTRACT FROM AUTHOR]

The classical Itô-Michler theorem states that the degree of every ordinary irreducible character of a finite group G is coprime to a prime p if and only if the Sylow p -subgroups of G are abelian and normal. In an earlier paper [8] , we used the notion of average character degree to prove an improvement of this theorem for the prime p = 2. In this follow-up paper, we obtain a full improvement for all primes. [ABSTRACT FROM AUTHOR]

Let X be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus g ≥ 2 defined over an algebraically closed field K of odd characteristic p ≥ 0 , and let Aut (X) be the group of all automorphisms of X which fix K element-wise. For any a subgroup G of Aut (X) whose order is a power of an odd prime d other than p , the bound proven by Zomorrodian for Riemann surfaces is | G | ≤ 9 (g − 1) where the extremal case can only be obtained for d = 3 and g ≥ 10. We prove Zomorrodian's result for any K. The essential part of our paper is devoted to extremal 3-Zomorrodian curves X. Two cases are distinguished according as the quotient curve X / Z for a central subgroup Z of Aut (X) of order 3 is either elliptic, or not. For elliptic type extremal 3-Zomorrodian curves X , we completely determine the two possibilities for the abstract structure of G using deeper results on finite 3-groups. We also show infinite families of extremal 3-Zomorrodian curves for both types, of elliptic or non-elliptic. Our paper does not adapt methods from the theory of Riemann surfaces, nevertheless it sheds a new light on the connection between Riemann surfaces and their automorphism groups. [ABSTRACT FROM AUTHOR]

This is the first paper of three papers in a row aiming to study cohomology of group theoretic Dehn fillings. In the present paper, we prove a particular free product structure, which is termed the Cohen-Lyndon property, of Dehn filling kernels. As an application, we describe the structure of relative relation modules of Dehn fillings. [ABSTRACT FROM AUTHOR]

Two classical results in algebraic geometry are that the branch curve of a del Pezzo surface of degree 1 can be embedded as a space sextic curve in P 3 and that every space sextic curve has exactly 120 tritangents corresponding to its odd theta characteristics. In this paper we revisit both results from the computational perspective. Specifically, we give an algorithm to construct space sextic curves that arise from blowing up P 2 at eight points and provide algorithms to compute the 120 tritangents and their Steiner system of any space sextic. Furthermore, we develop efficient inverses to the aforementioned methods. We present an algorithm to either reconstruct the original eight points in P 2 from a space sextic or certify that this is not possible. Moreover, we extend a construction of Lehavi [8] which recovers a space sextic from its tritangents and Steiner system. All algorithms in this paper have been implemented in magma. [ABSTRACT FROM AUTHOR]

An algebra A over a field K is said to be invertible if it has a basis B consisting only of units; if B − 1 is again a basis, A is invertible-2 , or I2. The question of when an invertible algebra is necessarily I2 arises naturally. The study of these algebras was recently initiated by López-Permouth, Moore, Szabo, Pilewski [13] , [14]. In this paper, we prove several positive results on this problem, answering also some questions and generalizing a few results from these papers. We show that every field is an I2 algebra over any subfield, and that any subalgebra of the rational functions field K (x) which strictly contains K [ x ] , with K an algebraically closed field, has a symmetric basis B = B − 1. Using this, we expand the class of examples of algebras known to be invertible or I2 with several classes, such as semiprimary rings over fields K ≠ F 2 satisfying some additional mild condition. We also show that every commutative affine invertible algebra is almost I2 in the sense that it becomes I2 after localization at a single element. [ABSTRACT FROM AUTHOR]

This paper is dedicated to new methods of constructing weight structures and weight-exact localizations; our arguments generalize their bounded versions considered in previous papers of the authors. We start from a class of objects P of a triangulated category C _ that satisfies a certain (countable) negativity condition (there are no C _ -extensions of positive degrees between elements of P ; we actually need a somewhat stronger condition of this sort) to obtain a weight structure both "halves" of which are closed either with respect to C _ -coproducts of less than α objects (where α is a fixed regular cardinal) or with respect to all coproducts (provided that C _ is closed with respect to coproducts of this sort). This construction gives all "reasonable" weight structures satisfying the latter conditions. In particular, one can obtain certain weight structures on spectra (in SH) consisting of less than α cells, and on certain localizations of SH; these results are new. [ABSTRACT FROM AUTHOR]

The cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. This paper examines the cyclic shift graphs of 'plactic-like' monoids, whose elements can be viewed as combinatorial objects of some type: aside from the plactic monoid itself (the monoid of Young tableaux), examples include the hypoplactic monoid (quasi-ribbon tableaux), the sylvester monoid (binary search trees), the stalactic monoid (stalactic tableaux), the taiga monoid (binary search trees with multiplicities), and the Baxter monoid (pairs of twin binary search trees). It was already known that for many of these monoids, connected components of the cyclic shift graph consist of elements that have the same evaluation (that is, contain the same number of each generating symbol). This paper focuses on the maximum diameter of a connected component of the cyclic shift graph of these monoids in the rank- n case. For the hypoplactic monoid, this is n − 1 ; for the sylvester and taiga monoids, at least n − 1 and at most n ; for the stalactic monoid, 3 (except for ranks 1 and 2, when it is respectively 0 and 1); for the plactic monoid, at least n − 1 and at most 2 n − 3. The current state of knowledge, including new and previously-known results, is summarized in a table. [ABSTRACT FROM AUTHOR]

Abstract: This is the final paper in a series of four on fixed point ratios in non-subspace actions of finite classical groups. Our main result states that if G is a finite almost simple classical group and Ω is a faithful transitive non-subspace G-set then either for all elements of prime order, or is one of a small number of known exceptions. In this paper we assume is either an almost simple irreducible subgroup in Aschbacher''s collection, or a subgroup in a small additional set which arises when G has socle (q even) or . This completes the proof of the main theorem. [Copyright &y& Elsevier]

The main aim of this paper is to characterize ideals I in the power series ring R = K [ [ x 1 , ... , x s ] ] that are finitely determined up to contact equivalence by proving that this is the case if and only if I is an isolated complete intersection singularity, provided dim (R / I) > 0 and K is an infinite field (of arbitrary characteristic). Here two ideals I and J are contact equivalent if the local K –algebras R / I and R / J are isomorphic. If I is minimally generated by a 1 , ... , a m , we call I finitely contact determined if it is contact equivalent to any ideal J that can be generated by b 1 , ... , b m with a i − b i ∈ 〈 x 1 , ... , x s 〉 k for some integer k. We give also computable and semicontinuous determinacy bounds. The above result is proved by considering left–right equivalence on the ring M m , n of m × n matrices A with entries in R and we show that the Fitting ideals of a finitely determined matrix in M m , n have maximal height, a result of independent interest. The case of ideals is treated by considering 1-column matrices. Fitting ideals together with a special construction are used to prove the characterization of finite determinacy for ideals in R. Some results of this paper are known in characteristic 0, but they need new (and more sophisticated) arguments in positive characteristic partly because the tangent space to the orbit of the left-right group cannot be described in the classical way. In addition we point out several other oddities, including the concept of specialization for power series, where the classical approach (due to Krull) does not work anymore. We include some open problems and a conjecture. [ABSTRACT FROM AUTHOR]

This paper is the first one in a series of three dealing with the concept of injective stabilization of the tensor product and its applications. Its primary goal is to collect known facts and establish a basic operational calculus that will be used in the subsequent parts. This is done in greater generality than is necessary for the stated goal. Several results of independent interest are also established. They include, among other things, connections with satellites, an explicit construction of the stabilization of a finitely presented functor, various exactness properties of the injectively stable functors, a construction, from a functor and a short exact sequence, of a doubly-infinite exact sequence by splicing the injective stabilization of the functor and its derived functors. When specialized to the tensor product with a finitely presented module, the injective stabilization with coefficients in the ring is isomorphic to the 1-torsion functor. The Auslander-Reiten formula is extended to a more general formula, which holds for arbitrary (i.e., not necessarily finite) modules over arbitrary associative rings with identity. Weakening of the assumptions in the theorems of Eilenberg and Watts leads to characterizations of the requisite zeroth derived functors. The subsequent papers, provide applications of the developed techniques. Part II deals with new notions of torsion module and cotorsion module of a module. This is done for arbitrary modules over arbitrary rings. Part III introduces a new concept, called the asymptotic stabilization of the tensor product. The result is closely related to different variants of stable homology (these are generalizations of Tate homology to arbitrary rings). A comparison transformation from Vogel homology to the asymptotic stabilization of the tensor product is constructed and shown to be epic. [ABSTRACT FROM AUTHOR]

Abstract Lower bounds on Hilbert–Samuel multiplicity are known for several types of commutative noetherian local rings, and rings with multiplicities which achieve these lower bounds are said to have minimal multiplicity. The first part of this paper gives characterizations of rings of minimal multiplicity in terms of the Ext-algebra of the ring; in particular, we show that minimal multiplicity can be detected via an Ext-algebra which is Gorenstein or Koszul AS-regular. The second part of this paper characterizes rings of minimal multiplicity via a numerical homological invariant introduced by J. Herzog and S. B. Iyengar called linearity defect. Our characterizations allow us to answer in two special cases a question raised by Herzog and Iyengar. [ABSTRACT FROM AUTHOR]

Abstract In this paper we establish the relation between key polynomials (as defined in [12]) and minimal pairs of definition of a valuation. We also discuss truncations of valuations on a polynomial ring K [ x ]. We prove that a valuation ν is equal to its truncation on some polynomial if and only if ν is valuation-transcendental. Another important result of this paper is that if μ is any extension of ν to K ‾ [ x ] and Λ is a complete sequence of key polynomials for ν , without last element, then for each Q ∈ Λ there exists a suitable root a Q ∈ K ‾ of Q such that { a Q } Q ∈ Λ is a pseudo-convergent sequence defining μ. [ABSTRACT FROM AUTHOR]

Abstract Let G be an infinite group and let X be a finite generating set for G such that the growth series of G with respect to X is a rational function; in this case G is said to have rational growth with respect to X. In this paper a result on sizes of spheres (or balls) in the Cayley graph Γ (G , X) is obtained: namely, the size of the sphere of radius n is bounded above and below by positive constant multiples of n α λ n for some integer α ≥ 0 and some λ ≥ 1. As an application of this result, a calculation of degree of commutativity (d. c.) is provided: for a finite group F , its d. c. is defined as the probability that two randomly chosen elements in F commute, and Antolín, Martino and Ventura have recently generalised this concept to all finitely generated groups. It has been conjectured that the d. c. of a group G of exponential growth is zero. This paper verifies the conjecture (for certain generating sets) when G is a right-angled Artin group or, more generally, a graph product of groups of rational growth in which centralisers of non-trivial elements are "uniformly small". [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to present some contributions to the theory of finite transformation monoids. The dominating influence that permutation groups have on transformation monoids is used to describe and characterise transitive transformation monoids and primitive transitive transformation monoids. We develop a theory that not only includes the analogs of several important theorems of the classical theory of permutation groups but also contains substantial information about the algebraic structure of the transformation monoids. Open questions naturally arising from the substantial paper of Steinberg (2010) [11] have been answered. Our results can also be considered as a further development in the hunt for a solution of the Černý conjecture. [ABSTRACT FROM AUTHOR]

Highlights • Let G be a finite group and ψ (G) = ∑ g ∈ G o (g) , where o (g) denotes the order of g ∈ G. • M. Herzog et al. in [J. Algebra, 2018] give two new criteria for solvability of finite groups. • They proved that if G is a group of order n and ψ (G) ≥ ψ (C n) / 6.68 , then G is solvable. • They conjectured that: Conjecture. If G is a group of order n and ψ (G) > 211 1617 ψ (C n) , then G is solvable. • As the main result of this paper we prove the validity of this conjecture. Abstract Let G be a finite group and ψ (G) = ∑ g ∈ G o (g) , where o (g) denotes the order of g ∈ G. In [M. Herzog, et al., Two new criteria for solvability of finite groups, J. Algebra, 2018], the authors put forward the following conjecture: Conjecture. If G is a group of order n and ψ (G) > 211 ψ (C n) / 1617 , where C n is the cyclic group of order n, then G is solvable. In this paper we prove the validity of this conjecture. [ABSTRACT FROM AUTHOR]

Abstract In this paper we classify all blocks with defect group C 2 n × C 2 × C 2 up to Morita equivalence. Together with a recent paper of Wu, Zhang and Zhou, this completes the classification of Morita equivalence classes of 2-blocks with abelian defect groups of rank at most 3. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. The case considered in this paper is significant because it involves comparison of Morita equivalence classes between a group and a normal subgroup of index 2, so requires novel reduction techniques which we hope will be of wider interest. We note that this also completes the classification of blocks with abelian defect groups of order dividing 16 up to Morita equivalence. A consequence is that Broue's abelian defect group conjecture holds for all blocks mentioned above. [ABSTRACT FROM AUTHOR]

Abstract Let G be a finite group, N a nilpotent normal subgroup of G and let V (Z G , N) denote the group formed by the units of the integral group ring Z G of G which map to the identity under the natural homomorphism Z G → Z (G / N). Sehgal asked whether any torsion element of V (Z G , N) is conjugate in the rational group algebra of G to an element of G. This is a special case of the Zassenhaus Conjecture. By results of Cliff and Weiss and Hertweck, Sehgal's Problem has a positive solution if N has at most one non-cyclic Sylow subgroup. We present some algorithms to study Sehgal's Problem when N has at most one non-abelian Sylow subgroup. They are based on the Cliff–Weiss inequalities introduced by the authors in a previous paper. With the help of these algorithms we obtain some positive answers to Sehgal's Problem and use them to show that for units in V (Z G , N) our method is strictly stronger than the well known HeLP Method. We then present a method to use the output of one of the algorithms to construct explicit metabelian groups which are candidates to a negative solution to Sehgal's Problem. Recently Eisele and Margolis showed that some of the examples proposed in this paper are indeed counterexamples to the Zassenhaus Conjecture. These are the first known counterexamples. Moreover, we prove that every metabelian negative solution of Sehgal's Problem satisfying some minimal conditions is given by our construction. [ABSTRACT FROM AUTHOR]

Abstract Let A be a commutative ring, and let a be a finitely generated ideal in it. It is known that a necessary and sufficient condition for the derived a -torsion and a -adic completion functors to be nicely behaved is the weak proregularity of a. In particular, the MGM Equivalence holds. Because weak proregularity is defined in terms of elements of the ring (it involves limits of Koszul complexes), it is not suitable for noncommutative ring theory. In this paper we introduce a new condition on a torsion class T in a module category: weak stability. Our first main theorem says that in the commutative case, the ideal a is weakly proregular if and only if the corresponding torsion class T is weakly stable. We then study weak stability of torsion classes in module categories over noncommutative rings. There are three main theorems in this context: ▷ For a torsion class T that is weakly stable, quasi-compact and finite dimensional, the right derived torsion functor is isomorphic to a left derived tensor functor. ▷ The Noncommutative MGM Equivalence , that holds under the same assumptions on T. ▷ A theorem about symmetric derived torsion for complexes of bimodules. This last theorem is a generalization of a result of Van den Bergh from 1997, and corrects an error in a paper of Yekutieli & Zhang from 2003. [ABSTRACT FROM AUTHOR]

Abstract There is the paper by H. Tietze published in 1905 on differential transcendence of solutions of difference Riccati equations. In this paper, we clarify the essence of Tietze's treatment and make it purely algebraic. As an application, the q -Airy equation is studied. [ABSTRACT FROM AUTHOR]

A permutation group is said to be quasiregular if each of its transitive constituents is regular, and a quasiregular coherent configuration can be thought as a combinatorial analog of such a group: the transitive constituents are replaced by the homogeneous components. In this paper, we are interested in the question when the configuration is schurian, i.e., formed by the orbitals of a permutation group, or/and separable, i.e., uniquely determined by the intersection numbers. In these terms, an old result of Hanna Neumann is, in a sense, dual to the statement that the quasiregular coherent configurations with cyclic homogeneous components are schurian. In the present paper, we (a) establish the duality in a precise form and (b) generalize the latter result by proving that a quasiregular coherent configuration is schurian and separable if the groups associated with the homogeneous components have distributive lattices of normal subgroups. [ABSTRACT FROM AUTHOR]

In connection with our previous work on semi-galois categories [1,2] , this paper proves an arithmetic analogue of Christol's theorem concerning an automata-theoretic characterization of when a formal power series ξ = ∑ ξ n t n ∈ F q [ [ t ] ] over finite field F q is algebraic over the polynomial ring F q [ t ] . There are by now several variants of Christol's theorem, all of which are concerned with rings of positive characteristic. This paper provides an arithmetic (or F 1 -) variant of Christol's theorem in the sense that it replaces the polynomial ring F q [ t ] with the ring O K of integers of a number field K and the ring F q [ [ t ] ] of formal power series with the ring of Witt vectors. We also study some related problems. [ABSTRACT FROM AUTHOR]

This paper studies separating invariants of finite groups acting on affine varieties through automorphisms. Several results, proved by Serre, Dufresne, Kac–Watanabe and Gordeev, and Jeffries and Dufresne exist that relate properties of the invariant ring or a separating subalgebra to properties of the group action. All these results are limited to the case of linear actions on vector spaces. The goal of this paper is to lift this restriction by extending these results to the case of (possibly) non-linear actions on affine varieties. Under mild assumptions on the variety and the group action, we prove that polynomial separating algebras can exist only for reflection groups. The benefit of this gain in generality is demonstrated by an application to the semigroup problem in multiplicative invariant theory. Then we show that separating algebras which are complete intersections in a certain codimension can exist only for 2-reflection groups. Finally we prove that a separating set of size n + k − 1 (where n is the dimension of X ) can exist only for k -reflection groups. Several examples show that most of the assumptions on the group action and the variety that we make cannot be dropped. [ABSTRACT FROM AUTHOR]

The first Weyl algebra over k , A 1 = k 〈 x , y 〉 / ( x y − y x − 1 ) admits a natural Z -grading by letting deg ⁡ x = 1 and deg ⁡ y = − 1 . Smith showed that gr ­ A 1 is equivalent to the category of quasicoherent sheaves on a certain quotient stack. Using autoequivalences of gr ­ A 1 , Smith constructed a commutative ring C , graded by finite subsets of the integers. He then showed gr ­ A 1 ≡ gr ­ ( C , Z fin ) . In this paper, we prove analogues of Smith's results by using autoequivalences of a graded module category to construct rings with equivalent graded module categories. For certain generalized Weyl algebras, we use autoequivalences defined in a companion paper so that these constructions yield commutative rings. [ABSTRACT FROM AUTHOR]

We work over an algebraically closed field of characteristic zero. The purpose of this paper is to prove that the complete linear system of the double point divisors of smooth birational-divisors on conical rational scrolls are base-point-free. A smooth birational-divisor on a conical rational scroll has a nonbirational inner center, that is a point on it from which the linear projection gives nonbirational map to its image. In the previous paper by the author, it was shown that for a projective variety without nonbirational inner centers, its double point divisor is base-point-free. [ABSTRACT FROM AUTHOR]

Let OSp ( V ) be the orthosymplectic supergroup on an orthosymplectic vector superspace V of superdimension ( m | 2 n ) . Lehrer and Zhang showed that there is a surjective algebra homomorphism F r r : B r ( m − 2 n ) → End OSp ( V ) ( V ⊗ r ) , where B r ( m − 2 n ) is the Brauer algebra of degree r with parameter m − 2 n . The second fundamental theorem of invariant theory in this setting seeks to describe the kernel Ker F r r of F r r as a 2-sided ideal of B r ( m − 2 n ) . In this paper, we show that Ker F r r ≠ 0 if and only if r ≥ r c : = ( m + 1 ) ( n + 1 ) , and give a basis and a dimension formula for Ker F r r . We show that Ker F r r as a 2-sided ideal of B r ( m − 2 n ) is generated by Ker F r c r c for any r ≥ r c , and we provide an explicit set of generators for Ker F r c r c . These generators coincide in the classical case with those obtained in recent papers of Lehrer and Zhang on the second fundamental theorem of invariant theory for the orthogonal and symplectic groups. As an application we obtain the necessary and sufficient conditions for the endomorphism algebra End osp ( V ) ( V ⊗ r ) over the orthosymplectic Lie superalgebra osp ( V ) to be isomorphic to B r ( m − 2 n ) . [ABSTRACT FROM AUTHOR]

A classical problem, originated by Cohn's 1966 paper [1] , is to characterize the integral domains R satisfying the property: ( GE n ) “every invertible n × n matrix with entries in R is a product of elementary matrices”. Cohn called these rings generalized Euclidean, since the classical Euclidean rings do satisfy ( GE n ) for every n > 0 . Important results on algebraic number fields motivated a natural conjecture: a non-Euclidean principal ideal domain R does not satisfy ( GE n ) for some n > 0 . We verify this conjecture for two important classes of non-Euclidean principal ideal domains: (1) the coordinate rings of special algebraic curves, among them the elliptic curves having only one rational point; (2) the non-Euclidean PID's constructed by a fixed procedure, described in Anderson's 1988 paper [2] . [ABSTRACT FROM AUTHOR]

In the first part of this paper, we study Koszul property of directed graded categories. In the second part of this paper, we prove a general criterion for an infinite directed category to be Koszul. We show that infinite directed categories in the theory of representation stability are Koszul over a field of characteristic zero. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new concept of key polynomials for a given valuation ν on K [ x ] . We prove that such polynomials have many of the expected properties of key polynomials as those defined by MacLane and Vaquié, for instance, that they are irreducible and that the truncation of ν associated to each key polynomial is a valuation. Moreover, we prove that every valuation ν on K [ x ] admits a sequence of key polynomials that completely determines ν (in the sense which we make precise in the paper). We also establish the relation between these key polynomials and pseudo-convergent sequences defined by Kaplansky. [ABSTRACT FROM AUTHOR]

Recently, we introduced the notion of flow (depending on time) of finite-dimensional algebras. A flow of algebras (FA) is a particular case of a continuous-time dynamical system whose states are finite-dimensional algebras with (cubic) matrices of structural constants satisfying an analogue of the Kolmogorov–Chapman equation (KCE). Since there are several kinds of multiplications between cubic matrices one has fix a multiplication first and then consider the KCE with respect to the fixed multiplication. The existence of a solution for the KCE provides the existence of an FA. In this paper our aim is to find sufficient conditions on the multiplications under which the corresponding KCE has a solution. Mainly our conditions are given on the algebra of cubic matrices (ACM) considered with respect to a fixed multiplication of cubic matrices. Under some assumptions on the ACM (e.g. power associative, unital, associative, commutative) we describe a wide class of FAs, which contain algebras of arbitrary finite dimension. In particular, adapting the theory of continuous-time Markov processes, we construct a class of FAs given by the matrix exponent of cubic matrices. Moreover, we remarkably extend the set of FAs given with respect to the Maksimov's multiplications of our paper [8] . For several FAs we study the time-dependent behavior (dynamics) of the algebras. We derive a system of differential equations for FAs. [ABSTRACT FROM AUTHOR]

In this paper, we used the free fields of Wakimoto to construct a class of irreducible representations for the general linear Lie superalgebra gl m | n ( C ) . The structures of the representations over the general linear Lie superalgebra and the special linear Lie superalgebra are studied in this paper. Then we extend the construction to the affine Kac–Moody Lie superalgebra gl m | n ˆ ( C ) on the tensor product of a polynomial algebra and an exterior algebra with infinitely many variables involving one parameter μ , and we also obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter μ is nonzero. [ABSTRACT FROM AUTHOR]

The thin residue is an important concept in the theory of association schemes. Association schemes whose thin residues are thin have been studied in several papers. In this paper we construct a new class of association schemes whose thin residues are thin, and give a characterization of their automorphism groups. In particular, we give a formula for the order of the automorphism group, and a necessary and sufficient condition under which the association scheme is Schurian. Schur rings are used as a tool in our approach. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of the theory of permutation groups. However, this is another instance of a situation common in mathematics in which a very natural problem turns out to be extremely difficult. Fortunately, the enormous progresses of the last few decades seem to allow a new momentum on the attack to this problem. In this paper we prove that there are infinite families of primitive groups contained in the union of imprimitive groups and propose a new hierarchy for primitive groups based on that fact. In addition we introduce some algorithms to handle permutations, provide the corresponding GAP implementation, solve some open problems, and propose a large list of open problems. [ABSTRACT FROM AUTHOR]

Poisson algebras are, just like Lie algebras, particular cases of Lie–Rinehart algebras. The latter were introduced by Rinehart in his seminal 1963 paper, where he also introduces the notion of an enveloping algebra and proves — under some mild conditions — that the enveloping algebra of a Lie–Rinehart algebra satisfies a Poincaré–Birkhoff–Witt theorem (PBW theorem). In the case of a Poisson algebra ( A , ⋅ , { ⋅ , ⋅ } ) over a commutative ring R (with unit), Rinehart's result boils down to the statement that if A is smooth (as an algebra), then gr ( U ( A ) ) and Sym ( Ω ( A ) ) are isomorphic as graded algebras; in this formula, U ( A ) stands for the Poisson enveloping algebra of A and Ω ( A ) is the A -module of Kähler differentials of A (viewing A as an R -algebra). In this paper, we give several new constructions of the Poisson enveloping algebra in some general and in some particular contexts. Moreover, we show that for an important class of singular Poisson algebras, the PBW theorem still holds. In geometrical terms, these Poisson algebras correspond to (singular) Poisson hypersurfaces of arbitrary smooth affine Poisson varieties. Throughout the paper we give several examples and present some first applications of the main theorem; applications to deformation theory and to Poisson and Hochschild (co-) homology will be worked out in a future publication. [ABSTRACT FROM AUTHOR]

Ulf Rehmann and Jun Morita, in their 1989 paper A Matsumoto Type Theorem for Kac–Moody Groups , gave a presentation of K 2 ( A , F ) for any generalised Cartan matrix A and field F . The purpose of this paper is to use this presentation to compute K 2 ( A , F ) more explicitly in the case when A is hyperbolic. In particular, we shall show that these K 2 ( A , F ) can always be expressed as a product of quotients of K 2 ( F ) and K 2 ( 2 , F ) . Along the way, we shall also prove a similar result in the case when A has an odd entry in each column. [ABSTRACT FROM AUTHOR]

In 2006 Sommers and Tymoczko defined so called arrangements of ideal type A I stemming from ideals I in the set of positive roots of a reduced root system. They showed in a case by case argument that A I is free if the root system is of classical type or G 2 and conjectured that this is also the case for all types. This was established only very recently in a uniform manner by Abe, Barakat, Cuntz, Hoge and Terao. The set of non-zero exponents of the free arrangement A I is given by the dual of the height partition of the roots in the complement of I in the set of positive roots, generalizing the Shapiro–Steinberg–Kostant theorem which asserts that the dual of the height partition of the set of positive roots gives the exponents of the associated Weyl group. Our first aim in this paper is to investigate a stronger freeness property of the A I . We show that all A I are inductively free, with the possible exception of some cases in type E 8 . In the same paper from 2006, Sommers and Tymoczko define a Poincaré polynomial I ( t ) associated with each ideal I which generalizes the Poincaré polynomial W ( t ) for the underlying Weyl group W . Solomon showed that W ( t ) satisfies a product decomposition depending on the exponents of W for any Coxeter group W . Sommers and Tymoczko showed in a case by case analysis in types A n , B n and C n , and some small rank exceptional types that a similar factorization property holds for the Poincaré polynomials I ( t ) generalizing the formula of Solomon for W ( t ) . They conjectured that their multiplicative formula for I ( t ) holds in all types. In our second aim to investigate this conjecture further, the same inductive tools we develop to obtain inductive freeness of the A I are also employed. Here we also show that this conjecture holds inductively in almost all instances with only a small number of possible exceptions. [ABSTRACT FROM AUTHOR]

This note is motivated by the problem to understand, given a commutative ring F , which G -sets X , Y give rise to isomorphic F [ G ] -representations F [ X ] ≅ F [ Y ] . A typical step in such investigations is an argument that uses induction theorems to give very general sufficient conditions for all such relations to come from proper subquotients of G . In the present paper we axiomatise the situation, and prove such a result in the generality of Mackey functors and Green functors with inflation. Our result includes, as special cases, a result of Deligne on monomial relations, a result of the first author and Tim Dokchitser on Brauer relations in characteristic 0, and a new result on Brauer relations in characteristic p > 0 . We will need the new result in a forthcoming paper on Brauer relations in positive characteristic. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove some theorems about vanishing of Euler class groups. For example, suppose where is a polynomial ring, is a non-zero divisor and . Then we prove that the Euler class group for any rank one projective B module L. [Copyright &y& Elsevier]

We present a practical algorithm which, given a non-archimedean local field K and any two elements A , B ∈ S L 2 (K) , determines after finitely many steps whether or not the subgroup 〈 A , B 〉 ≤ S L 2 (K) is discrete and free of rank two. This makes use of the Ping Pong Lemma applied to the action of S L 2 (K) by isometries on its Bruhat-Tits tree. The algorithm itself can also be used for two-generated subgroups of the isometry group of any locally finite simplicial tree, and has applications to the constructive membership problem. In an appendix joint with Frédéric Paulin, we give an erratum to his 1989 paper 'The Gromov topology on R -trees', which details some translation length formulae that are fundamental to the algorithm. [ABSTRACT FROM AUTHOR]

If a Nakayama algebra is not cyclic, it has finite global dimension. For a cyclic Nakayama algebra, there are many characterizations of when it has finite global dimension. In [17] , Shen gave such a characterization using Ringel's resolution quiver. In [11] , the second author, with Zacharia, gave a cyclic homology characterization for when a monomial relation algebra has finite global dimension. We show directly that these criteria are equivalent for all Nakayama algebras. Our comparison result also reproves both characterizations. In a separate paper we discuss an interesting example that came up in our attempt to generalize this comparison result to arbitrary monomial relation algebras [8]. [ABSTRACT FROM AUTHOR]

By a φ -variety V , we mean a supervariety or a ⁎-variety generated by an associative algebra over a field F of characteristic zero. In this case, we consider its sequence of φ -codimensions c n φ (V) and say that V is minimal of polynomial growth n k if c n φ (V) grows like n k , but any proper φ -subvariety grows like n t with t < k. In this paper, we deal with minimal φ -varieties generated by unitary algebras and prove that for k ≤ 2 there is only a finite number of them. We also explicit a list of finite dimensional algebras generating such minimal φ -varieties. For k ≥ 3 , we show that the number of minimal φ -varieties can be infinity and we classify all minimal φ -varieties of polynomial growth n k by giving a recipe for the construction of their T φ -ideals. [ABSTRACT FROM AUTHOR]

In this paper we present an algorithm to compute a Standard Basis for a fractional ideal I of the local ring O of an n -space algebroid curve with several branches. This allows us to determine the semimodule of values of I. When I = O , we may obtain a (finite) set of generators of the semiring of values of the curve, which determines its classical semigroup. In the complex context, identifying the Kähler differential module Ω O / C of a plane curve with a fractional ideal of O and applying our algorithm, we can compute the set of values of Ω O / C , which is an important analytic invariant associated to the curve. [ABSTRACT FROM AUTHOR]