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There are several classical characterisations of the valuative dimension of a commutative ring. Constructive versions of this dimension have been given and proven to be equivalent to the classical notion within classical mathematics, and they can be used for the usual examples of commutative rings. To the contrary of the classical versions, the constructive versions have a clear computational content. This paper investigates the computational relationship between three possible constructive definitions of the valuative dimension of a commutative ring. In doing so, it proves these constructive versions to be equivalent within constructive mathematics. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic 0. We study some cohomological properties of Lie subalgebras of the Witt algebra W = Der (k [ t , t − 1 ]) and the one-sided Witt algebra W ≥ − 1 = Der (k [ t ]). In the first part of the paper, we consider finite codimension subalgebras of W ≥ − 1. We compute derivations and one-dimensional extensions of such subalgebras. These correspond to Ext U (L) 1 (M , L) , where L is a subalgebra of W ≥ − 1 and M is a one-dimensional representation of L. We find that these subalgebras exhibit a kind of rigidity: their derivations and extensions are controlled by the full one-sided Witt algebra. As an application of these computations, we prove that any isomorphism between finite codimension subalgebras of W ≥ − 1 extends to an automorphism of W ≥ − 1. The second part of the paper is devoted to explaining the observed rigidity. We define a notion of "completely non-split extension" and prove that W ≥ − 1 is the universal completely non-split extension of any of its subalgebras of finite codimension. In some sense, this means that even when studying subalgebras of W ≥ − 1 as abstract Lie algebras, they remember that they are contained in W ≥ − 1. We also consider subalgebras of infinite codimension, explaining the similarities and differences between the finite and infinite codimension situations. Almost all of the results above are also true for subalgebras of the Witt algebra. We summarise results for W at the end of the paper. [ABSTRACT FROM AUTHOR]

Complete filtered A ∞ -algebras model certain deformation problems in the noncommutative setting. The formal deformation theory of a group representation is a classical example. With such applications in mind, we provide the A ∞ analogs of several key theorems from the Maurer-Cartan theory for L ∞ -algebras. In contrast with the L ∞ case, our results hold over a field of arbitrary characteristic. We first leverage some abstract homotopical algebra to give a concise proof of the A ∞ -Goldman-Millson theorem: The nerve functor, which assigns a simplicial set N • (A) to an A ∞ -algebra A , sends filtered quasi-isomorphisms to homotopy equivalences. We then characterize the homotopy groups of N • (A) in terms of the cohomology algebra H (A) , and its group of quasi-invertible elements. Finally, we return to the characteristic zero case and show that the nerve of A is homotopy equivalent to the simplicial Maurer-Cartan set of its commutator L ∞ -algebra. This answers a question posed by N. de Kleijn and F. Wierstra in [26]. [ABSTRACT FROM AUTHOR]

We exhibit an adjunction between a category of abstract algebras of partial functions and a category of set quotients. The algebras are those atomic algebras representable as a collection of partial functions closed under relative complement and domain restriction; the morphisms are the complete homomorphisms. This generalises the discrete adjunction between the atomic Boolean algebras and the category of sets. We define the compatible completion of a representable algebra, and show that the monad induced by our adjunction yields the compatible completion of any atomic representable algebra. As a corollary, the adjunction restricts to a duality on the compatibly complete atomic representable algebras, generalising the discrete duality between complete atomic Boolean algebras and sets. We then extend these adjunction, duality, and completion results to representable algebras equipped with arbitrary additional completely additive and compatibility preserving operators. [ABSTRACT FROM AUTHOR]

Let B be a ring, not necessarily commutative, having an involution ⁎ and let U 2 m (B) be the unitary group of rank 2 m associated to a hermitian or skew hermitian form relative to ⁎. When B is finite, we construct a Weil representation of U 2 m (B) via Heisenberg groups and find its explicit matrix form on the Bruhat elements. As a consequence, we derive information on generalized Gauss sums. On the other hand, there is an axiomatic method to define a Weil representation of U 2 m (B) , and we compare the two Weil representations thus obtained under fairly general hypotheses. When B is local, not necessarily finite, we compute the index of the subgroup of U 2 m (B) generated by its Bruhat elements. Besides the independent interest, this subgroup and index are involved in the foregoing comparison of Weil representations. [ABSTRACT FROM AUTHOR]

Abstract Motivated by the notion of extremal vertex operator algebras, we investigate cyclic orbifolds of vertex operator algebras coming from extremal even self-dual lattices in d = 48 and d = 72. In this way we construct about one hundred new examples of holomorphic VOAs with a small number of low weight states. [ABSTRACT FROM AUTHOR]

Abstract We show a dichotomy for groups of cardinality less than continuum. The number of homomorphisms from the Hawaiian earring group to such a group G is either the cardinality of G in case G is noncommutatively slender, or the number is 2 2 ℵ 0 in case G is not noncommutatively slender. An example of a noncommutatively slender group with nontrivial divisible element is also exhibited. [ABSTRACT FROM AUTHOR]

Abstract This paper has two main parts. First, we give a classification of the ℓ -blocks of finite special linear and unitary groups SL n (ϵ q) in the non-defining characteristic ℓ ≥ 3. Second, we describe how the ℓ -weights of SL n (ϵ q) can be obtained from the ℓ -weights of GL n (ϵ q) when ℓ ∤ gcd (n , q − ϵ) , and verify the Alperin weight conjecture for SL n (ϵ q) under the condition ℓ ∤ gcd (n , q − ϵ). As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for any unipotent ℓ -block of SL n (ϵ q) if ℓ ∤ gcd (n , q − ϵ). [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 k be a possibly non-perfect field of characteristic p > 0. In this work we prove the local existence of absolute p -bases for regular algebras of finite type over k. Namely, consider a regular variety Z over k. Kimura and Niitsuma proved that, for every ξ ∈ Z , the local ring O Z , ξ has a p -basis over O Z , ξ p. Here we show that, for every ξ ∈ Z , there exists an open affine neighborhood of ξ , say ξ ∈ Spec (A) ⊂ Z , so that A admits a p -basis over A p. This passage from the local ring to an affine neighborhood of ξ has geometrical consequences, some of which will be discussed in the second part of the article. As we will see, given a p -basis B of the algebra A over A p , there is a family of differential operators on A naturally associated to B. These differential operators will enable us to give a Jacobian criterion for regularity for varieties defined over k , as well as a method to compute the order of an ideal I ⊂ A. [ABSTRACT FROM AUTHOR]

Abstract Let X be any scheme defined over a Dedekind scheme S with a given section x ∈ X (S). We prove the existence of a pro-finite S -group scheme ℵ (X , x) and a universal ℵ (X , x) -torsor dominating all the pro-finite pointed torsors over X. Though ℵ (X , x) may not be unique in general it still can provide useful information in order to better understand X. In a similar way we prove the existence of a pro-algebraic S -group scheme ℵ alg (X , x) and a ℵ alg (X , x) -torsor dominating all the pro-algebraic and affine pointed torsors over X. The case where X → S has no sections is also considered. [ABSTRACT FROM AUTHOR]

Abstract Let (R , m) be a commutative complete Gorenstein local ring and let Λ be a Gorenstein order, that is to say, Λ is a maximal Cohen–Macaulay R -module and Hom R (Λ , R) is a projective Λ-module. The main theme of this paper is to study the representation-theoretic properties of generalized lattices, i.e. those Λ-modules which are Gorenstein projective over R. It is proved that Λ has only finitely many isomorphism classes of indecomposable lattices if and only if every generalized lattice is the direct sum of finitely generated ones. It is also turn out that, if R is one-dimensional, then a generalized lattice M which is not the direct sum of copies of a finite number of lattices, contains indecomposable sublattices of arbitrarily large finite h _ -length, an invariant assigned to each generalized lattice which measures Hom modulo projectives. [ABSTRACT FROM AUTHOR]

Abstract Let l be a prime number. In the present paper, we discuss a pro-l version of the congruence subgroup problem for mapping class groups of genus one. Our main result is that the pro-2 version has an affirmative answer, but the pro- l version for l ≥ 11 has a negative answer. [ABSTRACT FROM AUTHOR]

Abstract Let R be an associative ring with unity and let [ R ] and U (R) denote the associated Lie ring (with [ a , b ] = a b − b a) and the group of units of R , respectively. In 1983 Gupta and Levin proved that if [ R ] is a nilpotent Lie ring of class c then U (R) is a nilpotent group of class at most c. The aim of the present note is to show that, in general, a similar statement does not hold if [ R ] is n -Engel. We construct an algebra R over a field of characteristic ≠ 2 , 3 such that the Lie algebra [ R ] is 5-Engel but the group U (R) is not. [ABSTRACT FROM AUTHOR]

Abstract We introduce a certain differential graded bialgebra, neither commutative nor cocommutative, that governs perturbations of a differential on complexes supplied with an abstract Hodge decomposition. This leads to a conceptual treatment of the Homological Perturbation Lemma and its multiplicative version. As an application we give an explicit form of the decomposition theorem for A ∞ algebras and A ∞ modules and, more generally, for twisted objects in differential graded categories. [ABSTRACT FROM AUTHOR]

Abstract Howie and Duncan observed that a word in a free product with length at least two, which is not a proper power and involves no letter of order two can be decomposed as a product of two cyclic subwords each of which is uniquely positioned. Using this property, they proved various important results about a one-relator product of groups with such word as the relator. In this paper, we show that similar results hold in a more general setting where we allow a certain number of elements of order two. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study Virasoro vertex algebras and affine vertex algebras over a general field of characteristic p > 2. More specifically, we study certain quotients of the universal Virasoro and affine vertex algebras by ideals related to the p -centers of the Virasoro algebra and affine Lie algebras. Among the main results, we classify their irreducible N -graded modules by explicitly determining their Zhu algebras and show that these vertex algebras have only finitely many irreducible N -graded modules and they are C 2 -cofinite. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the algebra of global differential operators D X on wonderful compactifications X of symmetric spaces G / H of type A 1 and A 2. We will first construct a global differential operator on these varieties that does not come from the infinitesimal action of the Lie algebra of G. We will then focus on type A 2 , where we will show that D X is an algebra of finite type, and that for any invertible sheaf L on X , H 0 (X , L) is either 0 or a simple left D X , L -module. Finally, we will show with the help of local cohomology that this still holds for higher cohomology groups H i (X , L). [ABSTRACT FROM AUTHOR]

Abstract A correspondence functor is a functor from the category of finite sets and correspondences to the category of k -modules, where k is a commutative ring. A main tool for this study is the construction of a correspondence functor associated to any finite lattice T. We prove for instance that this functor is projective if and only if the lattice T is distributive. Moreover, it has quotients which play a crucial role in the analysis of simple functors. The special case of total orders yields some more specific and complete results. [ABSTRACT FROM AUTHOR]

Abstract We describe the structure and homological properties of arbitrary generalized standard Auslander–Reiten components of artin algebras. In particular, we prove that for all but finitely many indecomposable modules in such components the Euler characteristic is defined and nonnegative. Further, we provide a handy criterion for an infinite Auslander–Reiten component of an artin algebra to be generalized standard. We solve also the long standing open problem concerning the structure of artin algebras admitting a separating family of Auslander–Reiten components. [ABSTRACT FROM AUTHOR]

Abstract We investigate the Rees algebra and the toric ring of the squarefree monomial ideal associated to the three-dimensional Ferrers diagram. Under the projection property condition, we describe explicitly the presentation ideals of the Rees algebra and the toric ring. We show that the toric ring is a Koszul Cohen–Macaulay normal domain, while the Rees algebra is Koszul and the defining ideal is of fiber type. [ABSTRACT FROM AUTHOR]

Abstract It is well known that for two univariate polynomials over the complex number field the number of their common roots is equal to the order of their resultant. In this paper, we show that this fundamental relationship still holds for the tropical polynomials under suitable adaptation of the notion of order, if the roots are simple and non-zero. [ABSTRACT FROM AUTHOR]

Abstract Given a normal toric algebra R , we compute a uniform integer D = D (R) > 0 such that the symbolic power P (D N) ⊆ P N for all N > 0 and all monomial primes P. We compute the multiplier D explicitly in terms of the polyhedral cone data defining R , illustrating the output for Segre–Veronese algebras. [ABSTRACT FROM AUTHOR]

Abstract We show that for k ≥ 6 and g sufficiently large, the mapping class group of a surface of genus g can be generated by three elements of order k. We also show that this can be done with four elements of order 5 when g is at least 8. [ABSTRACT FROM AUTHOR]

Abstract This note provides an extension of a classical result of Springer in the Algebraic Theory of Quadratic Forms for discretely valued fields to the dyadic case. [ABSTRACT FROM AUTHOR]

Abstract Let R be a perfect ring of characteristic p. We show that the group of continuous R -linear automorphisms of the perfect power series ring over R is generated by the automorphisms of the ordinary power series ring together with Frobenius; this answers a question of Jared Weinstein. [ABSTRACT FROM AUTHOR]

Abstract In this paper we revisit two concepts which were originally introduced by Aschbacher and are crucial in the theory of saturated fusion systems: Firstly, we give a new approach to defining the centra-lizer of a normal subsystem. Secondly, we revisit the construction of the product of two normal subsystems which centralize each other. [ABSTRACT FROM AUTHOR]

Abstract We begin the study of the notion of diameter of an ideal I ⊂ S = k [ x 1 , ... , x n ] , an invariant measuring the distance between the minimal primes of I. We provide large classes of Hirsch ideals, i.e. ideals satisfying diam (I) ≤ height (I) , such as: quadratic radical ideals such that S / I is Gorenstein and height (I) ≤ 4 , or ideals admitting a square-free complete intersection initial ideal. [ABSTRACT FROM AUTHOR]

We give presentations for the C-groups of rank n − 1 of the symmetric group S n . We also classify C-groups of rank n − 2 for S n . We show that all these C-groups correspond to regular hypertopes, that is, thin, residually connected flag-transitive geometries. Therefore we generalise some similar results obtained in the framework of string C-groups that are in one-to-one correspondence with abstract regular polytopes. [ABSTRACT FROM AUTHOR]

We establish new structures on Grothendieck–Witt rings, including a G W ( k ) -module structure on the unit group G W ( k ) × and a presentation of G W _ × as an infinite G m -loop sheaf. Even though our constructions are motivated by speculations in stable A 1 -homotopy theory, our arguments are purely algebraic. [ABSTRACT FROM AUTHOR]

This note has three contributions. The first one is to give elementary proofs about finite free resolution using only the notion of latent regular element (regular element which may be obtained after addition of indeterminates) and no notion from homological algebra. The second contribution is to use this to provide a simple proof of Wiebe's Theorem, a fundamental result in the theory of resultants [5] . Finally the third contribution is to explain how to “automatically” prove results or obtain counter-examples about regular sequences of a given length. We obtain in this way a new example of a local ring with a regular sequence b 1 , b 2 such that b 2 , b 1 is not regular. [ABSTRACT FROM AUTHOR]

In [8] we introduced the notion of multi-Koszul algebra : it is an extension of the definition of generalized Koszul algebra given by R. Berger in [1] for homogeneous algebras (see also [7] ) that can be applied to any nonnegatively graded connected algebra over a field k . The goal of this article is on the one hand to properly extend the notion of multi-Koszul algebra to the case where the base ring K is a product of copies of a field k , which a priori allows us to treat quiver algebras, and on the other hand to introduce the notion of multi-Koszul module such that it extends the usual definition of generalized Koszul module over a generalized Koszul algebra. We show eventually that multi-Koszul algebras and multi-Koszul modules are strongly linked via the notion of one-point extensions, as in the case of generalized Koszul algebras. Moreover, we describe the complete structure of right A ∞ -module on E x t A • ( M , K ) over E x t A • ( K , K ) , where M is a multi-Koszul module over a multi-Koszul algebra A , extending a result in [9] . As a corollary, we obtain that the underlying right module structure of E x t A • ( M , K ) over E x t A • ( K , K ) is generated by the component of cohomological degree zero, as in the case of generalized Koszul modules over generalized Koszul algebras. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to construct new families of irreducible Gelfand–Tsetlin modules for U q ( gl n ) . These modules have arbitrary singularity and Gelfand–Tsetlin multiplicities bounded by 2. Most previously known irreducible modules had all Gelfand–Tsetlin multiplicities bounded by 1 [16] , [17] . In particular, our method provides new families of irreducible Gelfand–Tsetlin modules for gl n . This generalizes the results of [10] and [15] . [ABSTRACT FROM AUTHOR]

We classify finite irreducible conformal modules over a class of infinite Lie conformal algebras B ( p ) of Block type, where p is a nonzero complex number. In particular, we obtain that a finite irreducible conformal module over B ( p ) may be a nontrivial extension of a finite conformal module over Vir if p = − 1 , where Vir is a Virasoro conformal subalgebra of B ( p ) . As a byproduct, we also obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal algebras b ( n ) for n ≥ 1 . [ABSTRACT FROM AUTHOR]

Noncommutative rational functions, i.e., elements of the universal skew field of fractions of a free algebra, can be defined through evaluations of noncommutative rational expressions on tuples of matrices. This interpretation extends their traditionally important role in the theory of division rings and gives rise to their applications in other areas, from free real algebraic geometry to systems and control theory. If a noncommutative rational function is regular at the origin, it can be described by a linear object, called a realization . In this article we present an extension of the realization theory that is applicable to arbitrary noncommutative rational functions and is well-adapted for studying matrix evaluations. Of special interest are the minimal realizations, which compensate the absence of a canonical form for noncommutative rational functions. The non-minimality of a realization is assessed by obstruction modules associated with it; they enable us to devise an efficient method for obtaining minimal realizations. With them we describe the stable extended domain of a noncommutative rational function and define a numerical invariant that measures its complexity. Using these results we determine concrete size bounds for rational identity testing, construct minimal symmetric realizations and prove an effective local–global principle for linear dependence of noncommutative rational functions. [ABSTRACT FROM AUTHOR]

Nonassociative algebras satisfying the polynomial identities x 1 ( x 2 x 3 ) = x 2 ( x 1 x 3 ) and ( x 1 x 2 ) x 3 = ( x 1 x 3 ) x 2 are called bicommutative. We prove the following results: (i) Finitely generated bicommutative algebras are weakly noetherian, i.e., satisfy the ascending chain condition for two-sided ideals. (ii) We give the positive solution to the Specht problem (or the finite basis problem) for varieties of bicommutative algebras over an arbitrary field of any characteristic. [ABSTRACT FROM AUTHOR]

We study p -Brauer characters of a finite group G which are restrictions of generalized characters vanishing on p -singular elements for a fixed prime p dividing the order of G . Such Brauer characters are called quasi-projective. We show that for each irreducible Brauer character φ there exists a minimal p -power, say p a ( φ ) , such that p a ( φ ) φ is quasi-projective. The exponent a ( φ ) only depends on the Cartan matrix of the block to which φ belongs. Moreover p a ( φ ) is bounded by the vertex of the module affording φ , and equality holds in case that G is p -solvable. We give some evidence for the conjecture that a ( φ ) = 0 occurs if and only if φ belongs to a block of defect 0. Finally, we study indecomposable quasi-projective Brauer characters of a block B . This set is finite and corresponds to a minimal Hilbert basis of the rational cone defined by the Cartan matrix of B . [ABSTRACT FROM AUTHOR]

Let n ≥ 3 be a positive integer. We show that there exist nilpotent groups of class 2 n in which every proper subgroup has class at most n . (Such groups are necessarily finite.) We also show that there exists a torsion-free nilpotent group of class 2 n in which every subgroup of infinite index has class at most n . These results are proved by establishing analogous result for nilpotent Lie algebras and then using the Lazard correspondence or the Mal'cev correspondence as appropriate. [ABSTRACT FROM AUTHOR]

In this paper, the authors apply a stratification of moduli spaces of complex Lie algebras to analyzing the moduli spaces of n × n matrices under scalar similarity and bilinear forms under the cogredient action. For similar matrices, we give a complete description of a stratification of the space by some very simple projective orbifolds of the form P n / G , where G is a subgroup of the symmetric group Σ n + 1 acting on P n by permuting the projective coordinates. For bilinear forms, we give a similar stratification up to dimension 4. [ABSTRACT FROM AUTHOR]

A ring R is said to be left pure-hereditary (resp. RD -hereditary) if every left ideal of R is pure-projective (resp. RD -projective). In this paper, some properties and examples of these rings, which are nontrivial generalizations of hereditary rings, are given. For instance, we show that if R is a left RD -hereditary left nonsingular ring, then R is left Noetherian if and only if u . dim ( R R ) < ∞ . Also, we show that a ring R is quasi-Frobenius if and only if R is a left FGF, left coherent right pure-injective ring. A ring R is said to be left FP-hereditary if every left ideal of R is FP-projective. It is shown that if R is a left CF ring, then R is left Noetherian if and only if R is left pure-hereditary, if and only if R is left FP-hereditary, if and only if R is left coherent. It is shown that every left self-injective left FP-hereditary ring is semiperfect. Finally, it is shown that a ring R is left FP-hereditary (resp. left coherent) if and only if every submodule (resp. finitely generated submodule) of a projective left R -module is FP-projective, if and only if every pure factor module of an injective left R -module is injective (resp. FP-injective), if and only if for each FP-injective left R -module U , E ( U ) / U is injective (resp. FP-injective). [ABSTRACT FROM AUTHOR]

We define two different simplicial complexes, the common divisor simplicial complex and the prime divisor simplicial complex, from a set of integers, and explore their similarities. We will define a map between the two simplicial complexes, and use this map to show that for any set of integers, the fundamental groups of the resulting simplicial complexes are isomorphic. [ABSTRACT FROM AUTHOR]

Let g be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We provide necessary and also some sufficient conditions in order for its Poisson center and semi-center to be polynomial algebras over k . This occurs for instance if g is quadratic of index 2 with [ g , g ] ≠ g and also if g is nilpotent of index at most 2. The converse holds for filiform Lie algebras of type L n , Q n , R n and W n . We show how Dixmier's fourth problem for an algebraic Lie algebra g can be reduced to that of its canonical truncation g Λ . Moreover, Dixmier's statement holds for all Lie algebras of dimension at most eight. The nonsolvable ones among them possess a polynomial Poisson center and semi-center. [ABSTRACT FROM AUTHOR]

The algebras U T n of the n × n upper triangular matrices over a field K are of significant importance in the theory of algebras with polynomial identities. Group gradings on algebras appear in various areas and provide an indispensable tool in the study of the algebraic and combinatorial properties of the algebras in question. We classify the group gradings on the Lie algebra U T n ( − ) . It was proved by Valenti and Zaicev in 2007 that every group grading on the associative algebra U T n is isomorphic to an elementary grading. The elementary gradings on U T n are also well understood, see [6] . It follows from our results that there are nonelementary gradings on U T n ( − ) . Thus the gradings on the Lie algebra U T n ( − ) are much more intricate than those in the associative case. [ABSTRACT FROM AUTHOR]

We give a non-constructive proof that fusion rings attached to a simple complex Lie algebra of rank 2 are complete intersections. [ABSTRACT FROM AUTHOR]

Fermat ideals define planar point configurations that are closely related to the intersection locus of the members of a specific pencil of curves. These ideals have gained recent popularity as counterexamples to some proposed containments between symbolic and ordinary powers [6] . We give a systematic treatment of the family of Fermat ideals, describing explicitly the minimal generators and the minimal free resolutions of all their ordinary powers as well as many symbolic powers. We use these to study the ordinary and the symbolic Rees algebra of Fermat ideals. Specifically, we show that the symbolic Rees algebras of Fermat ideals are Noetherian. Along the way, we give formulas for the Castelnuovo–Mumford regularity of powers of Fermat ideals and we determine their reduction ideals. [ABSTRACT FROM AUTHOR]

In [5,6] Kraśkiewicz and Pragacz introduced representations of the upper-triangular Lie algebra b whose characters are Schubert polynomials. In [12] the author studied the properties of Kraśkiewicz–Pragacz modules using the theory of highest weight categories. From the results there, in particular we obtain a certain highest weight category whose standard modules are KP modules. In this paper we show that this highest weight category is self Ringel-dual. This leads to an interesting symmetry relation on Ext groups between KP modules. We also show that the tensor product operation on b -modules is compatible with Ringel duality functor. [ABSTRACT FROM AUTHOR]

We introduce the notion of operadic torsors and operadic quasi-torsors. We show that if an operadic (quasi-)torsor between two operads exists, then these operads are (quasi-)isomorphic. As an application we present the (arguably) shortest known proof of the Deligne conjecture. [ABSTRACT FROM AUTHOR]

We classify Alt n and its perfect covers for n ≥ 6 up to conjugacy as subgroups of the exceptional complex Lie groups (with some exceptions where we give lower bounds and discuss the possibilities of Lie primitive subgroups). [ABSTRACT FROM AUTHOR]

A general fiber of the Gauss map of a projective variety in P N coincides with a linear subvariety of P N in characteristic zero. In positive characteristic, S. Fukasawa showed that a general fiber of the Gauss map can be a non-linear variety. In this paper, we show that each irreducible component of such a possibly non-linear fiber of the Gauss map is contracted to one point by the degeneracy map, and is contained in a linear subvariety corresponding to the kernel of the differential of the Gauss map. We also show the inseparability of Gauss maps of strange varieties not being cones. [ABSTRACT FROM AUTHOR]

In this study, we investigate a new class of Lie algebras, i.e., tame locally extended affine Lie algebras of nullity 1, which are an infinite-rank analog of affine Lie algebras. This type of algebra is called a locally affine Lie algebra. A certain ideal of a locally affine Lie algebra, called a core, is a universal extension of a local version of a loop algebra, which is called a locally loop algebra. We classify locally loop algebras and locally affine Lie algebras. [ABSTRACT FROM AUTHOR]