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In this paper we present a characterization for the defect of simple algebraic extensions of valued fields. This characterization generalizes the known result for the henselian case, namely that the defect is the product of the relative degrees of limit augmentations. The main tool used here is the graded algebra associated to a valuation on a polynomial ring. Let K h be a henselization of a valued field K. Another relevant result proved in this paper is that for every valuation μ h on K h [ x ] , with restriction μ on K [ x ] , the corresponding map G μ ↪ G μ h of graded algebras is an isomorphism. [ABSTRACT FROM AUTHOR]

Let A be a k -algebra over a commutative ring k. By the renowned Tannaka-Kreĭn reconstruction, liftings of the monoidal structure from M k to M A correspond to bialgebra structures on A and liftings of the closed monoidal structure correspond to Hopf algebra structures on A. In this paper, we determine conditions on A that correspond to liftings of the closed structure alone, i.e. without considering the monoidal one, which lead to the notion of what we call a gabi-algebra. First, we tackle the question from the general perspective of monads, then we focus on the set-theoretic and the linear setting. Our main and most surprising result is that a normal gabi-algebra , that is an algebra A whose category of modules is (associative and unital normal) closed with closed forgetful functor to M k , is automatically a Hopf algebra (thus justifying our title). [ABSTRACT FROM AUTHOR]

We resolve the long-standing problem of constructing the action of the operad of framed (stable) genus-0 curves on Hamiltonian Floer theory; this operad is equivalent to the framed E 2 operad. We formulate the construction in the following general context: we associate to each compact subset of a closed symplectic manifold a new chain-level model for symplectic cohomology with support, which we show carries an action of a model for the chains on the moduli space of framed genus 0 curves. This construction turns out to be strictly functorial with respect to inclusions of subsets, and the action of the symplectomorphism group. In the general context, we appeal to virtual fundamental chain methods to construct the operations over fields of characteristic 0, and we give a separate account, over arbitrary rings, in the special settings where Floer's classical transversality approach can be applied. We perform all constructions over the Novikov ring, so that the algebraic structures we produce are compatible with the quantitative information that is contained in Floer theory. Over fields of characteristic 0, our construction can be combined with results in the theory of operads to produce explicit operations encoding the structure of a homotopy BV algebra. In an appendix, we explain how to extend the results of the paper from the class of closed symplectic manifolds to geometrically bounded ones. [ABSTRACT FROM AUTHOR]

The aim of this paper is to unify classification theories of torsion classes of finite dimensional algebras and commutative Noetherian rings. For a commutative Noetherian ring R and a module-finite R -algebra Λ, we study the set tors Λ (respectively, torf Λ) of torsion (respectively, torsionfree) classes of the category of finitely generated Λ-modules. We construct a bijection from torf Λ to ∏ p torf (κ (p) ⊗ R Λ) , and an embedding Φ t from tors Λ to T R (Λ) : = ∏ p tors (κ (p) ⊗ R Λ) , where p runs over all prime ideals of R. When Λ = R , these give classifications of torsionfree classes, torsion classes and Serre subcategories of mod R due to Takahashi, Stanley-Wang and Gabriel. To give a description of Im Φ t , we introduce the notion of compatible elements in T R (Λ) , and prove that all elements in Im Φ t are compatible. We give a sufficient condition on (R , Λ) such that all compatible elements belong to Im Φ t (we call (R , Λ) compatible in this case). For example, if R is semi-local and dim ⁡ R ≤ 1 , then (R , Λ) is compatible. We also give a sufficient condition in terms of silting Λ-modules. As an application, for a Dynkin quiver Q , (R , R Q) is compatible and we have a poset isomorphism tors R Q ≃ Hom poset (Spec R , C Q) for the Cambrian lattice C Q of Q. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new approach that allows to identify the Gelfand spectrum of weighted Fourier algebras as a subset of an abstract complexification of the corresponding group for a wide class of groups and weights. This generalizes recent related results of Ghandehari-Lee-Ludwig-Spronk-Turowska [11] about the spectrum of Beurling-Fourier algebras on some Lie groups. In the case of discrete groups we show that the spectrum of Beurling-Fourier algebra is homeomorphic to the group itself. [ABSTRACT FROM AUTHOR]

Cluster algebra structures for Grassmannians and their (open) positroid strata are controlled by a Postnikov diagram D or, equivalently, a dimer model on the disc, as encoded by either a bipartite graph or the dual quiver (with faces). The associated dimer algebra A , determined directly by the quiver with a certain potential, can also be realised as the endomorphism algebra of a cluster-tilting object in an associated Frobenius cluster category. In this paper, we introduce a class of A -modules corresponding to perfect matchings of the dimer model of D and show that, when D is connected, the indecomposable projective A -modules are in this class. Surprisingly, this allows us to deduce that the cluster category associated to D embeds into the cluster category for the appropriate Grassmannian. We show that the indecomposable projectives correspond to certain matchings which have appeared previously in work of Muller–Speyer. This allows us to identify the cluster-tilting object associated to D , by showing that it is determined by one of the standard labelling rules constructing a cluster of Plücker coordinates from D. By computing a projective resolution of every perfect matching module, we show that Marsh–Scott's formula for twisted Plücker coordinates, expressed as a dimer partition function, is a special case of the general cluster character formula, and thus observe that the Marsh–Scott twist can be categorified by a particular syzygy operation in the Grassmannian cluster category. [ABSTRACT FROM AUTHOR]

In this paper we develop a new groupoid-based structure theory for the class of regular ⁎-semigroups. This class occupies something of a 'sweet spot' between the important classes of inverse and regular semigroups, and contains many natural examples. Some of the most significant families include the partition, Brauer and Temperley-Lieb monoids, among other diagram monoids. Our main result is that the category of regular ⁎-semigroups is isomorphic to the category of so-called 'chained projection groupoids'. Such a groupoid is in fact a triple (P , G , ε) , where: • P is a projection algebra (in the sense of Imaoka and Jones), • G is an ordered groupoid with object set P , and • ε : C → G is a special functor, where C is a certain natural 'chain groupoid' constructed from P. Roughly speaking: the groupoid G = G (S) remembers only the 'easy' products in a regular ⁎-semigroup S ; the projection algebra P = P (S) remembers only the 'conjugation action' of the projections of S ; and the functor ε = ε (S) tells us how G and P 'fit together' in order to recover the entire structure of S. In this way, we obtain the first completely general structure theorem for regular ⁎-semigroups. As a consequence of our main result, we give a new proof of the celebrated Ehresmann–Schein–Nambooripad Theorem, which establishes an isomorphism between the categories of inverse semigroups and inductive groupoids. Other applications will be given in future works. We consider several examples along the way, and pose a number of problems that we believe are worthy of further attention. [ABSTRACT FROM AUTHOR]

We relate the group structure of van der Kallen on orbit sets of unimodular rows ([26]) with values in a smooth algebra A over a field k with the motivic cohomotopy groups of X = Spec A with coefficients in A n ∖ 0 in the sense of [5]. In the last section, we compare the motivic cohomotopy theory studied in this paper and defined by A n + 1 ∖ 0 or, equivalently, by an A 1 -weakly equivalent quadric Q 2 n + 1 to that considered in [5] , defined by a quadric Q 2 n , by means of explicit morphisms Q 2 n + 1 → Q 2 n , Q 2 n × G m → Q 2 n + 1 of quadrics. [ABSTRACT FROM AUTHOR]

It is well-known that the "pre-2-category" C a t dg coh (k) of small dg categories over a field k , with 1-morphisms defined as dg functors, and 2-morphisms defined as the complexes of coherent natural transformations, fails to be a strict 2-category. The question "What do dg categories form?", raised by V. Drinfeld in [14] , is interpreted in this context as a question of finding a weak 2-category structure on C a t dg coh (k). In [32] , D. Tamarkin proposed an answer to this question, by constructing a contractible 2-operad in the sense of M. Batanin [3] , acting on C a t dg coh (k). In this paper, we construct another contractible 2-operad, acting on C a t dg coh (k). Our main tool is the twisted tensor product of small dg categories, introduced in [25]. We establish a one-side associativity for the twisted tensor product, making (C a t dg coh (k) , ⊗ ∼) a skew monoidal category in the sense of [30] , and construct a twisted composition C o h dg (D , E) ⊗ ∼ C o h dg (C , D) → C o h dg (C , E) , and prove some compatibility between these two structures. Taken together, the two structures give rise to a 2-operad O , acting on C a t dg coh (k). Its contractibility is a consequence of a general result of [25]. [ABSTRACT FROM AUTHOR]

We introduce a formalism for derived moduli functors on differential graded associative algebras, which leads to non-commutative enhancements of derived moduli stacks and naturally gives rise to structures such as Hall algebras. Descent arguments are not available in the non-commutative context, so we establish new methods for constructing various kinds of atlases. The formalism permits the development of the theory of shifted bi-symplectic and shifted double Poisson structures in the companion paper [27]. [ABSTRACT FROM AUTHOR]

Let G be an abelian group acting on a smooth algebraic variety X. We investigate the product structure and the bigrading on the cohomology of polyvector fields on the orbifold [ X / G ] , as introduced by Căldăraru and Huang. In this paper, we provide many new examples given by quotients of Fermat hypersurfaces, where the product is shown to be associative. This is expected due to the conjectural isomorphism at the level of algebras between the cohomology of polyvector fields and the Hochschild cohomology of orbifolds. We prove this conjecture for the Calabi-Yau Fermat hypersurface orbifold. We also show that for Calabi-Yau orbifolds, the multiplicative bigrading on the cohomology of polyvector fields agrees with what is expected in homological mirror symmetry. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is a mathematical construction of non-perturbative deformations of two-dimensional conformal field theories. We introduce the notion of a full vertex algebra which formulates a compact two-dimensional conformal field theory. Then, we construct a deformation family of full vertex algebras which serves as a current-current deformation of conformal field theory in physics. The parameter space of the deformations is expressed as a quotient of an orthogonal Grassmannian. This parameter space is a part of the CFT moduli space expected in physics. We demonstrate that nontrivial quotients of Grassmannians are obtained from holomorphic vertex operator algebras of central charge 24. [ABSTRACT FROM AUTHOR]

Cluster categories were introduced in 2006 by Buan-Marsh-Reineke-Reiten-Todorov in order to categorify acyclic cluster algebras without coefficients. Their construction was generalized by Amiot (2009) [3] and Plamondon (2011) [50] to arbitrary cluster algebras associated with quivers. A higher dimensional generalization is due to Guo (2011) [17]. Cluster algebras with coefficients are important since they appear in nature as coordinate algebras of varieties like Grassmannians, double Bruhat cells, unipotent cells, etc. The work of Geiss-Leclerc-Schröer often yields Frobenius exact categories which allow us to categorify such cluster algebras. In this paper, we generalize the construction of (higher) cluster categories by Claire Amiot and by Lingyan Guo to the relative context. We prove the existence of an n -cluster tilting object in a Frobenius extriangulated category, namely the Higgs category (generalizing the Frobenius categories of Geiss-Leclerc-Schröer), which is stably n -Calabi–Yau and Hom-finite, arising from a left (n + 1) -Calabi–Yau morphism. Our results apply in particular to relative Ginzburg dg algebras coming from ice quivers with potential and higher Auslander algebras associated to n -representation-finite algebras. [ABSTRACT FROM AUTHOR]

Let L be the noncrossing partition lattice associated to a finite Coxeter group W. In this paper we construct explicit bases for the top homology groups of intervals and rank-selected subposets of L. We define a multiplicative structure on the Whitney homology of L in terms of the basis, and the resulting algebra has similarities to the Orlik-Solomon algebra. As an application, we obtain four chain complexes which compute the integral homology of the Milnor fibre of the reflection arrangement of W , the Milnor fibre of the discriminant of W , the hyperplane complement of W and the Artin group of type W , respectively. We also tabulate some computational results on the integral homology of the Milnor fibres. [ABSTRACT FROM AUTHOR]

In this paper we establish Koszul duality between dg categories and a class of curved coalgebras, generalizing the corresponding result for dg algebras and conilpotent curved coalgebras. We show that the normalized chain complex functor transforms the Quillen equivalence between quasicategories and simplicial categories into this Koszul duality. This allows us to give a conceptual interpretation of the dg nerve of a dg category and its adjoint. As an application, we prove that the category of representations of a quasicategory K is equivalent to the coderived category of comodules over C ⁎ (K) , the chain coalgebra of K. A corollary of this is a characterization of the category of constructible dg sheaves on a stratified space as the coderived category of a certain dg coalgebra. [ABSTRACT FROM AUTHOR]

Trialities of W -algebras are isomorphisms between the affine cosets of three different W -(super)algebras, and were first conjectured in the physics literature by Gaiotto and Rapčák. In this paper we prove trialities among eight families of W -(super)algebras of types B , C , and D. The key idea is to identify the affine cosets of these algebras with one-parameter quotients of the universal two-parameter even spin W ∞ -algebra which was recently constructed by Kanade and the second author. Our result is a vast generalization of both Feigin-Frenkel duality in types B , C , and D , and the coset realization of principal W -algebras of type D due to Arakawa and us. It also provides a new coset realization of principal W -algebras of types B and C. As an application, we prove the rationality of the affine vertex superalgebra L k (o s p 1 | 2 n) , the minimal W -algebra W k − 1 / 2 (s p 2 n + 2 , f min) , and the coset Com (L k (s p 2 m) , L k (s p 2 n)) , for all integers k , n , m ≥ 1 with m < n. We also prove the rationality of some families of principal W -superalgebras of o s p 1 | 2 n and o s p 2 | 2 n , and subregular W -algebras of s o 2 n + 1. [ABSTRACT FROM AUTHOR]

Given a link in the thickened annulus, its annular Khovanov homology carries an action of the Lie algebra sl 2 , which is natural with respect to annular link cobordisms. We consider the problem of lifting this action to the stable homotopy refinement of the annular homology. As part of this program, the actions of the standard generators of sl 2 are lifted to maps of spectra. In particular, it follows that the sl 2 action on homology commutes with the action of the Steenrod algebra. The main new technical ingredients developed in this paper, which may be of independent interest, concern certain types of cancellations in the cube of resolutions and the resulting more intricate structure of the moduli spaces in the framed flow category. [ABSTRACT FROM AUTHOR]

In this paper, we study a compact analogue of Schur-Weyl duality for real orthogonal and symplectic groups. We analyse the functors F μ , k defined in [12] , and introduce a new algebra B k θ. This algebra's finite dimensional modules are naturally a co-domain for the functors F μ , k. With this new insight, we prove that principle series modules map to principles series modules of graded Hecke algebras and these functors preserve Langlands quotients and unitarity of Harish-Chandra modules. [ABSTRACT FROM AUTHOR]

The question of which separable C*-algebras have abelian central sequence algebras was raised and studied by Phillips ([17]) and Ando-Kirchberg ([2]). In this paper we give a complete answer to their question: Moreover, we introduce a higher-dimensional analogue of Fell's condition and show that it completely characterizes subhomogeneity of central sequence algebras. In contrast, we show that any non-trivial extension by compact operators has not only non-abelian but not even residually type I central sequence algebra. In particular its central sequence algebra is not type I and not residually finite-dimensional (RFD). Our techniques extensively use properties of nilpotent elements in C*-algebras. [ABSTRACT FROM AUTHOR]

In this paper, we study equivalences between the categories of quasi–coherent sheaves on non–commutative noetherian schemes. In particular, we give a new proof of Căldăraru's conjecture about Morita equivalences of Azumaya algebras on noetherian schemes. Moreover, we derive necessary and sufficient condition for two reduced non–commutative curves to be Morita equivalent. [ABSTRACT FROM AUTHOR]

Generalized affine Grassmannian slices provide geometric realizations for weight spaces of representations of semisimple Lie algebras. They are also Coulomb branches, symplectic dual to Nakajima quiver varieties. In this paper, we prove that neighboring generalized affine Grassmannian slices are related by Hamiltonian reduction by the action of the additive group. We also prove a weaker version of the same result for their quantizations, algebras known as truncated shifted Yangians. [ABSTRACT FROM AUTHOR]

For a length abelian category, we show that all torsion-free classes can be classified by using only the information on bricks, including non functorially-finite ones. The idea is to consider the set of simple objects in a torsion-free class, which has the following property: it is a set of bricks where every non-zero map between them is an injection. We call such a set a monobrick. In this paper, we provide a uniform method to study torsion-free classes and wide subcategories via monobricks. We show that monobricks are in bijection with left Schur subcategories, which contains all subcategories closed under extensions, kernels and images, thus unifies torsion-free classes and wide subcategories. Then we show that torsion-free classes bijectively correspond to cofinally closed monobricks. Using monobricks, we deduce several known results on torsion(-free) classes and wide subcategories (e.g. finiteness result and bijections) in length abelian categories, without using τ -tilting theory. For Nakayama algebras, left Schur subcategories are the same as subcategories closed under extensions, kernels and images, and we show that its number is related to the large Schröder number. [ABSTRACT FROM AUTHOR]

A loop is a rather general algebraic structure that has an identity element and division, but is not necessarily associative. Smooth loops are a direct generalization of Lie groups. A key example of a non-Lie smooth loop is the loop of unit octonions. In this paper, we study properties of smooth loops and their associated tangent algebras, including a loop analog of the Maurer-Cartan equation. Then, given a manifold, we introduce a loop bundle as an associated bundle to a particular principal bundle. Given a connection on the principal bundle, we define the torsion of a loop bundle structure and show how it relates to the curvature, and also develop aspects of a non-associative gauge theory. Throughout, we see how some of the known properties of G 2 -structures can be seen from this more general setting. [ABSTRACT FROM AUTHOR]

The ı quiver algebras were introduced recently by the authors to provide a Hall algebra realization of universal ı quantum groups, which is a generalization of Bridgeland's Hall algebra construction for (Drinfeld doubles of) quantum groups; here an ı quantum group and a corresponding Drinfeld-Jimbo quantum group form a quantum symmetric pair. In this paper, the Dynkin ı quiver algebras are shown to arise as new examples of singular Nakajima-Keller-Scherotzke categories. Then we provide a geometric construction of the universal ı quantum groups and their "dual canonical bases" with positivity, via the quantum Grothendieck rings of Nakajima-Keller-Scherotzke quiver varieties, generalizing Qin's geometric realization of quantum groups of type ADE. [ABSTRACT FROM AUTHOR]

Let G be connected reductive algebraic group defined over an algebraically closed field of characteristic p > 0 and suppose that p is a good prime for the root system of G , the derived subgroup of G is simply connected and the Lie algebra g = Lie (G) admits a non-degenerate (Ad G) -invariant symmetric bilinear form. Given a linear function χ on g we denote by U χ (g) the reduced enveloping algebra of g associated with χ. By the Kac–Weisfeiler conjecture (now a theorem), any irreducible U χ (g) -module has dimension divisible by p d (χ) where 2 d (χ) is the dimension of the coadjoint G -orbit containing χ. In this paper we give a positive answer to the natural question raised in the 1990s by Kac, Humphreys and the first-named author and show that any algebra U χ (g) admits a module of dimension p d (χ). [ABSTRACT FROM AUTHOR]

The paper lays the foundation for the study of unimodular rows using Spin groups. We show that elementary orbits of unimodular rows (of any length n ≥ 3) are equivalent to elementary Spin orbits on the unit sphere. (This bijection is proved over all commutative rings.) In the special case n = 3 , this leads to an interpretation of the Vaserstein symbol using Spin groups. In addition, we introduce a new composition law that operates on certain subspaces of the underlying quadratic space (using the multiplication in composition algebras). In particular, the special case of split-quaternions leads to the composition of unimodular rows (discovered by L. Vaserstein and later generalized by W. van der Kallen). Strikingly, with this approach, we now see the possibility of new orbit structures not only for unimodular rows (using octonion multiplication) but also for more general quadratic spaces. [ABSTRACT FROM AUTHOR]

We prove a Chevalley formula for anti-dominant weights in the torus-equivariant K -group of semi-infinite flag manifolds, which is described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or equivalently, quantum Lakshmibai-Seshadri paths); in contrast to the Chevalley formula for dominant weights in our previous paper [17] , the formula for anti-dominant weights has a significant finiteness property. Based on geometric results established in [17] , our proof is representation-theoretic, and the Chevalley formula for anti-dominant weights follows from a certain identity for the graded characters of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra; in the proof of this identity, we make use of the (combinatorial) standard monomial theory for semi-infinite Lakshmibai-Seshadri paths, and also a string property of Demazure-like subsets of the set of semi-infinite Lakshmibai-Seshadri paths of a fixed shape, which gives an explicit realization of the crystal basis of a level-zero extremal weight module. [ABSTRACT FROM AUTHOR]