Search

Let G be a topological group. Then the based loopspace OG of G is an algebra over the cacti operad, while the double loopspace O2BG of the classifying space of G is an algebra over the framed little discs operad. This paper shows that these two algebras are equivalent, in the sense that they are weakly equivalent E-algebras, where E is an operad weakly equivalent to both framed little discs and cacti. We recover the equivalence between cacti and framed little discs, and Menichi's isomorphism between the BV-algebras H*(OG) and H*(O2BG). [ABSTRACT FROM AUTHOR]

In this paper we study the stack τg of smooth triple covers of a conic; when g ≥ 5 this stack is embedded Mg as the locus of trigonal curves. We show that τ is a quotient [Ug/Γg/], where Γg/ is a certain algebraic group and Ug/ is an open subscheme of a Γg/-equivariant vector bundle over an open subscheme of a representation of Γg/. Using this, we compute the integral Picard group of τg when g/ > 1. τhe main tools are a result of Miranda that describes a flat finite triple cover of a scheme S as given by a locally free sheaf E of rank two on S, with a section of Sym3E ⊗detEv, and a new description of the stack of globally generated locally free sheaves of fixed rank and degree on a projective line as a quotient stack. [ABSTRACT FROM AUTHOR]

The modular variety of nonsingular and complete hyperelliptic curves with level-two structure of genus $ 3$ It has 36 irreducible (isomorphic) components. One of the purposes of this paper will be to describe the equations of one of these components. Two further models use the fact that hyperelliptic curves of genus three can be obtained as coverings of a projective line with $ 8$, uses the semistable degenerated point configurations in $ (P^1)^8$ $ Y=\overline{\mathcal{B}/\Gamma[1-{\textrm i}]}.$ We use the standard notation $ \bar M_{0,8}$}[rr]& &X\;.} \end{displaymath} --> $\displaystyle \xymatrix{ &\bar M_{0,8}\ar[dl]\ar[dr]&\\ Y\ar@{-->}[rr]& &X\;.}$ The horizontal arrow is only birational but not everywhere regular. In this paper we find another realization of this triangle which uses the fact that there are graded algebras (closely related to algebras of modular forms) $ A,B$ $ X=\mathop{\rm proj}\nolimits(A), Y=\mathop{\rm proj}\nolimits(B).$ [ABSTRACT FROM AUTHOR]

This paper concerns the homotopy type of hyperplane arrangements associated to infinite Coxeter groups acting as reflection groups on $\mathbb C^n$. A long-standing conjecture states that the complement of such an arrangement should be aspherical. Some partial results on this conjecture were previously obtained by the author and M. Davis. In this paper, we extend those results to another class of Coxeter groups. The key technical result is that the spherical Deligne complex for the 4-strand braid group is CAT(1). [ABSTRACT FROM AUTHOR]

Test ideals are an important concept in tight closure theory and their behavior via flat base change can be very difficult to understand. Our paper presents results regarding this behavior under flat maps with reasonably nice (but far from smooth) fibers. This involves analyzing, in depth, a special type of ideal of test elements, called the CS test ideal. Besides providing new results, the paper also contains extensions of a theorem by G. Lyubeznik and K. E. Smith on the completely stable test ideal and of theorems by F. Enescu and, independently, M. Hashimoto on the behavior of $F$-rationality under flat base change. [ABSTRACT FROM AUTHOR]

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring. [ABSTRACT FROM AUTHOR]

An $SL_n$-character of a group $G$ is the trace of an $SL_n$-represen\-ta\-tion of $G.$ We show that all algebraic relations between $SL_n$-characters of $G$ can be visualized as relations between graphs (resembling Feynman diagrams) in any topological space $X,$ with $\pi_1(X)=G.$ We also show that all such relations are implied by a single local relation between graphs. In this way, we provide a topological approach to the study of $SL_n$-representations of groups. \par The motivation for this paper was our work with J. Przytycki on invariants of links in 3-manifolds which are based on the Kauffman bracket skein relation. These invariants lead to a notion of a skein module of $M$ which, by a theorem of Bullock, Przytycki, and the author, is a deformation of the $SL_2$-character variety of $\pi_1(M).$ This paper provides a generalization of this result to all $SL_n$-character varieties. [ABSTRACT FROM AUTHOR]

To date, integral bases for the centre of the Iwahori-Hecke algebra of a finite Coxeter group have relied on character theoretical results and the isomorphism between the Iwahori-Hecke algebra when semisimple and the group algebra of the finite Coxeter group. In this paper, we generalize the minimal basis approach of an earlier paper, to provide a way of describing and calculating elements of the minimal basis for the centre of an Iwahori-Hecke algebra which is entirely combinatorial in nature, and independent of both the above mentioned theories. This opens the door to further generalization of the minimal basis approach to other cases. In particular, we show that generalizing it to centralizers of parabolic subalgebras requires only certain properties in the Coxeter group. We show here that these properties hold for groups of type $A$ and $B$, giving us the minimal basis theory for centralizers of any parabolic subalgebra in these types of Iwahori-Hecke algebra. [ABSTRACT FROM AUTHOR]

We consider the class of compact Hermitian manifolds whose Chern connection is Ambrose-Singer, namely, it has parallel torsion and curvature. We prove structure theorems for such manifolds. [ABSTRACT FROM AUTHOR]

We generalize Renault's notion of measurewise amenability to actions of second countable, Hausdorff, étale groupoids on separable C^*-algebras and show that measurewise amenability characterizes nuclearity of the crossed product whenever the C^*-algebra acted on is nuclear. In the more general context of Fell bundles over second countable, Hausdorff, étale groupoids, we introduce a version of Exel's approximation property. We prove that the approximation property implies nuclearity of the cross-sectional algebra whenever the unit bundle is nuclear. For Fell bundles associated to groupoid actions, we show that the approximation property implies measurewise amenability of the underlying action. [ABSTRACT FROM AUTHOR]

In this paper we give some topological characterizations of bounded Baire-1 functions using some ranks. Kechris and Louveau classified the Baire-1 functions to the subclasses $\mathbb{B}^\xi_1(K)$ for every $\xi<\omega_1$ (where $K$ is a compact metric space). The first basic result of this paper is that for $\xi<\omega$, $f\in \mathbb{B}^{\xi+1}_1(K)$ iff there exists a sequence $(f_n)$ of differences of bounded semicontinuous functions on $K$ with $f_n\to f$ pointwise and $\gamma((f_n))\le \omega^\xi$ (where ``$\gamma$'' denotes the convergence rank). This extends the work of Kechris and Louveau who obtained this result for $\xi=1$. We also show that the result fails for $\xi\ge \omega$. The second basic result of the paper involves the introduction of a new ordinal-rank on sequences $(f_n)$, called the $\delta$-rank, which is smaller than the convergence rank $\gamma$. This result yields the following characterization of $\mathbb{B}^\xi_1(K): f\in \mathbb{B}^\xi_1(K)$ iff there exists a sequence $(f_n)$ of continuous functions with $f_n\to f$ pointwise and $\delta((f_n))\le \omega^{\xi-1}$ if $1\le \xi<\omega$, resp. $\delta((f_n))\le \omega^\xi$ if $\xi\ge \omega$. [ABSTRACT FROM AUTHOR]

This paper completes the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one. In particular, we study the elliptic algebras $A(+)$, $A(-)$, and $A({\mathbf{a}})$, where ${\mathbf{a}}\in \BbP^{2}$, which were first defined in an earlier paper. We omit a set $S\subset \BbP^2$ consisting of 11 specified points where the algebras $A({\mathbf{a}})$ become too degenerate to be regular. \begin{thm} Let $A$ represent $A(+)$, $A(-)$ or $A({\mathbf{a}})$, where ${\mathbf{a}} \in \BbP^2\setminus S$. Then $A$ is an Artin-Schelter regular algebra of global dimension three. Moreover, $A$ is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables. \end{thm} This, combined with our earlier results, completes the classification. [ABSTRACT FROM AUTHOR]

{ An (ordinary) $P$-partition is an order-preserving map from a partially ordered set to a chain, with special rules specifying where equal values may occur. Examples include number-theoretic partitions (ordered and unordered, strict or unrestricted), plane partitions, and the semistandard tableaux associated with Schur's $S$-functions. In this paper, we introduce and develop a theory of enriched $P$-partitions; like ordinary $P$-partitions, these are order-preserving maps from posets to chains, but with different rules governing the occurrence of equal values. The principal examples of enriched $P$-partitions given here are the tableaux associated with Schur's $Q$-functions. In a sequel to this paper, further applications related to commutation monoids and reduced words in Coxeter groups will be presented. } [ABSTRACT FROM AUTHOR]

Let $G$ be a reductive $ \mathfrak{p}$-adic group. In his paper, ``The Plancherel Formula for Reductive $\mathfrak{p}$-adic Groups", Harish-Chandra summarized the theory underlying the Plancherel formula for $G$ and sketched a proof of the Plancherel theorem for $G$. One step in the proof, stated as Theorem 11 in Harish-Chandra's paper, has seemed an elusively difficult step for the reader to supply. In this paper we prove the Plancherel theorem, essentially, by proving a special case of Theorem 11. We close by deriving a version of Theorem 11 from the Plancherel theorem. [ABSTRACT FROM AUTHOR]

We obtain explicit formulas for the 1-point functions of all states in the symmetrized Heisenberg algebra M^+ and symmetrized lattice vertex operator algebras V_L^+. For this we employ a new \mathbf {Z}_2-twisted variant of so-called Zhu recursion. [ABSTRACT FROM AUTHOR]

The Frobenius-Perron theory of an endofunctor of a \Bbbk-linear category (recently introduced in Chen et al. [Algebra Number Theory 13 (2019), pp. 2005–2055]) provides new invariants for abelian and triangulated categories. Here we study Frobenius-Perron type invariants for derived categories of commutative and noncommutative projective schemes. In particular, we calculate the Frobenius-Perron dimension for domestic and tubular weighted projective lines, define Frobenius-Perron generalizations of Calabi-Yau and Kodaira dimensions, and provide examples. We apply this theory to the derived categories associated to certain Artin-Schelter regular and finite-dimensional algebras. [ABSTRACT FROM AUTHOR]

We show that the number of non-isotopic commutative semifields of odd order p^{n} is exponential in n when n = 4t and t is not a power of 2. We introduce a new family of commutative semifields and a method for proving isotopy results on commutative semifields that we use to deduce the aforementioned bound. The previous best bound on the number of non-isotopic commutative semifields of odd order was quadratic in n and given by Zhou and Pott [Adv. Math. 234 (2013), pp. 43–60]. Similar bounds in the case of even order were given in Kantor [J. Algebra 270 (2003), pp. 96–114] and Kantor and Williams [Trans. Amer. Math. Soc. 356 (2004), pp. 895–938]. [ABSTRACT FROM AUTHOR]

We study the category of representations of \mathfrak {sl}_{m+2n} in positive characteristic, with p-character given by a nilpotent with Jordan type (m+n,n). Recent work of Bezrukavnikov-Mirkovic [Ann. of Math. (2) 178 (2013), pp. 835–919] implies that this representation category is equivalent to \mathcal {D}_{m,n}^0, the heart of the exotic t-structure on the derived category of coherent sheaves on a Springer fibre for that nilpotent. Using work of Cautis and Kamnitzer [Duke Math. J. 142 (2008), pp. 511–588], we construct functors indexed by affine tangles, between these categories \mathcal {D}_{m,n} (i.e. for different values of n). This allows us to describe the irreducible objects in \mathcal {D}_{m,n}^0 and enumerate them by crossingless (m,m+2n) matchings. We compute the \mathrm {Ext} spaces between the irreducible objects, and conjecture that the resulting Ext algebra is an annular variant of Khovanov's arc algebra. In subsequent work, we use these results to give combinatorial dimension formulae for the irreducible representations. These results may be viewed as a positive characteristic analogue of results about two-block parabolic category \mathcal {O} due to Lascoux-Schutzenberger [Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 249–266], Bernstein-Frenkel-Khovanov [Selecta Math. (N.S.) 5 (1999), pp. 199–241], Brundan-Stroppel [Represent. Theory 15 (2011), pp. 170–243], et al. [ABSTRACT FROM AUTHOR]

The \imathHall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of 1-periodic complexes of coherent sheaves on the projective line. This \imathHall algebra is shown to realize the universal q-Onsager algebra (i.e., \imathquantum group of split affine A_1 type) in its Drinfeld type presentation. The \imathHall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these two \imathHall algebras, explaining the isomorphism of the q-Onsager algebra under the two presentations. [ABSTRACT FROM AUTHOR]

Using the theory of Macdonald [ Symmetric functions and Hall polynomials , The Clarendon Press, Oxford University Press, New York, 1995], Gordon [Bull. London Math. Soc. 35 (2003), pp. 321–336] showed that the graded characters of the simple modules for the restricted rational Cherednik algebra by Etingof and Ginzburg [Invent. Math. 147 (2002), pp. 243–348] associated to the symmetric group \mathfrak {S}_n are given by plethystically transformed Macdonald polynomials specialized at q=t. We generalize this to restricted rational Cherednik algebras of wreath product groups C_\ell \wr \mathfrak {S}_n and prove that the corresponding characters are given by a specialization of the wreath Macdonald polynomials defined by Haiman in [ Combinatorics, symmetric functions, and Hilbert schemes , Int. Press, Somerville, MA, 2003, pp. 39–111]. [ABSTRACT FROM AUTHOR]

We prove that all K-homology classes of the stable (and unstable) Ruelle algebra of a Smale space have explicit Fredholm module representatives that are finitely summable on the same smooth subalgebra and with the same degree of summability. The smooth subalgebra is induced by a metric on the underlying Smale space groupoid and fine transversality relations between stable and unstable sets. The degree of summability is related to the fractal dimension of the Smale space. Further, the Fredholm modules are obtained by taking Kasparov products with a fundamental class of the Spanier-Whitehead K-duality between the Ruelle algebras. Finally, we obtain general results on stability under holomorphic functional calculus and construct Lipschitz algebras on étale groupoids. [ABSTRACT FROM AUTHOR]

Starting from an integrable rank-n vertex model, we construct an explicit family of partition functions indexed by compositions \mu = (\mu _1,\dots,\mu _n). Using the Yang–Baxter algebra of the model and a certain rotation operation that acts on our partition functions, we show that they are eigenfunctions of the Cherednik–Dunkl operators Y_i for all 1 \leqslant i \leqslant n, and are thus equal to nonsymmetric Macdonald polynomials E_{\mu }. Our partition functions have the combinatorial interpretation of ensembles of coloured lattice paths which traverse a cylinder. Applying a simple bijection to such path ensembles, we show how to recover the well-known combinatorial formula for E_{\mu } due to Haglund–Haiman–Loehr. [ABSTRACT FROM AUTHOR]

Motivic equivalence for algebraic groups was recently introduced by De Clercq [Compos. Math. 153 (2017), pp. 2195–2213], where a characterization of motivic equivalent groups in terms of higher Tits indexes is given. As a consequence, if the quadrics associated to two quadratic forms have the same Chow motives with coefficients in \mathbb {F}_2, this remains true for any two projective homogeneous varieties of the same type under the orthogonal groups of those two quadratic forms. Our main result extends this to all groups of classical type, and to some exceptional groups, introducing a notion of critical variety. On the way, we prove that motivic equivalence of the automorphism groups of two involutions can be checked after extending scalars to some index reduction field, which depends on the type of the involutions. In addition, we describe conditions on the base field which guarantee that motivic equivalent involutions actually are isomorphic, extending a result of Hoffmann on quadratic forms. [ABSTRACT FROM AUTHOR]

We construct ample groupoids from certain categories of paths, and prove that their C^*-algebras coincide with the continued fraction approximately finite dimensional (AF) algebras of Effros and Shen. The proof relies on recent classification results for simple nuclear C^*-algebras. The groupoids are not principal. This provides examples of Cartan subalgebras in the continued fraction AF algebras that are isomorphic, but not conjugate, to the standard diagonal subalgebras. [ABSTRACT FROM AUTHOR]

To each submonoid P of a group we associate a universal Toeplitz \mathrm {C}^*-algebra \mathcal {T}_u(P) defined via generators and relations; \mathcal {T}_u(P) is a quotient of Li's semigroup \mathrm {C}^*-algebra \mathrm {C}^*_s(P) and they are isomorphic iff P satisfies independence. We give a partial crossed product realization of \mathcal {T}_u(P) and show that several results known for \mathrm {C}^*_s(P) when P satisfies independence are also valid for \mathcal {T}_u(P) when independence fails. At the level of the reduced semigroup \mathrm {C}^*-algebra \mathcal {T}_\lambda (P), we show that nontrivial ideals have nontrivial intersection with the reduced crossed product of the diagonal subalgebra by the action of the group of units of P, generalizing a result of Li for monoids with trivial unit group. We characterize when the action of the group of units is topologically free, in which case a representation of \mathcal {T}_\lambda (P) is faithful iff it is jointly proper. This yields a uniqueness theorem that generalizes and unifies several classical results. We provide a concrete presentation for the covariance algebra of the product system over P with one-dimensional fibers in terms of a new notion of foundation sets of constructible ideals. We show that the covariance algebra is a universal analogue of the boundary quotient and give conditions on P for the boundary quotient to be purely infinite simple. We discuss applications to a numerical semigroup and to the ax+b-monoid of an integral domain. This is particularly interesting in the case of nonmaximal orders in number fields, for which we show independence always fails. [ABSTRACT FROM AUTHOR]

We study homogeneous Besov and Triebel–Lizorkin spaces defined on doubling metric measure spaces in terms of a self-adjoint operator whose heat kernel satisfies Gaussian estimates together with its derivatives. When the measure space is a smooth manifold and such operator is a sum of squares of smooth vector fields, we prove that their intersection with L^\infty is an algebra for pointwise multiplication. Our results apply to nilpotent Lie groups and Grushin settings. [ABSTRACT FROM AUTHOR]

We give a purely complex geometric proof of the existence of the Bergman kernel expansion. Our method actually provides a sharper estimate, and in the case that the metrics are real analytic, we prove that the remainder decays faster than any polynomial. [ABSTRACT FROM AUTHOR]

We prove the Tits-Weiss conjecture for Albert division algebras over fields of arbitrary characteristic in the affirmative. The conjecture predicts that every norm similarity of an Albert division algebra is a product of a scalar homothety and U-operators. This conjecture is equivalent to the Kneser-Tits conjecture for simple, simply connected algebraic groups with Tits index E^{78}_{8,2}. We prove that a simple, simply connected algebraic group with Tits index E_{8,2}^{78} or E_{7,1}^{78}, defined over a field of arbitrary characteristic, is R-trivial, in the sense of Manin, thereby proving the Kneser-Tits conjecture for such groups. The Tits-Weiss conjecture follows as a consequence. [ABSTRACT FROM AUTHOR]

Borcherds lift for an even lattice of signature (p, q) is a lifting from weakly holomorphic modular forms of weight (p-q)/2 for the Weil representation. We introduce a new product operation on the space of such modular forms and develop a basic theory. The product makes this space a finitely generated filtered associative algebra, without unit element and noncommutative in general. This is functorial with respect to embedding of lattices by the quasi-pullback. Moreover, the rational space of modular forms with rational principal part is closed under this product. In some examples with p=2, the multiplicative group of Borcherds products of integral weight forms a subring. [ABSTRACT FROM AUTHOR]

We study the category of algebras of substitudes (also known to be equivalent to the regular patterns of Getzler and operads coloured by a category) equipped with a (semi)model structure lifted from the model structure on the underlying presheaves. We are especially interested in the case when the model structure on presheaves is a Cisinski style localisation with respect to a proper Grothendieck fundamental localiser. For example, for W = W∞ the minimal fundamental localiser, the local objects in such a localisation are locally constant presheaves, and local algebras of substitudes are exactly algebras whose underlying presheaves are locally constant. We investigate when this localisation has nice properties. We single out a class of such substitudes which we call left localisable and show that the substitudes for n-operads, symmetric, and braided operads are in this class. As an application we develop a homotopy theory of higher braided operads and prove a stabilisation theorem for their Wk-localisations. This theorem implies, in particular, a generalisation of the Baez-Dolan stabilisation hypothesis for higher categories. [ABSTRACT FROM AUTHOR]

As an application of the double affine Hecke algebra with unequal parameters on Weyl orbits to representation theory of semisimple Lie algebras, we find the graded multiplicities of the trivial module and of the little adjoint module in the exterior algebra of the little adjoint module of a simple Lie algebra \mathfrak {g} with a non-simply laced Dynkin diagram. We prove that in type B, C or F these multiplicities can be expressed in terms of special exponents of positive long roots in the dual root system of \mathfrak {g}. [ABSTRACT FROM AUTHOR]

A Lie bracket defined on the linear span of the free homotopy classes of undirected closed curves on surfaces was discovered in stages passing through Thurston's earthquake deformations, Wolpert's corresponding calculations with Hamiltonian vector fields and Goldman's algebraic treatment of the latter leading to a Lie bracket on the span of directed closed curves. The purpose of this work is to deepen the understanding of the former Lie bracket which will be referred to as the Thurston-Wolpert-Goldman Lie bracket or, briefly, the TWG bracket. We give a local direct geometric definition of the TWG bracket and use this geometric point of view to prove three results: firstly, the center of the TWG bracket is the Lie sub algebra generated by the class of the trivial loop and the classes of loops parallel to boundary components or punctures; secondly the analogous result holds for the centers of the universal enveloping algebra and of the symmetric algebra determined by the TWG Lie algebra; and thirdly, in terms of the natural basis, the TWG bracket of two non-central curves is always a linear combination of non-central curves. These three results hold for surfaces with or without boundary. We also give a brief and more illuminating proof of a known result, namely, the TWG bracket counts intersection between a simple closed curve and any other closed curve. We conclude by discussing substantial computer evidence suggesting an unexpected and strong conjectural statement relating the intersection structure of curves and the TWG bracket, namely, if the TWG bracket of two distinct undirected curves is zero then these curve classes have disjoint representatives. The computer experiments were performed on curves on surfaces with boundary. The main tools are basic hyperbolic geometry and Thurston's earthquake theory. [ABSTRACT FROM AUTHOR]

We consider the finite generation property for cohomology of a finite tensor category \mathscr {C}, which requires that the self-extension algebra of the unit \operatorname {Ext}^\text {\tiny∙}_\mathscr {C}(\mathbf {1},\mathbf {1}) is a finitely generated algebra and that, for each object V in \mathscr {C}, the graded extension group \operatorname {Ext}^\text {\tiny∙}_\mathscr {C}(\mathbf {1},V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on \mathscr {C}. For example, the stated result holds when \mathscr {C} is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0, we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes. [ABSTRACT FROM AUTHOR]

We describe the E-infinity algebra structure on the complex of singular cochains of a topological space, in the context of sheaf theory. As a first application, for any algebraic variety we define a weight filtration compatible with its E-infinity structure. This naturally extends the theory of mixed Hodge structures in rational homotopy to p-adic homotopy theory. The spectral sequence associated to the weight filtration gives a new family of algebraic invariants of the varieties for any coefficient ring, carrying Steenrod operations. As a second application, we promote Deligne's intersection complex computing intersection cohomology, to a sheaf carrying E-infinity structures. This allows for a natural interpretation of the Steenrod operations defined on the intersection cohomology of any topological pseudomanifold. [ABSTRACT FROM AUTHOR]

We introduce and study iterated primitives of meromorphic quasimodular forms for SL2(Z), generalizing work of Manin and Brown for holomorphic modular forms. We prove that the algebra of iterated primitives of meromorphic quasimodular forms is naturally isomorphic to a certain explicit shuffle algebra. We deduce from this an Ax–Lindemann–Weierstrass type algebraic independence criterion for primitives of meromorphic quasimodular forms which includes a recent result of Paşol–Zudilin as a special case. We also study spaces of meromorphic modular forms with restricted poles, generalizing results of Guerzhoy in the weakly holomorphic case. [ABSTRACT FROM AUTHOR]

Let A be an associative algebra with involution * over a field of characteristic zero. A central *-polynomial of A is a polynomial in non-commutative variables that takes central values in A. Here we prove the existence of two limits called the central *-exponent and the proper central *-exponent that give a measure of the growth of the central *-polynomials and proper central *-polynomials, respectively. Moreover, we compare them with the PI-*-exponent of the algebra. [ABSTRACT FROM AUTHOR]

The article presents a study to examine the solvability of second-order equations with partially BMO coefficients. It analyses the Lp-solvability of second-order elliptic and parabolic equations in nondivergence form using divergence-form equations. It mentions the leading coefficients has been assumed to be in locally BMO spaces.

In this paper, we first give a slight improvement of Yamanoi's truncated second main theorem for holomorphic maps into abelian varieties. We then use the result to study the uniqueness problem for such maps. The results obtained generalize and improve E. M. Schmid's uniqueness theorem for holomorphic maps into elliptic curves. In the last section, we consider algebraic dependence for a finite collection of holomorphic curves into an abelian variety. [ABSTRACT FROM AUTHOR]

This paper studies quadratic and Koszul duality for modules over positively graded categories. Typical examples are modules over a path algebra, which is graded by the path length, of a not necessarily finite quiver with relations. We present a very general definition of quadratic and Koszul duality functors backed up by explicit examples. This generalizes the work of Beilinson, Ginzburg, and Soergel, 1996, in two substantial ways: We work in the setup of graded categories, i.e. we allow infinitely many idempotents and also define a ``Koszul'' duality {it functor} for not necessarily Koszul categories. As an illustration of the techniques we reprove the Koszul duality (Ryom-Hansen, 2004) of translation and Zuckerman functors for the classical category $mathcal {O}$ in a quite elementary and explicit way. From this we deduce a conjecture of Bernstein, Frenkel, and Khovanov, 1999. As applications we propose a definition of a ``Koszul'' dual category for integral blocks of Harish-Chandra bimodules and for blocks outside the critical hyperplanes for the Kac-Moody category $mathcal {O}$. [ABSTRACT FROM AUTHOR]

This paper is dedicated to a description of the poles of the Igusa local zeta function $Z(s,f,v)$ when $f(x,y)$ satisfies a new non-degeneracy condition called \textit{arithmetic non-degeneracy}. More precisely, we attach to each polynomial $f(x,y)$ a collection of convex sets $\Gamma ^{A}(f)=\left{ \Gamma _{f,1},\dots ,\Gamma _{f,l_{0}}\right} $ called the \textit{arithmetic Newton polygon} of $f(x,y)$, and introduce the notion of \textit{arithmetic non-degeneracy with respect to }$\Gamma ^{A}(f)$. If $L_{v}$ is a $p$-adic field, and $f(x,y)\in L_{v}\left[ x,y \right] $ is arithmetically non-degenerate, then the poles of $Z(s,f,v)$ can be described explicitly in terms of the equations of the straight segments that form the boundaries of the convex sets $\Gamma _{f,1},\dots , \Gamma _{f,l_{0}}$. Moreover, the proof of the main result gives an effective procedure for computing $Z(s,f,v)$. [ABSTRACT FROM AUTHOR]

In this paper we let $G$ be a simple algebraic group and $r$ be a natural number, and consider the codimension in $G$ of the variety of elements $g\in G$ satisfying $g^r=1$. We shall obtain a lower bound for this codimension which is independent of characteristic, and show that it is attained if $G$ is of adjoint type. [ABSTRACT FROM AUTHOR]

In this paper we show that the $\varrho$-th order variation operator, for both the Fejér and Poisson kernels, are bounded from $L^p$ to $L^p$, $12$. Counterexamples are given if $\varrho =2$. [ABSTRACT FROM AUTHOR]

Let $A=\oplus_{g\in G}A_g$ be a $G$-graded associative algebra over a field of characteristic zero. In this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component $A_e$ to that of the whole of $A$, in the case where the support of the grading is finite. We prove the conjecture in several natural cases, one of them being the case where $A$ is finite dimensional and $A_e$ has polynomial growth. [ABSTRACT FROM AUTHOR]

This paper studies coordinates in two variables over a $\mathbb{Q}$-algebra. It gives several ways to characterize such coordinates. Also, various results about coordinates in two variables that were previously only known for fields, are extended to arbitrary $\mathbb{Q}$-algebras. [ABSTRACT FROM AUTHOR]

Let $\mathcal{G}\to B$ be a bundle of compact Lie groups acting on a fiber bundle $Y \to B$. In this paper we introduce and study gauge-equivariant $K$-theory groups $K_\mathcal{G}^i(Y)$. These groups satisfy the usual properties of the equivariant $K$-theory groups, but also some new phenomena arise due to the topological non-triviality of the bundle $\mathcal{G}\to B$. As an application, we define a gauge-equivariant index for a family of elliptic operators $(P_b)_{b \in B}$ invariant with respect to the action of $\mathcal{G}\to B$, which, in this approach, is an element of $K_\mathcal{G}^0(B)$. We then give another definition of the gauge-equivariant index as an element of $K_0(C^*(\mathcal{G}))$, the $K$-theory group of the Banach algebra $C^*(\mathcal{G})$. We prove that $K_0(C^*(\mathcal{G})) \simeq K^0_\mathcal{G}(\mathcal{G})$ and that the two definitions of the gauge-equivariant index are equivalent. The algebra $C^*(\mathcal{G})$ is the algebra of continuous sections of a certain field of $C^*$-algebras with non-trivial Dixmier-Douady invariant. The gauge-equivariant $K$-theory groups are thus examples of twisted $K$-theory groups, which have recently turned out to be useful in the study of Ramond--Ramond fields. [ABSTRACT FROM AUTHOR]

We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras. [ABSTRACT FROM AUTHOR]

We study modules over the commutative ring spectrum HF2 ∧ HF2, whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator ξk in the category of associative algebras freely kills the higher generators ξk+n. Using new information about the conjugation operation in the dual Steenrod algebra, we also consider quotients by families of Milnor generators and their conjugates. This allows us to produce a family of associative HF2 ∧ HF2-algebras whose coefficient rings are finite-dimensional and exhibit unexpected duality features. We then use these algebras to give detailed computations of the homotopy groups of several modules over this ring spectrum. [ABSTRACT FROM AUTHOR]

In the present article, we study the conjecture of Sharifi on the surjectivity of the map πθ. Here θ is a primitive even Dirichlet character of conductor Np, which is exceptional in the sense of Ohta. After localizing at the prime ideal p of the Iwasawa algebra related to the trivial zero of the Kubota–Leopoldt p-adic L-function Lp(s,θ−1ω2), we compute the image of πθ,p in a local Galois cohomology group and prove that it is an isomorphism. Also, we prove that the residual Galois representations associated to the cohomology of modular curves are decomposable after taking the same localization. [ABSTRACT FROM AUTHOR]

We show that the subgraph of the concave transform of a multiplicative filtration on a section ring is the Newton–Okounkov body of a certain semigroup, and if the filtration is induced by a divisorial valuation, then the associated graded algebra is the algebra of sections of a concrete line bundle in higher dimension. We use this description to give a rationality criterion for certain Seshadri constants. [ABSTRACT FROM AUTHOR]

We show that several apparently unrelated formulas involving left or right Bousfield localizations in homotopy theory are induced by comparison maps associated with pairs of adjoint functors. Such comparison maps are used in the article to discuss the existence of functorial liftings of homotopical localizations and cellularizations to categories of algebras over monads acting on model categories, with emphasis on the cases of module spectra and algebras over simplicial operads. Some of our results hold for algebras up to homotopy as well; for example, if T is the reduced monad associated with a simplicial operad and ƒ is any map of pointed simplicial sets, then ƒ-localization coincides with Tƒ-localization on spaces underlying homotopy T-algebras, and similarly for cellularizations. [ABSTRACT FROM AUTHOR]