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We equip the type A diagrammatic Hecke category with a special derivation, so that after specialization to characteristic p it becomes a p-dg category. We prove that the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. We conjecture that the p-dg Grothendieck group is isomorphic to the Iwahori-Hecke algebra, equipping it with a basis which may differ from both the Kazhdan-Lusztig basis and the p-canonical basis. More precise conjectures will be found in the sequel. Here are some other results contained in this paper. We provide an incomplete proof of the classification of all degree +2 derivations on the diagrammatic Hecke category, and a complete proof of the classification of those derivations for which the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. In particular, our special derivation is unique up to duality and equivalence. We prove that no such derivation exists in simply-laced types outside of finite and affine type A. We also examine a particular Bott-Samelson bimodule in type A_7, which is indecomposable in characteristic 2 but decomposable in all other characteristics. We prove that this Bott-Samelson bimodule admits no nontrivial fantastic filtrations in any characteristic, which is the analogue in the p-dg setting of being indecomposable. [ABSTRACT FROM AUTHOR]

In this paper, for an arbitrary Kac-Moody Lie algebra {\mathfrak g} and a diagram automorphism \mu of {\mathfrak g} satisfying certain natural linking conditions, we introduce and study a \mu-twisted quantum affinization algebra {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) of {\mathfrak g}. When {\mathfrak g} is of finite type, {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) is Drinfeld's current algebra realization of the twisted quantum affine algebra. When \mu =\mathrm {id} and {\mathfrak g} in affine type, {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) is the quantum toroidal algebra introduced by Ginzburg, Kapranov and Vasserot. As the main results of this paper, we first prove a triangular decomposition for {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right). Second, we give a simple characterization of the affine quantum Serre relations on restricted {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right)-modules in terms of "normal order products". Third, we prove that the category of restricted {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right)-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right). Last, we study the classical limit of {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) and abridge it to the quantization theory of extended affine Lie algebras. In particular, based on a classification result of Allison-Berman-Pianzola, we obtain the \hbar-deformation of all nullity 2 extended affine Lie algebras. [ABSTRACT FROM AUTHOR]

Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz (1991) and was further developed in the work of S. Ault (2010). In this paper, we show that, for algebras defined over a field of characteristic 0, the symmetric homology is naturally equivalent to the (one-dimensional) representation homology introduced by the authors in joint work with G. Khachatryan (2013). Using known results on representation homology, we compute symmetric homology explicitly for basic algebras, such as polynomial algebras and universal enveloping algebras of (DG) Lie algebras. As an application, we prove two conjectures of Ault and Fiedorowicz, including their main conjecture (2007) on topological interpretation of symmetric homology of polynomial algebras. [ABSTRACT FROM AUTHOR]

Using Jack polynomials, Goulden and Jackson have introduced a one parameter deformation \tau _b of the generating series of bipartite maps, which generalizes the partition function of \beta-ensembles of random matrices. The Matching-Jack conjecture suggests that the coefficients c^\lambda _{\mu,\nu } of the function \tau _b in the power-sum basis are non-negative integer polynomials in the deformation parameter b. Dołęga and Féray have proved in 2016 the "polynomiality" part in the Matching-Jack conjecture, namely that coefficients c^\lambda _{\mu,\nu } are in \mathbb {Q}[b]. In this paper, we prove the "integrality" part, i.e. that the coefficients c^\lambda _{\mu,\nu } are in \mathbb {Z}[b]. The proof is based on a recent work of the author that deduces the Matching-Jack conjecture for marginal sums \overline { c}^\lambda _{\mu,l} from an analog result for the b-conjecture, established in 2020 by Chapuy and Dołęga. A key step in the proof involves a new connection with the graded Farahat-Higman algebra. [ABSTRACT FROM AUTHOR]

In [Math. Ann. 367 (2017), pp. 1517–1586] Bar-Natan and the first author show that solutions to the Kashiwara–Vergne equations are in bijection with certain knot invariants: homomorphic expansions of welded foams. Welded foams are a class of knotted tubes in \mathbb {R}^4, which can be finitely presented algebraically as a circuit algebra , or equivalently, a wheeled prop. In this paper we describe the Kashiwara-Vergne groups \mathsf {KV} and \mathsf {KRV}—the symmetry groups of Kashiwara-Vergne solutions—as automorphisms of the completed circuit algebras of welded foams, and their associated graded circuit algebras of arrow diagrams, respectively. Finally, we provide a description of the graded Grothendieck-Teichmüller group \mathsf {GRT}_1 as automorphisms of arrow diagrams. [ABSTRACT FROM AUTHOR]

Majorana representations have been introduced by Ivanov [ Cambridge Tracts in Mathematics , Cambridge University Press, Cambridge, 2009] in order to provide an axiomatic framework for studying the actions on the Griess algebra of the Monster and of its subgroups generated by Fischer involutions. A crucial step in this programme is to obtain an explicit description of the Majorana representations of A_{12} (by Franchi, Ivanov, and Mainardis [J. Algebraic Combin. 44 (2016), pp. 265-292], the largest alternating group admitting a Majorana representation) for this might eventually lead to a new and independent construction of the Monster group (see A.A Ivanov [ Group theory and computation , Indian Stat. Inst. Ser., Springer, Singapore, 2018, Section 4, page 115]). In this paper we prove that A_{12} has two possible Majorana sets, one of which is the set \mathcal X_b of involutions of cycle type 2^2, the other is the union of \mathcal X_b with the set \mathcal X_s of involutions of cycle type 2^6. The latter case (the saturated case) is most interesting, since the Majorana set is precisely the set of involutions of A_{12} that fall into the class of Fischer involutions when A_{12} is embedded in the Monster. We prove that A_{12} has a unique saturated Majorana representation and we determine its degree and decomposition into irreducibles. As consequences we get that the Harada-Norton group has, up to equivalence, a unique Majorana representation and every Majorana algebra, affording either a Majorana representation of the Harada-Norton group or a saturated Majorana representation of A_{12}, satisfies the Straight Flush Conjecture (see A. A. Ivanov [ Contemp. Math. , Amer. Math. Soc., Providence, RI, 2017, pp. 11-17] and A. A. Ivanov [ Group theory and computation , Indian Stat. Inst. Ser., Springer, Singapore, 2018, pp. 107-118]). As a by-product we also determine the degree and the decomposition into irreducibles of the Majorana representation induced on A_8, the four point stabilizer subgroup of A_{12}. We finally state a conjecture about Majorana representations of the alternating groups A_n, 8\leq n\leq 12. [ABSTRACT FROM AUTHOR]

For a primitive Dirichlet character \chi modulo q, we define M(\chi)=\max _{t } |\sum _{n \leq t} \chi (n)|. In this paper, we study this quantity for characters of a fixed odd order g\geq 3. Our main result provides a further improvement of the classical Pólya-Vinogradov inequality in this case. More specifically, we show that for any such character \chi we have \begin{equation*} M(\chi)\ll _{\varepsilon } \sqrt {q}(\log q)^{1-\delta _g}(\log \log q)^{-1/4+\varepsilon }, \end{equation*} where \delta _g ≔1-\frac {g}{\pi }\sin (\pi /g). This improves upon the works of Granville and Soundararajan [J. Amer. Math. Soc. 20 (2007), pp. 357–384] and of Goldmakher [Algebra Number Theory 6 (2012), pp. 123–163]. Furthermore, assuming the Generalized Riemann Hypothesis (GRH) we prove that \begin{equation*} M(\chi) \ll \sqrt {q} \left (\log _2 q\right)^{1-\delta _g} \left (\log _3 q\right)^{-\frac {1}{4}}\left (\log _4 q\right)^{O(1)}, \end{equation*} where \log _j is the j-th iterated logarithm. We also show unconditionally that this bound is best possible (up to a power of \log _4 q). One of the key ingredients in the proof of the upper bounds is a new Halász-type inequality for logarithmic mean values of completely multiplicative functions, which might be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper we give a version of Bergman's diamond lemma which applies to certain monoidal categories presented by generators and relations. In particular, it applies to: the Coxeter presentation of the symmetric groups, the quiver Hecke algebras of Khovanov-Lauda-Rouquier, the Webster tensor product algebras, and various generalizations of these. We also give an extension of Manin-Schechtmann theory to non-reduced expressions. [ABSTRACT FROM AUTHOR]

Eulerian polynomials are fundamental in combinatorics and algebra. In this paper we study the linear transformation \mathcal {A}: \mathbb {R}[t] \to \mathbb {R}[t] defined by \mathcal {A}(t^n) = A_n(t), where A_n(t) denotes the n-th Eulerian polynomial. We give combinatorial, topological and Ehrhart theoretic interpretations of the operator \mathcal {A}, and investigate questions of unimodality and real-rootedness. In particular, we disprove a conjecture by Brenti (1989) concerning the preservation of real zeros, and generalize and strengthen recent results of Haglund and Zhang (2019) on binomial Eulerian polynomials. [ABSTRACT FROM AUTHOR]

In this paper we develop new reduction techniques for testing the finiteness of the finitistic dimension of a finite dimensional algebra over a field. Viewing the latter algebra as a quotient of a path algebra, we propose two operations on the quiver of the algebra, namely arrow removal and vertex removal. The former gives rise to cleft extensions and the latter to recollements. These two operations provide us new practical methods to detect algebras of finite finitistic dimension. We illustrate our methods with many examples. [ABSTRACT FROM AUTHOR]

In this short note we correct the statement of the main result of [Trans. Amer. Math. Soc. 373 (2020), no. 3, 1909-1940]. That paper presented the rational cohomology ring of a toric arrangement by generators and relations. One of the series of relations given in the paper is indexed over the set circuits in the arrangement's arithmetic matroid. That series of relations should however be indexed over all sets X with |X| = rk(X)+1. Below we give the complete and correct presentation of the rational cohomology ring. [ABSTRACT FROM AUTHOR]

Zeta values in Tate algebras were introduced by Pellarin in 2012. They are generalizations of Carlitz zeta values and play an increasingly important role in function field arithmetic. In this paper we prove a conjecture of Pellarin on identities for these zeta values. The proof is based on arithmetic properties of Carlitz zeta values and an explicit formula for Bernoulli-type polynomials attached to zeta values in Tate algebras. [ABSTRACT FROM AUTHOR]

Zarhin has extensively studied restrictions placed on the endomorphism algebras of Jacobians J for which the Galois group associated to their 2-torsion is insoluble and "large" (relative to the dimension of J). In this paper we examine what happens when this Galois group merely contains an element of "large" prime order. [ABSTRACT FROM AUTHOR]

Consider a proper, isometric action by a unimodular, locally compact group G on a complete Riemannian manifold M. For equivariant elliptic operators that are invertible outside a cocompact subset of M, we show that a localised index in the K-theory of the maximal group C*-algebra of G is well-defined. The approach is based on the use of maximal versions of equivariant localised Roe algebras, and many of the technical arguments in this paper are used to handle the ways in which they differ from their reduced versions. By using the maximal group C*-algebra instead of its reduced counterpart, we can apply the trace given by integration over G to recover an index defined earlier by the last two authors, and developed further by Braverman, in terms of sections invariant under the group action. This leads to refinements of index-theoretic obstructions to Riemannian metrics of positive scalar curvature on noncompact manifolds, and also on orbifolds and other singular quotients of proper group actions. As a motivating application in another direction, we prove a version of Guillemin and Sternberg's quantisation commutes with reduction principle for equivariant indices of Spinc Callias-type operators. [ABSTRACT FROM AUTHOR]

It is natural to ask whether it is possible to construct Lévy-like processes where actions by random elements of a given semigroup play the role of increments. Such semigroups induce a convolution-like algebra structure in the space of finite measures. In this paper, we show that the Whittaker convolution operator, related with the Shiryaev process, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. We then introduce the class of Lévy processes with respect to the Whittaker convolution and study their basic properties. We obtain a martingale characterization of the Shiryaev process analogous to Lévy's characterization of Brownian motion. Our results demonstrate that a nice theory of Lévy processes with respect to generalized convolutions can be developed for differential operators whose associated convolution does not satisfy the usual compactness assumption on the support of the convolution. [ABSTRACT FROM AUTHOR]

Let R be a standard graded Gorenstein algebra over a field presented by quadrics. In [Compositio Math. 129 (2001), no. 1, 95-121], Conca-Rossi-Valla show that such a ring is Koszul if reg R ≤ 2 or if reg R = 3 and c = codim R ≤ 4, and they ask whether this is true for reg R = 3 in general. We determine sufficient conditions on a non-Koszul quadratic Cohen-Macaulay ring R that guarantee the Nagata idealization ~ R = R × ωR(−a−1) is a non-Koszul quadratic Gorenstein ring. We prove there exist rings of regularity 3 satisfying our conditions for all c ≥ 9; this yields a negative answer to the question from the above-mentioned paper. [ABSTRACT FROM AUTHOR]

Let g be the derived subalgebra of a Kac-Moody Lie algebra of finite-type or affine-type, let μ be a diagram automorphism of g, and let L(g,μ) be the loop algebra of g associated to μ. In this paper, by using the vertex algebra technique, we provide a general construction of current-type presentations for the universal central extension g[μ] of L(g,μ). The construction contains the classical limit of Drinfeld's new realization for (twisted and untwisted) quantum affine algebras [Soviet Math. Dokl. 36 (1988), pp. 212-216] and the Moody-Rao-Yokonuma presentation for toroidal Lie algebras [Geom. Dedicata 35 (1990), pp. 283-307] as special examples. As an application, when g is of simply-laced-type, we prove that the classical limit of the μ-twisted quantum affinization of the quantum Kac-Moody algebra associated to g introduced in [J. Math. Phys. 59 (2018), 081701] is the universal enveloping algebra of g[μ]. [ABSTRACT FROM AUTHOR]

One of the main results stated in Theorem 4.4 of our article, which appears in Trans. Amer. Math. Soc. 372 (2019), no. 10, 7191-7223, is that the structure algebra K[M(X,r)], over a field K, of a finite bijective left non-degenerate solution (X,r) of the Yang-Baxter equation is a module-finite central extension of a commutative affine subalgebra. This is proven by showing that the structure monoid M(X,r) is central-by-finite. This however is not true, even in case (X,r) is a (left and right) non-degenerate involutive solution. The proof contains a subtle mistake. However, it turns out that the monoid M(X,r) is abelian-by-finite and thus the conclusion that K[M(X,r)] is a module-finite normal extension of a commutative affine subalgebra remains valid. In particular, K[M(X,r)] is Noetherian and satisfies a polynomial identity. The aim of this paper is to give a proof of this result. In doing so, we also strengthen Lemma 5.3 (and its consequences, namely Lemma 5.4 and Proposition 5.5) showing that these results on the prime spectrum of the structure monoid hold even if the assumption that the solution (X,r) is square-free is omitted. [ABSTRACT FROM AUTHOR]

For a finite involutive non-degenerate solution (X, r) of the Yang- Baxter equation it is known that the structure monoid M(X, r) is a monoid of I-type, and the structure algebra K[M(X, r)] over a field K shares many properties with commutative polynomial algebras; in particular, it is a Noetherian PI-domain that has finite Gelfand-Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid M(X, r) and the algebra K[M(X, r)] is much more complicated than in the involutive case, we provide some deep insights. In this general context, using a realization of Lebed and Vendramin of M(X, r) as a regular submonoid in the semidirect product A(X, r)Sym(X), where A(X, r) is the structure monoid of the rack solution associated to (X, r), we prove that K[M(X, r)] is a finite module over a central affine subalgebra. In particular, K[M(X, r)] is a Noetherian PI-algebra of finite Gelfand-Kirillov dimension bounded by |X|. We also characterize, in ring-theoretical terms of K[M(X, r)], when (X, r) is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of M(X, r). These results allow us to control the prime spectrum of the algebra K[M(X, r)] and to describe the Jacobson radical and prime radical of K[M(X, r)]. Finally, we give a matrix-type representation of the algebra K[M(X, r)]/P for each prime ideal P of K[M(X, r)]. As a consequence, we show that if K[M(X, r)] is semiprime, then there exist finitely many finitely generated abelian-by-finite groups, G1, . . . ,Gm, each being the group of quotients of a cancellative subsemigroup of M(X, r) such that the algebra K[M(X, r)] embeds into Mv1 (K[G1])×· · ·×Mvm(K[Gm]), a direct product of matrix algebras. [ABSTRACT FROM AUTHOR]

The main result of this paper is the following: if F is any field and R any F-subalgebra of the algebra Mn(F) of n × n matrices over F with Lie nilpotence index m, then. dimFR ≼ M(m + 1, n), where M(m + 1, n) is the maximum of 1/2(n² -Σ m+1 i=1 k² i) + 1 subject to the constraint Σ m+1 i=1 ki = n and k1, k2, . . . , km+1 nonnegative integers. This answers in the affirmative a conjecture by the first and third authors. The case m = 1 reduces to a classical theorem of Schur (1905), later generalized by Jacobson (1944) to all fields, which asserts that if F is an algebraically closed field of characteristic zero and R is any commutative F-subalgebra of Mn(F), then dimFR ≼ n²/4+ 1. Examples constructed from block upper triangular matrices show that the upper bound of M(m+1, n) cannot be lowered for any choice of m and n. An explicit formula for M(m + 1, n) is also derived. [ABSTRACT FROM AUTHOR]

Let R = C[x1, x2, . . ., xn]/(f1, . . ., fm) be a positively graded Artinian algebra. A long-standing conjecture in algebraic geometry, differential geometry, and rational homotopy theory is the non-existence of negative weight derivations on R. Alexsandrov conjectured that there are no negative weight derivations when R is a complete intersection algebra, and Yau conjectured there are no negative weight derivations on R when R is the moduli algebra of a weighted homogeneous hypersurface singularity. This problem is also important in rational homotopy theory and differential geometry. In this paper we prove the non-existence of negative weight derivations on R when the degrees of f1, . . .,fm are bounded below by a constant C depending only on the weights of x1, . . ., xn. Moreover this bound C is improved in several special cases. [ABSTRACT FROM AUTHOR]

This paper looks for Frobenius subcategories, via the separated monomorphism category smon(Q, I,X), and on the other hand, aims to establish an RSS equivalence from smon(Q, I,X) to its dual sepi(Q, I,X). For a bound quiver (Q, I) and an algebra A, where Q is acyclic and I is generated by monomial relations, let Λ = A⊗k kQ/I. For any additive subcategory X of A-mod, we introduce smon(Q, I,X) combinatorially. It describes Gorenstein-projective Λ-modules as GP(Λ) = smon(Q, I,GP(A)). It admits a homological interpretation and enjoys a reciprocity smon(Q, I, ⊥T) = ⊥(T⊗kQ/I) for a cotilting A-module T. As an application, smon(Q, I,X) has Auslander-Reiten sequences if X is resolving and contravariantly finite with X =A-mod. In particular, smon(Q, I,A) has Auslander-Reiten sequences. It also admits a filtration interpretation as smon(Q, I,X) = Fil(X ⊗P(kQ/I)), provided that X is extension-closed. As an application, smon(Q, I,X) is an extension-closed Frobenius subcategory if and only if so is X. This gives "new" Frobenius subcategories of Λ-mod in the sense that they may not be GP(Λ). Ringel-Schmidmeier-Simson equivalence smon(Q, I,X)≅ sepi(Q, I,X) is introduced and the existence is proved for arbitrary extension-closed subcategories X. In particular, the Nakayama functor NΛ gives an RSS equivalence smon(Q, I,A)≅ sepi(Q, I,A) if and only if A is Frobenius. For a chain Q with arbitrary I, an explicit formula of an RSS equivalence is found for arbitrary additive subcategories X. [ABSTRACT FROM AUTHOR]

Nicolas Monod introduced the class of groups with the fixed-point property for cones, characterized by always admitting a nonzero fixed-point when acting (suitably) on proper weakly complete cones. He proved that his class of groups contains the class of groups with subexponential growth and is contained in the class of supramenable groups. In this paper we investigate what Monod’s results say about the existence of invariant traces on (typically nonunital) C∗-algebras equipped with an action of a group with the fixed-point property for cones. As an application of these results, we provide results on the existence (and nonexistence) of traces on the (nonuniform) Roe algebra. [ABSTRACT FROM AUTHOR]

The 4-dimensional Sklyanin algebras, over ℂ., A(E, τ), are constructed from an elliptic curve E and a translation automorphism τ of E. The Klein vierergruppe Γ acts as graded algebra automorphisms of A(E, τ). There is also an action of Γ as automorphisms of the matrix algebra M2(ℂ.) making it isomorphic to the regular representation. The main object of study in this paper is the invariant subalgebra à := A(E, τ)⊗M2(ℂ))Γ. Like A(E, τ), à is noetherian, generated by 4 degree-one elements modulo six quadratic relations, Koszul, Artin-Schelter regular of global dimension 4, has the same Hilbert series as the polynomial ring on 4 variables, satisfies the x condition, and so on. These results are special cases of general results proved for a triple (A, T, H) consisting of a Hopf algebra H, an (often graded) H-comodule algebra A, and an H-torsor T. Those general results involve transferring properties between A, A ⊗ T, and (A ⊗ T)coH. We then investigate à from the point of view of non-commutative projective geometry. We examine its point modules, line modules, and a certain quotient ... := Ã/(Θ,Θ') where Θ and Θ' are homogeneous central elements of degree two. In doing this we show that à differs from A in interesting ways. For example, the point modules for A are parametrized by E and 4 more points, whereas à has exactly 20 point modules. Although ... is not a twisted homogeneous coordinate ring in the sense of Artin and Van den Bergh, a certain quotient of the category of graded ...-modules is equivalent to the category of quasi-coherent sheaves on the curve E/E[2] where E[2] is the 2-torsion subgroup. We construct line modules for à that are parametrized by the disjoint union (E/(ξ1)) ⊔ (E/(ξ2)) ⊔ (E/(ξ3)) of the quotients of E by its three subgroups of order 2. [ABSTRACT FROM AUTHOR]

Magnus proved in 1930 that, given two elements x and y of a finitely generated free group F with equal normal closures (x)F = (y)F, x is conjugated either to y or y-1. More recently, this property, called the Magnus property, has been generalized to oriented surface groups. In this paper, we consider an analogue property for profinite surface groups. While the Magnus property, in general, does not hold in the profinite setting, it does hold in some restricted form. In particular, for L a class of finite groups, we prove that if x and y are algebraically simple elements of the pro-L completion ...L of an orientable surface group Π such that, for all n ∈ ℕ, there holds (xn)... = (yn)..., then x is conjugated to ys for some s ∈ (...L)*. As a matter of fact, a much more general property is proved and further extended to a wider class of profinite completions. The most important application of the theory above is a generalization of the description of centralizers of profinite Dehn twists given in [Marco Boggi, Trans. Amer. Math. Soc. 366 (2014), 5185-5221] to profinite Dehn multitwists. [ABSTRACT FROM AUTHOR]

Let μ be a Borel measure on a compactum X. The main objects in this paper are s-ideals Ipdimq, J0pμq, Jf pμq of Borel sets in X that can be covered by countably many compacta which are finite-dimensional, or of μ-measure null, or of finite μ-measure, respectively. Answering a question of J. Zapletal, we shall show that for the Hilbert cube, the s-ideal Ipdimq is not homogeneous in a strong way. We shall also show that in some natural instances of measures μ with nonhomogeneous s-ideals J0pμq or Jf pμq, the completions of the quotient Boolean algebras BorelpXq{J0pμq or BorelpXq{Jf pμq may be homogeneous. We discuss the topic in a more general setting, involving calibrated s-ideals. [ABSTRACT FROM AUTHOR]

This paper generalizes the classical Sz.-Nagy-Foias H∞(𝔻) functional calculus for Hilbert space contractions. In particular, we replace the single contraction T with a tuple T = (T1, . . ., Td) of commuting bounded operators on a Hilbert space and replace H∞(𝔻) with a large class of multiplier algebras of Hilbert function spaces on the unit ball in ℂd. [ABSTRACT FROM AUTHOR]

This paper continues the investigation of quasilength, of content of local cohomology with respect to generators of the support ideal, and of robust algebras begun in joint work of Hochster and Huneke. We settle several questions raised by Hochster and Huneke. In particular, we give a family of examples of top local cohomology modules both in equal characteristic 0 and in positive prime characteristic that are nonzero but have content 0. We use the notion of a robust forcing algebra (the condition turns out to be strictly stronger than the notion of a solid forcing algebra in, for example, equal characteristic 0) to define a new closure operation on ideals. We prove that this new notion of closure coincides with tight closure for ideals in complete local domains of positive characteristic, which requires proving that forcing algebras for instances of tight closure are robust, and study several related problems. This gives, in effect, a new characterization of tight closure in complete local domains of positive characteristic. As a byproduct, we also answer a question of Lyubeznik in the negative. [ABSTRACT FROM AUTHOR]

In this paper, we give a representation-theoretic interpretation of the specialization Ew◦λ(q,∞) of the nonsymmetric Macdonald polynomial Ew◦λ(q, t) at t = ∞ in terms of the Demazure submodule Vw◦- (λ) of the level-zero extremal weight module V (λ) over a quantum affine algebra of an arbitrary untwisted type. Here, λ is a dominant integral weight, and w◦ denotes the longest element in the finite Weyl group W. Also, for each x ∈ W, we obtain a combinatorial formula for the specialization Exλ(q,∞) at t = ∞ of the nonsymmetric Macdonald polynomial Exλ(q, t) and also a combinatorical formula for the graded character gch Vx- (λ) of the Demazure submodule Vx- (λ) of V (λ). Both of these formulas are described in terms of quantum Lakshmibai-Seshadri paths of shape λ. [ABSTRACT FROM AUTHOR]

We work over an algebraically closed field of characteristic zero. The purpose of this paper is to characterize a nondegenerate projective variety X with a linear projection which induces a nonbirational map to its image. As an application, for smooth X of degree d and codimension e, we prove the "semiampleness" of the (d - e + 1)th twist of the ideal sheaf. This improves a linear bound of the regularity of smooth projective varieties by Bayer-Mumford-Bertram-Ein-Lazarsfeld, and gives an asymptotic regularity bound. [ABSTRACT FROM AUTHOR]

The Lambert W function, giving the solutions of a simple transcendental equation, has become a famous function and arises in many applications in combinatorics, physics, or population dyamics just to mention a few. In this paper we construct and study in great detail a generalization of the Lambert W which involves some special polynomials and even combinatorial aspects. [ABSTRACT FROM AUTHOR]

We study the description of the crystal structure on the set of Mirković-Vilonen polytopes. Anderson and Mirković defined an operator and conjectured that it coincides with the Kashiwara operator. Kamnitzer proved the conjecture for type A and gave a counterexample for type C3. He also gave an explicit formula to calculate the Kashiwara operator for type A. In this paper we prove that a part of the AM conjecture still holds in general, answering an open question of Kamnitzer (2007). Moreover, we show that although the formula given by Kamnitzer does not hold in general, it is still valid in many cases regardless of the type. The main tool is the connection between MV polytopes and preprojective algebras developed by Baumann and Kamnitzer. [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]

There are many classes of nonsimple graph C*-algebras that are classified by the six-term exact sequence in K-theory. In this paper we consider the range of this invariant and determine which cyclic six-term exact sequences can be obtained by various classes of graph C*-algebras. To accomplish this, we establish a general method that allows us to form a graph with a given sixterm exact sequence of K-groups by splicing together smaller graphs whose C*- algebras realize portions of the six-term exact sequence. As rather immediate consequences, we obtain the first permanence results for extensions of graph C*-algebras. We are hopeful that the results and methods presented here will also prove useful in more general cases, such as situations where the C*-algebras under investigation have more than one ideal and where there are currently no relevant classification theories available. [ABSTRACT FROM AUTHOR]

In this paper we prove relationships between two generalizations of commutative valuation theory for noncommutative central simple algebras: (1) Dubrovin valuation rings; and (2) the value functions called gauges introduced by Tignol and Wadsworth. We show that if v is a valuation on a field F with associated valuation ring V and v is defectless in a central simple F-algebra A, and C is a subring of A, then the following are equivalent: (a) C is the gauge ring of some minimal v-gauge on A, i.e., a gauge with the minimal number of simple components of C/J(C); (b) C is integral over V with C = B1 ∩...∩ Bξ where each Bi is a Dubrovin valuation ring of A with center V, and the Bi satisfy Gräter's Intersection Property. Along the way we prove the existence of minimal gauges whenever possible and we show how gauges on simple algebras are built from gauges on central simple algebras. [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]

In this paper we describe a new method of defining C*-algebras from oriented combinatorial data, thereby generalizing the construction of algebras from directed graphs, higher-rank graphs, and ordered groups. We show that only the most elementary notions of concatenation and cancellation of paths are required to define versions of Cuntz-Krieger and Toeplitz-Cuntz- Krieger algebras, and the presentation by generators and relations follows naturally. We give sufficient conditions for the existence of an AF core, hence of the nuclearity of the C*-algebras, and for aperiodicity, which is used to prove the standard uniqueness theorems. [ABSTRACT FROM AUTHOR]

Let F be a quadratic form in N ≥ 2 variables defined on a vector space V ⊆ KN over a global field K, and Ƶ ⊆ KN be a finite union of varieties defined by families of homogeneous polynomials over K. We show that if V \ Ƶ contains a nontrivial zero of F, then there exists a linearly independent collection of small-height zeros of F in V \ Ƶ, where the height bound does not depend on the height of Ƶ, only on the degrees of its defining polynomials. As a corollary of this result, we show that there exists a small-height maximal totally isotropic subspace W of the quadratic space (V, F) such that W is not contained in Ƶ. Our investigation extends previous results on small zeros of quadratic forms, including Cassels' theorem and its various generalizations. The paper also contains an appendix with two variations of Siegel's lemma. All bounds on height are explicit. [ABSTRACT FROM AUTHOR]

Let H be a hypergroup with left Haar measure. The amenability of H can be characterized by the existence of nets of positive, norm one functions in L¹(H) which tend to left invariance in any of several ways. In this paper we present a characterization of the amenability of H using configuration equations. Extending work of Rosenblatt and Willis, we construct, for a certain class of hypergroups, nets in L1(H) which tend to left invariance weakly, but not in norm. We define the semidirect product of H with a locally compact group. We show that the semidirect product of an amenable hypergroup and an amenable locally compact group is an amenable hypergroup and show how to construct Reiter nets for this semidirect product. These results are generalized to Lau algebras, providing a new characterization of left amenability of a Lau algebra and a notion of a semidirect product of a Lau algebra with a locally compact group. The semidirect product of a left amenable Lau algebra with an amenable locally compact group is shown to be a left amenable Lau algebra. [ABSTRACT FROM AUTHOR]

In this two-part paper we prove an existence result for affine buildings arising from exceptional algebraic reductive groups. Combined with earlier results on classical groups, this gives a complete and positive answer to the conjecture concerning the existence of affine buildings arising from such groups defined over a (skew) field with a complete valuation, as proposed by Jacques Tits. This second part builds upon the results of the first part and deals with the remaining cases. [ABSTRACT FROM AUTHOR]

In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis, there exists an outer automorphism of the Calkin algebra. (The Calkin algebra is the algebra of bounded operators on a separable complex Hilbert space, modulo the compact operators.) In this paper we establish that the analogous conclusion holds for a broad family of quotient algebras. Specifically, we will show that assuming the Continuum Hypothesis, if A is a separable algebra which is either simple or stable, then the corona of A has nontrivial automorphisms. We also discuss a connection with cohomology theory, namely, that our proof can be viewed as a computation of the cardinality of a particular derived inverse limit. [ABSTRACT FROM AUTHOR]

In this paper, we first define a new family of Yoneda algebras, called F-Auslander-Yoneda algebras, in triangulated categories by introducing the notion of admissible sets F in N, which includes higher cohomologies indexed by F, and then present a general method to construct a family of new derived equivalences for these F-Auslander-Yoneda algebras (not necessarily Artin algebras), where the choices of the parameters F are rather abundant. Among applications of our method are the following results: (1) if A is a self-injective Artin algebra, then, for any A-module X and for any admissible set F in N, the F-Auslander-Yoneda algebras of A ⊕ X and A ⊕ OA(X) are derived equivalent, where O is the Heller loop operator. (2) Suppose that A and B are representation-finite self-injective algebras with additive generators AX and BY, respectively. If A and B are derived equivalent, then so are the F-Auslander-Yoneda algebras of X and Y for any admissible set F. In particular, the Auslander algebras of A and B are derived equivalent. The converse of this statement is open. Further, motivated by these derived equivalences between F-Auslander-Yoneda algebras, we show, among other results, that a derived equivalence between two basic self-injective algebras may transfer to a derived equivalence between their quotient algebras obtained by factoring out socles. [ABSTRACT FROM AUTHOR]