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The aim of this paper is to study a poset isomorphism between two support \tau-tilting posets. We take several information from combinatorial properties of support \tau-tilting posets and give an analogue of Happel-Unger's reconstruction theorem. [ABSTRACT FROM AUTHOR]

Building on work of Ruelle and Putnam in the Smale space case, Thomsen defined the homoclinic and heteroclinic C^\ast-algebras for an expansive dynamical system. In this paper we define a class of expansive dynamical systems, called synchronizing dynamical systems, that exhibit hyperbolic behavior almost everywhere. Synchronizing dynamical systems generalize Smale spaces (and even finitely presented systems). Yet they still have desirable dynamical properties such as having a dense set of periodic points. We study various C^\ast-algebras associated with a synchronizing dynamical system. Among other results, we show that the homoclinic algebra of a synchronizing system contains an ideal which behaves like the homoclinic algebra of a Smale space. [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]

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]

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]

The Witt ring of a complex flag variety describes the interesting – i.e. torsion – part of its topological KO-theory. We show that for a large class of flag varieties, these Witt rings are exterior algebras, and that the degrees of the generators can be determined by Dynkin diagram combinatorics. Besides a few well-studied examples such as full flag varieties and projective spaces, this class includes many flag varieties whose Witt rings were previously unknown, including many flag varieties of exceptional types. In particular, it includes all flag varieties of types G_2 and F_4. The results also extend to flag varieties over other algebraically closed fields. [ABSTRACT FROM AUTHOR]

We extend Nekrashevych's KK-duality for C^*-algebras of regular, recurrent, contracting self-similar group actions to regular, contracting self-similar groupoid actions on a graph, removing the recurrence condition entirely and generalising from a finite alphabet to a finite graph. More precisely, given a regular and contracting self-similar groupoid (G,E) acting faithfully on a finite directed graph E, we associate two C^*-algebras, \mathcal {O}(G,E) and \widehat {\mathcal {O}}(G,E), to it and prove that they are strongly Morita equivalent to the stable and unstable Ruelle C*-algebras of a Smale space arising from a Wieler solenoid of the self-similar limit space. That these algebras are Spanier-Whitehead dual in KK-theory follows from the general result for Ruelle algebras of irreducible Smale spaces proved by Kaminker, Putnam, and the last author. [ABSTRACT FROM AUTHOR]

We determine the defining equations of the Rees algebra and of the special fiber ring of the ideal of maximal minors of a 2\times n sparse matrix. We prove that their initial algebras are ladder determinantal rings. This allows us to show that the Rees algebra and the special fiber ring are Cohen-Macaulay domains, they are Koszul, they have rational singularities in characteristic zero and are F-rational in positive characteristic. [ABSTRACT FROM AUTHOR]

We construct Cartan subalgebras and hence groupoid models for classes of AH-algebras. Our results cover all AH-algebras whose building blocks have base spaces of dimension at most one as well as Villadsen algebras, and thus go beyond classifiable simple C*-algebras. [ABSTRACT FROM AUTHOR]

Let A be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded A-modules, providing weighted versions of Castelnuovo–Mumford regularity, Tor-regularity, Artin–Schelter regularity, and concavity. In some cases an invariant (such as Tor-regularity) that is infinite can be replaced with a weighted invariant that is finite, and several homological invariants of complexes can be expressed as weighted homological regularities. We prove a few weighted homological identities some of which unify different classical homological identities and produce interesting new ones. [ABSTRACT FROM AUTHOR]

We prove that large classes of algebras in the framework of root of unity quantum cluster algebras have the structures of maximal orders in central simple algebras and Cayley–Hamilton algebras in the sense of Procesi. We show that every root of unity upper quantum cluster algebra is a maximal order and obtain an explicit formula for its reduced trace. Under mild assumptions, inside each such algebra we construct a canonical central subalgebra isomorphic to the underlying upper cluster algebra, such that the pair is a Cayley–Hamilton algebra; its fully Azumaya locus is shown to contain a copy of the underlying cluster \mathcal {A}-variety. Both results are proved in the wider generality of intersections of mixed quantum tori over subcollections of seeds. Furthermore, we prove that all monomial subalgebras of root of unity quantum tori are Cayley–Hamilton algebras and classify those ones that are maximal orders. Arbitrary intersections of those over subsets of seeds are also proved to be Cayley–Hamilton algebras. Previous approaches to constructing maximal orders relied on filtration and homological methods. We use new methods based on cluster algebras. [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]

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 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]

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]

An FI- or an OI-module M over a corresponding noetherian polynomial algebra P} may be thought of as a sequence of compatible modules Mn over a polynomial ring Pn whose number of variables depends linearly on n. In order to study invariants of the modules Mn in dependence of n, an equivariant Hilbert series is introduced if M is graded. If M is also finitely generated, it is shown that this series is a rational function. Moreover, if this function is written in reduced form rather precise information about the irreducible factors of the denominator is obtained. This is key for applications. It follows that the Krull dimension of the modules Mn grows eventually linearly in n, whereas the multiplicity of Mn grows eventually exponentially in n. Moreover, for any fixed degree j, the vector space dimensions of the degree j components of Mn grow eventually polynomially in n. As a consequence, any graded Betti number of Mn in a fixed homological degree and a fixed internal degree grows eventually polynomially in n. Furthermore, evidence is obtained to support a conjecture that the Castelnuovo-Mumford regularity and the projective dimension of Mn both grow eventually linearly in n. It is also shown that modules M whose width n components Mn are eventually Artinian can be characterized by their equivariant Hilbert series. Using regular languages and finite automata, an algorithm for computing equivariant Hilbert series is presented. [ABSTRACT FROM AUTHOR]