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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]

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]

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]

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]

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]

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 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]

This work studies the zeros of slice functions over the algebra of dual quaternions and it comprises applications to the problem of factorizing motion polynomials. The class of slice functions over a real alternative *-algebra A was defined by Ghiloni and Perotti [Adv. Math. 226 (2011), pp. 1662–1691], extending the class of slice regular functions introduced by Gentili and Struppa [C. R. Math. Acad. Sci. Paris 342 (2006), pp. 741–744]. Both classes strictly include the polynomials over A. We focus on the case when A is the algebra of dual quaternions DH. The specific properties of this algebra allow a full characterization of the zero sets, which is not available over general real alternative *-algebras. This characterization sheds some light on the study of motion polynomials over DH, introduced by Hegedüs, Schicho, and Schröcker [Mech. Mach. Theory 69 (2013), pp. 42–152] for their relevance in mechanism science. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and nonexistence of global positive solutions to $u_t=\Delta\ln(1+u)$ in $\Omega \times (0, +\infty)$, \[\frac{\partial\ln(1+u)}{\partial n}=\sqrt{1+u}(\ln(1+u))^{\alpha} \quad\text{on} \partial \Omega \times (0, +\infty),] and $u(x, 0)=u_0(x)$ in $\Omega$. Here $\alpha\geq 0$ is a parameter, $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain. After pointing out the mistakes in {\em Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary}, SIAM J. Math. Anal. {\bf 24} (1993), 317--326, by N. Wolanski, which claims that, for $\Omega=B_R$ the ball of $\mathbb{R}^N$, the positive solution exists globally if and only if $\alpha\leq 1$, we reconsider the same problem in general bounded domain $\Omega$ and obtain that every positive solution exists globally if and only if $\alpha\leq {1/2}$. [ABSTRACT FROM AUTHOR]

Let K be a complete discretely valued field of characteristic 0 with residue field \kappa of characteristic p. Let n be the p-rank of κ , i.e., pn = [κ : κp]. It was proved by Parimala and Suresh [Invent. Math. 197 (2014), no. 1, pp. 215-235] that the Brauer p-dimension of K lies between n/2 and 2n. For n ≤ 3, we improve the upper bound to n + 1 and provide examples to show that our bound is sharp. For n ≤ 2, we also improve the lower bound to n. For general n, we construct a family of fields Kn with residue fields of p-rank n such that Kn admits a central simple algebra Dn of period p and index pn+1. Our sharp lower bounds for n ≤ 2 and upper bounds for n ≤ 3 in combination with the nature of these examples motivate us to conjecture that the Brauer p-dimension of such fields always lies between n and n + 1. [ABSTRACT FROM AUTHOR]

In this article we present a unification of the theory of algebraic, singular, topological and directional transition matrices by introducing the (generalized) transition matrix which encompasses each of the previous four. Some transition matrix existence results are presented as well as verification that each of the previous transition matrices are cases of the (generalized) transition matrix. Furthermore we address how applications of the previous transition matrices to the Conley index theory carry over to the (generalized) transition matrix. [ABSTRACT FROM AUTHOR]

Motivated by a result of Ky Fan in 1965, we establish a characterization of a left amenable F-algebra (which includes the group algebra and the Fourier algebra of a locally compact group and quantum group algebras, or more generally the predual algebra of a Hopf von Neumann algebra) in terms of a finite-dimensional invariant subspace property. This is done by first revealing a fixed point property for the semigroup of norm one positive linear functionals in the algebra. Our result answers an open question posted in Tokyo in 1993 by the first author. We also show that the left amenability of an ideal in an F-algebra may determine the left amenability of the algebra. [ABSTRACT FROM AUTHOR]