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In synchronization problems, the goal is to estimate elements of a group from noisy measurements of their ratios. A popular estimation method for synchronization is the spectral method. It extracts the group elements from eigenvectors of a block matrix formed from the measurements. The eigenvectors must be projected, or 'rounded', onto the group. The rounding procedures are constructed ad hoc and increasingly so when applied to synchronization problems over non-compact groups. In this paper, we develop a spectral approach to synchronization over the non-compact group |$\mathrm{SE}(3)$|⁠ , the group of rigid motions of |$\mathbb{R}^{3}$|⁠. We based our method on embedding |$\mathrm{SE}(3)$| into the algebra of dual quaternions, which has deep algebraic connections with the group |$\mathrm{SE}(3)$|⁠. These connections suggest a natural rounding procedure considerably more straightforward than the current state of the art for spectral |$\mathrm{SE}(3)$| synchronization, which uses a matrix embedding of |$\mathrm{SE}(3)$|⁠. We show by numerical experiments that our approach yields comparable results with the current state of the art in |$\mathrm{SE}(3)$| synchronization via the spectral method. Thus, our approach reaps the benefits of the dual quaternion embedding of |$\mathrm{SE}(3)$| while yielding estimators of similar quality. [ABSTRACT FROM AUTHOR]

One of the questions investigated in deformation theory is to determine to which algebras can a given associative algebra be deformed. In this paper we investigate a different but related question, namely: for a given associative finite-dimensional |${\mathbb{C}}$| -algebra |$A$|⁠ , find algebras |$N$|⁠ , which can be deformed to |$A$|⁠. We develop a simple method that produces associative and flat deformations to investigate this question. As an application of this method we answer a question of Michael Wemyss about deformations of contraction algebras. [ABSTRACT FROM AUTHOR]

This erratum corrects the proof of the main result of a mathematical research paper. The proof contained errors, specifically in Lemmas 5.1 and 5.3. The authors provide a corrected proof for Theorem 5.2, which involves calculating a Gröbner basis for an ideal. The corrected proof is based on the concept of Motzkin paths and introduces the notion of exterior algebra elements. The authors conclude with a corollary stating that a specific set of elements forms a Gröbner basis for the ideal. [Extracted from the article]

Throughout its long history, mathematics has involved the use ofsystems of written signs, most notably, diagrams in Euclidean geometry and formulae in the symbolic language of arithmetic and algebra in the mathematics of Descartes, Euler, and others. Such systems of signs, I argue, enable one to embody chains of mathematical reasoning. I then show that, properly understood, Frege’s Begriffsschrift or concept-script similarly enables one to write mathematical reasoning. Much as a demonstration in Euclid or in early modern algebra does, a proof in Frege’s concept-script shows how it goes. [ABSTRACT FROM PUBLISHER]

In this article, we continue the study of a certain family of 2-Calabi–Yau tilted algebras, called dimer tree algebras. The terminology comes from the fact that these algebras can also be realized as quotients of dimer algebras on a disk. They are defined by a quiver with potential whose dual graph is a tree, and they are generally of wild representation type. Given such an algebra |$B$|⁠ , we construct a polygon |$\mathcal {S}$| with a checkerboard pattern in its interior, which defines a category |$\text {Diag}(\mathcal {S})$|⁠. The indecomposable objects of |$\text {Diag}(\mathcal {S})$| are the 2-diagonals in |$\mathcal {S}$|⁠ , and its morphisms are certain pivoting moves between the 2-diagonals. We prove that the category |$\text {Diag}(\mathcal {S})$| is equivalent to the stable syzygy category of the algebra |$B$|⁠. This result was conjectured by the authors in an earlier paper, where it was proved in the special case where every chordless cycle is of length three. As a consequence, we conclude that the number of indecomposable syzygies is finite, and moreover the syzygy category is equivalent to the 2-cluster category of type |$\mathbb {A}$|⁠. In addition, we obtain an explicit description of the projective resolutions, which are periodic. Finally, the number of vertices of the polygon |$\mathcal {S}$| is a derived invariant and a singular invariant for dimer tree algebras, which can be easily computed form the quiver. [ABSTRACT FROM AUTHOR]

This paper presents results of a study aimed at describing and discussing evidence/features of advanced algebraic thinking processes. To achieve these objectives, we analysed the written production of students enrolled in engineering courses working on mathematical modelling tasks related to differential equations. Our guiding question was as follows: how are students' advanced algebraic thinking processes expressed (or not) in modelling activities that involve differential equations? Additionally, we presented the concept of advanced algebraic thinking as an extension of the (elementary) algebraic thinking notion and as a relevant part of advanced mathematical thinking. We included written records of two models that were produced and used in two classes and concluded that practically all groups showed processes of symbolic representation and representation translation. We also observed that around half of the groups presented suitable algebraic synthetization and development processes to solve the model equation. Another finding was that none of the groups applied generalization processes. Therefore, not all processes involved in advanced algebraic thinking were developed. [ABSTRACT FROM AUTHOR]

In this paper, we prove a general result saying that under certain hypothesis an embedding of an affine vertex algebra into an affine |$W$| –algebra is conformal if and only if their central charges coincide. This result extends our previous result obtained in the case of minimal affine |$W$| -algebras [ 3 ]. We also find a sufficient condition showing that certain conformal levels are collapsing. This new condition enables us to find some levels |$k$| where |$W_{k}(sl(N), x, f)$| collapses to its affine part when |$f$| is of hook or rectangular type. Our methods can be applied to non-admissible levels. In particular, we prove Creutzig's conjecture [ 18 ] on the conformal embedding in the hook type |$W$| -algebra |$W_{k}(sl(n+m), x, f_{m,n})$| of its affine vertex subalgebra. Quite surprisingly, the problem of showing that certain conformal levels are not collapsing turns out to be very difficult. In the cases when |$k$| is admissible and conformal, we prove that |$W_{k}(sl(n+m), x, f_{m,n})$| is not collapsing. Then, by generalizing the results on semi-simplicity of conformal embeddings from [ 2 ], [ 5 ], we find many cases in which |$W_{k}(sl(n+m), x, f_{m,n})$| is semi-simple as a module for its affine subalgebra at conformal level and we provide explicit decompositions. [ABSTRACT FROM AUTHOR]

The center conjecture for the cyclotomic KLR algebras |$\mathscr{R}_{\beta }^{\Lambda }$| asserts that the center of |$\mathscr{R}_{\beta }^{\Lambda }$| consists of symmetric elements in its KLR |$x$| and |$e(\nu)$| generators. In this paper, we show that this conjecture is equivalent to the injectivity of some natural map |$\overline{\iota }_{\beta }^{\Lambda ,i}$| from the cocenter of |$\mathscr{R}_{\beta }^{\Lambda }$| to the cocenter of |$\mathscr{R}_{\beta }^{\Lambda +\Lambda _{i}}$| for all |$i\in I$| and |$\Lambda \in P^{+}$|⁠. We prove that the map |$\overline{\iota }_{\beta }^{\Lambda ,i}$| is given by multiplication with a center element |$z(i,\beta)\in \mathscr{R}_{\beta }^{\Lambda +\Lambda _{i}}$| and we explicitly calculate the element |$z(i,\beta)$| in terms of the KLR |$x$| and |$e(\nu)$| generators. We present explicit monomial bases for certain bi-weight spaces of the defining ideal of |$\mathscr{R}_{\beta }^{\Lambda }$|⁠. For |$\beta =\sum _{j=1}^{n}\alpha _{i_{j}}$| with |$\alpha _{i_{1}},\cdots , \alpha _{i_{n}}$| pairwise distinct, we construct an explicit monomial basis of |$\mathscr{R}_{\beta }^{\Lambda }$|⁠ , prove the map |$\overline{\iota }_{\beta }^{\Lambda ,i}$| is injective, and thus verify the center conjecture for these |$\mathscr{R}_{\beta }^{\Lambda }$|⁠. [ABSTRACT FROM AUTHOR]

In this paper, we give an alternative construction of a certain class of deformed double current algebras. These algebras are deformations of |$ U(\textrm {End}(\Bbbk ^r)[x,y]) $| and they were initially defined and studied by N. Guay in his papers. Here, we construct them as algebras of endomorphisms in Deligne category. We do this by taking an ultraproduct of spherical subalgebras of the extended Cherednik algebras of finite rank. [ABSTRACT FROM AUTHOR]

In this paper, we give a new geometric condition in terms of measured asymptotic expanders to ensure rigidity of Roe algebras. Consequently, we obtain the rigidity for all bounded geometry spaces that coarsely embed into some |$L^p$| -space for |$p\in [1,\infty)$|⁠. Moreover, we also verify rigidity for the box spaces constructed by Arzhantseva–Tessera and Delabie–Khukhro even though they do not coarsely embed into any |$L^p$| -space. The key step in our proof of rigidity is showing that a block-rank-one (ghost) projection on a sparse space |$X$| belongs to the Roe algebra |$C^{\ast }(X)$| if and only if |$X$| consists of (ghostly) measured asymptotic expanders. As a by-product, we also deduce that ghostly measured asymptotic expanders are new sources of counterexamples to the coarse Baum–Connes conjecture. [ABSTRACT FROM AUTHOR]

Associated with each finite subgroup |$\Gamma $| of |${\textrm {SL}}_2({\mathbb {C}})$| there is a family of noncommutative algebras |$O_\tau (\Gamma)$| quantizing |${\mathbb {C}}^2/\!\!/\Gamma $|⁠. Let |$G_\Gamma $| be the group of |$\Gamma $| -equivariant automorphisms of |$O_\tau $|⁠. In [ 16 ], one of the authors defined and studied a natural action of |$G_\Gamma $| on certain quiver varieties associated with |$\Gamma $|⁠. He established a |$G_\Gamma $| -equivariant bijective correspondence between quiver varieties and the space of isomorphism classes of |$O_\tau $| -ideals. In this paper we prove that the action of |$G_\Gamma $| on the quiver variety is transitive when |$\Gamma $| is a cyclic group. This generalizes an earlier result due to Berest and Wilson who showed the transitivity of the automorphism group of the 1st Weyl algebra on the Calogero–Moser spaces. Our result has two important implications. First, it confirms the Bocklandt–Le Bruyn conjecture for cyclic quiver varieties. Second, it will be used to give a complete classification of algebras Morita equivalent to |$O_\tau (\Gamma),$| thus answering the question of Hodges. At the end of the introduction we explain why the result of this paper does not extend when |$\Gamma $| is not cyclic. [ABSTRACT FROM AUTHOR]

We prove the following two results for a given uniformly locally finite metric space with Yu's property A: 1. The group of outer automorphisms of its uniform Roe algebra is isomorphic to its group of bijective coarse equivalences modulo closeness. 2. The group of outer automorphisms of its Roe algebra is isomorphic to its group of coarse equivalences modulo closeness. The main difficulty lies in the latter. To prove that, we obtain several uniform approximability results for maps between Roe algebras and use them to obtain a theorem about the "uniqueness" of Cartan masas of Roe algebras. We finish the paper with several applications of the results above to concrete metric spaces. [ABSTRACT FROM AUTHOR]

The original paper did not clearly state that Pierre-Guy Plamondon and Toshiya Yurikusa were the author of the paper, but that Bernhard Keller was the author of the appendix. One digram originally appeared as Graph It should have appeared as Graph In reference 12 "Yildirim" should have appeared as "Yildirim". [Extracted from the article]

Free algebras are always free as modules over the base ring in classical algebra. In equivariant algebra, free incomplete Tambara functors play the role of free algebras and Mackey functors play the role of modules. Surprisingly, free incomplete Tambara functors often fail to be free as Mackey functors. In this paper, we determine for all finite groups conditions under which a free incomplete Tambara functor is free as a Mackey functor. For solvable groups, we show that a free incomplete Tambara functor is flat as a Mackey functor precisely when these conditions hold. Our results imply that free incomplete Tambara functors are almost never flat as Mackey functors. However, we show that after suitable localizations, free incomplete Tambara functors are always free as Mackey functors. [ABSTRACT FROM AUTHOR]

This article gives a class of Nullstellensätze for noncommutative polynomials. The singularity set of a noncommutative polynomial |$f=f(x_1,\dots ,x_g)$| is |$\mathscr{Z}(\,f)=(\mathscr{Z}_n(\,f))_n$|⁠ , where |$\mathscr{Z}_n(\,f)=\{X \in{\operatorname{M}}_{n}({\mathbb{C}})^g \colon \det f(X) = 0\}.$| The 1st main theorem of this article shows that the irreducible factors of |$f$| are in a natural bijective correspondence with irreducible components of |$\mathscr{Z}_n(\,f)$| for every sufficiently large |$n$|⁠. With each polynomial |$h$| in |$x$| and |$x^*$| one also associates its real singularity set |$\mathscr{Z}^{{\operatorname{re}}}(h)=\{X\colon \det h(X,X^*)=0\}$|⁠. A polynomial |$f$| that depends on |$x$| alone (no |$x^*$| variables) will be called analytic. The main Nullstellensatz proved here is as follows: for analytic |$f$| but for |$h$| dependent on possibly both |$x$| and |$x^*$|⁠ , the containment |$\mathscr{Z}(\,f) \subseteq \mathscr{Z}^{{\operatorname{re}}} (h)$| is equivalent to each factor of |$f$| being "stably associated" to a factor of |$h$| or of |$h^*$|⁠. For perspective, classical Hilbert-type Nullstellensätze typically apply only to analytic polynomials |$f,h $|⁠ , while real Nullstellensätze typically require adjusting the functions by sums of squares of polynomials (sos). Since the above "algebraic certificate" does not involve a sos, it seems justified to think of this as the natural determinantal Hilbert Nullstellensatz. An earlier paper of the authors (Adv. Math. 331 (2018): 589–626) obtained such a theorem for special classes of analytic polynomials |$f$| and |$h$|⁠. This paper requires few hypotheses and hopefully brings this type of Nullstellensatz to near final form. Finally, the paper gives a Nullstellensatz for zeros |${\mathcal{V}}(\,f)=\{X\colon f(X,X^*)=0\}$| of a hermitian polynomial |$f$|⁠ , leading to a strong Positivstellensatz for quadratic free semialgebraic sets by the use of a slack variable. [ABSTRACT FROM AUTHOR]

This paper is a contribution to the construction of non-semisimple modular categories. We establish when Müger centralizers inside non-semisimple modular categories are also modular. As a consequence, we obtain conditions under which relative monoidal centers give (non-semisimple) modular categories, and we also show that examples include representation categories of small quantum groups. We further derive conditions under which representations of more general quantum groups, braided Drinfeld doubles of Nichols algebras of diagonal type, give (non-semisimple) modular categories. [ABSTRACT FROM AUTHOR]

We study trivial multiple zeta values in Tate algebras. These are particular examples of the multiple zeta values in Tate algebras introduced by the second author. If the number of variables involved is "not large" in a way that is made precise in the paper, we can endow the set of trivial multiple zeta values with a structure of module over a non-commutative polynomial ring with coefficients in the rational fraction field over |${\mathbb{F}}_q$|⁠. We determine the structure of this module in terms of generators and we show how in many cases, this is sufficient for the detection of linear relations between Thakur's multiple zeta values. [ABSTRACT FROM AUTHOR]

Gross–Hacking–Keel–Kontsevich [ 13 ] discuss compactifications of cluster varieties from positive subsets in the real tropicalization of the mirror. To be more precise, let |${\mathfrak {D}}$| be the scattering diagram of a cluster variety |$V$| (of either type– |${\mathcal {A}}$| or |${\mathcal {X}}$|⁠), and let |$S$| be a closed subset of |$\left (V^\vee \right)^{\textrm {trop}} \left ({\mathbb {R}}\right)$| —the ambient space of |${\mathfrak {D}}$|⁠. The set |$S$| is positive if the theta functions corresponding to the integral points of |$S$| and its |${\mathbb {N}}$| -dilations define an |${\mathbb {N}}$| -graded subalgebra of |$\Gamma (V, \mathcal {O}_V){ [x]}$|⁠. In particular, a positive set |$S$| defines a compactification of |$V$| through a Proj construction applied to the corresponding |${\mathbb {N}}$| -graded algebra. In this paper, we give a natural convexity notion for subsets of |$\left (V^\vee \right)^{\textrm {trop}} \left ({\mathbb {R}}\right)$|⁠ , called broken line convexity , and show that a set is positive if and only if it is broken line convex. The combinatorial criterion of broken line convexity provides a tractable way to construct positive subsets of |$\left (V^\vee \right)^{\textrm {trop}} \left ({\mathbb {R}}\right)$| or to check positivity of a given subset. [ABSTRACT FROM AUTHOR]

Let |$M$| be a compact 3-manifold and |$\Gamma =\pi _1(M)$|⁠. Work by Thurston and Culler–Shalen established the |${\operatorname{\textrm{SL}}}_2({\mathbb{C}})$| character variety |$X(\Gamma)$| as fundamental tool in the study of the geometry and topology of |$M$|⁠. This is particularly the case when |$M$| is the exterior of a hyperbolic knot |$K$| in |$S^3$|⁠. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand algebraic and number theoretic properties of the so-called canonical component of |$X(\Gamma)$|⁠ , as well as distinguished points on the canonical component, when |$\Gamma $| is a knot group. In particular, we study how the theory of quaternion Azumaya algebras can be used to obtain algebraic and arithmetic information about Dehn surgeries, and perhaps of most interest, to construct new knot invariants that lie in the Brauer groups of curves over number fields. [ABSTRACT FROM AUTHOR]

Coset constructions of |${{\mathcal{W}}}$| -algebras have many applications and were recently given for principal |${{\mathcal{W}}}$| -algebras of |$A$|⁠ , |$D$|⁠ , and |$E$| types by Arakawa together with the 1st and 3rd authors. In this paper, we give coset constructions of the large and small |$N=4$| superconformal algebras, which are the minimal |${{\mathcal{W}}}$| -algebras of |${{\mathfrak{d}}}(2,1;a)$| and |${{\mathfrak{p}}}{{\mathfrak{s}}}{{\mathfrak{l}}}(2|2)$|⁠ , respectively. From these realizations, one finds a remarkable connection between the large |$N=4$| algebra and the diagonal coset |$C^{k_1, k_2} = \textrm{Com}(V^{k_1+k_2}({{\mathfrak{s}}}{{\mathfrak{l}}}_2), V^{k_1}({{\mathfrak{s}}}{{\mathfrak{l}}}_2) \otimes V^{k_2}({{\mathfrak{s}}}{{\mathfrak{l}}}_2))$|⁠ , namely, as two-parameter vertex algebras, |$C^{k_1, k_2}$| coincides with the coset of the large |$N=4$| algebra by its affine subalgebra. We also show that at special points in the parameter space, the simple quotients of these cosets are isomorphic to various |${{\mathcal{W}}}$| -algebras. As a corollary, we give new examples of strongly rational principal |${{\mathcal{W}}}$| -algebras of type |$C$| at degenerate admissible levels. [ABSTRACT FROM AUTHOR]

This paper provides a definitive answer, based on considerations derived from first-order logic, to the question regarding the status of elementary geometry, whether elementary geometry can be reduced to algebra. The answer we arrive at is negative, and is based on a series of structural questions that can be asked only inside the geometric formal theory, as well as the consideration of reverse geometry, which is the art of finding minimal axiom systems strong enough to prove certain geometrical theorems, given that there are no algebraic structures that one could associate with those minimal axiom systems. [ABSTRACT FROM AUTHOR]

Let |$q = e^{i \theta } \in \mathbb{T}$| (where |$\theta \in \mathbb{R}$|⁠), and let |$u,v$| be |$q$| -commuting unitaries, that is, |$u$| and |$v$| are unitaries such that |$vu = quv$|⁠. In this paper, we find the optimal constant |$c = c_{\theta }$| such that |$u,v$| can be dilated to a pair of operators |$c U, c V$|⁠ , where |$U$| and |$V$| are commuting unitaries. We show that $$\begin{equation*} c_{\theta} = \frac{4}{\|u_{\theta}+u_{\theta}^*+v_{\theta}+v_{\theta}^*\|}, \end{equation*}$$ where |$u_{\theta }, v_{\theta }$| are the universal |$q$| -commuting pair of unitaries, and we give numerical estimates for the above quantity. In the course of our proof, we also consider dilating |$q$| -commuting unitaries to scalar multiples of |$q^{\prime}$| -commuting unitaries. The techniques that we develop allow us to give new and simple "dilation theoretic" proofs of well-known results regarding the continuity of the field of rotations algebras. In particular, for the so-called "almost Mathieu operator" |$h_{\theta } = u_{\theta }+u_{\theta }^*+v_{\theta }+v_{\theta }^*$|⁠ , we recover the fact that the norm |$\|h_{\theta }\|$| is a Lipschitz continuous function of |$\theta $|⁠ , as well as the result that the spectrum |$\sigma (h_{\theta })$| is a |$\frac{1}{2}$| -Hölder continuous function in |$\theta $| with respect to the Hausdorff metric. In fact, we obtain this Hölder continuity of the spectrum for every self-adjoint *-polynomial |$p(u_{\theta },v_{\theta })$|⁠ , which in turn endows the rotation algebras with the natural structure of a continuous field of C*-algebras. [ABSTRACT FROM AUTHOR]

This paper introduces methods for classifying actions of finite-dimensional Hopf algebras on path algebras of quivers and more generally on tensor algebras |$T_B(V)$| where |$B$| is semisimple. We work within the broader framework of finite (multi-)tensor categories |$\mathcal{C}$|⁠ , classifying tensor algebras in |$\mathcal{C}$| in terms of |$\mathcal{C}$| -module categories. We obtain two classification results for actions of semisimple Hopf algebras: the first for actions that preserve the ascending filtration on tensor algebras and the second for actions that preserve the descending filtration on completed tensor algebras. Extending to more general fusion categories, we illustrate our classification result for tensor algebras in the pointed fusion categories |$\textsf{Vec}_{G}^{\omega }$| and in group-theoretical fusion categories, especially for the representation category of the Kac–Paljutkin Hopf algebra. [ABSTRACT FROM AUTHOR]

Coset constructions of |${{\mathcal{W}}}$| -algebras have many applications and were recently given for principal |${{\mathcal{W}}}$| -algebras of |$A$|⁠ , |$D$|⁠ , and |$E$| types by Arakawa together with the 1st and 3rd authors. In this paper, we give coset constructions of the large and small |$N=4$| superconformal algebras, which are the minimal |${{\mathcal{W}}}$| -algebras of |${{\mathfrak{d}}}(2,1;a)$| and |${{\mathfrak{p}}}{{\mathfrak{s}}}{{\mathfrak{l}}}(2|2)$|⁠ , respectively. From these realizations, one finds a remarkable connection between the large |$N=4$| algebra and the diagonal coset |$C^{k_1, k_2} = \textrm{Com}(V^{k_1+k_2}({{\mathfrak{s}}}{{\mathfrak{l}}}_2), V^{k_1}({{\mathfrak{s}}}{{\mathfrak{l}}}_2) \otimes V^{k_2}({{\mathfrak{s}}}{{\mathfrak{l}}}_2))$|⁠ , namely, as two-parameter vertex algebras, |$C^{k_1, k_2}$| coincides with the coset of the large |$N=4$| algebra by its affine subalgebra. We also show that at special points in the parameter space, the simple quotients of these cosets are isomorphic to various |${{\mathcal{W}}}$| -algebras. As a corollary, we give new examples of strongly rational principal |${{\mathcal{W}}}$| -algebras of type |$C$| at degenerate admissible levels. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a notion of a self-similar action of a group |$G$| on a |$k$| -graph |$\Lambda $| and associate it a universal C |$^\ast $| -algebra |${{\mathcal{O}}}_{G,\Lambda }$|⁠. We prove that |${{\mathcal{O}}}_{G,\Lambda }$| can be realized as the Cuntz–Pimsner algebra of a product system. If |$G$| is amenable and the action is pseudo free, then |${{\mathcal{O}}}_{G,\Lambda }$| is shown to be isomorphic to a "path-like" groupoid C |$^\ast $| -algebra. This facilitates studying the properties of |${{\mathcal{O}}}_{G,\Lambda }$|⁠. We show that |${{\mathcal{O}}}_{G,\Lambda }$| is always nuclear and satisfies the universal coefficient theorem; we characterize the simplicity of |${{\mathcal{O}}}_{G,\Lambda }$| in terms of the underlying action, and we prove that, whenever |${{\mathcal{O}}}_{G,\Lambda }$| is simple, there is a dichotomy: it is either stably finite or purely infinite, depending on whether |$\Lambda $| has nonzero graph traces or not. Our main results generalize the recent work of Exel and Pardo on self-similar graphs. [ABSTRACT FROM AUTHOR]

We show that associated with any |$n$| -dimensional Bott–Samelson variety of a complex semi-simple Lie group |$G$|⁠ , one has |$2^n$| Poisson brackets on the polynomial algebra |$A={\mathbb{C}}[z_1, \ldots , z_n]$|⁠ , each an iterated Poisson Ore extension and one of them a symmetric Poisson Cauchon–Goodearl–Letzter (CGL) extension in the sense of Goodearl–Yakimov. We express the Poisson brackets in terms of root strings and structure constants of the Lie algebra of |$G$|⁠. It follows that the coordinate rings of all generalized Bruhat cells have presentations as symmetric Poisson CGL extensions. The paper establishes the foundation on generalized Bruhat cells and sets the stage for their applications to integrable systems, cluster algebras, total positivity, and toric degenerations of Poisson varieties, some of which are discussed in the Introduction. [ABSTRACT FROM AUTHOR]

Given a module |$M$| for the algebra |${\mathcal{D}}_{\mathtt{q}}(G)$| of quantum differential operators on |$G$|⁠ , and a positive integer |$n$|⁠ , we may equip the space |$F_n^G(M)$| of invariant tensors in |$V^{\otimes n}\otimes M$|⁠ , with an action of the double affine Hecke algebra of type |$A_{n-1}$|⁠. Here |$G= SL_N$| or |$GL_N$|⁠ , and |$V$| is the |$N$| -dimensional defining representation of |$G$|⁠. In this paper, we take |$M$| to be the basic |${\mathcal{D}}_{\mathtt{q}}(G)$| -module, that is, the quantized coordinate algebra |$M= {\mathcal{O}}_{\mathtt{q}}(G)$|⁠. We describe a weight basis for |$F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$| combinatorially in terms of walks in the type |$A$| weight lattice, and standard periodic tableaux, and subsequently identify |$F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$| with the irreducible "rectangular representation" of height |$N$| of the double affine Hecke algebra. [ABSTRACT FROM AUTHOR]

In this paper, we study two-sided tilting complexes of preprojective algebras of Dynkin type. We construct the most fundamental class of two-sided tilting complexes, which has a group structure by derived tensor products and induces a group of auto-equivalences of the derived category. We show that the group structure of the two-sided tilting complexes is isomorphic to the braid group of the corresponding folded graph. Moreover, we show that these two-sided tilting complexes induce tilting mutation and any tilting complex is given as the derived tensor products of them. Using these results, we determine the derived Picard group of preprojective algebras for type |$A$| and |$D$|⁠. [ABSTRACT FROM AUTHOR]

In this paper, we define the |$K$| -theoretic Hall algebra for dimension |$0$| coherent sheaves on a smooth projective surface, prove that the algebra is associative, and construct a homomorphism to a shuffle algebra introduced by Negut [ 10 ]. [ABSTRACT FROM AUTHOR]

In this paper, we initiate the generalization of the operadic calculus that governs the properties of homotopy algebras to a properadic calculus that governs the properties of homotopy gebras over a properad. In this first article of a series, we generalize the seminal notion of |${\infty }$| -morphisms and the ubiquitous homotopy transfer theorem. As an application, we recover the homotopy properties of involutive Lie bialgebras developed by Cieliebak–Fukaya–Latschev and we produce new explicit formulas. [ABSTRACT FROM AUTHOR]

Let |$A$| be a simple separable nuclear C |$^*$| -algebra with a unique tracial state and no unbounded traces, and let |$\alpha $| be a strongly outer action of a finite group |$G$| on |$A$|⁠. In this paper, we show that |$\alpha \otimes \textrm{id}$| on |$A\otimes \mathcal{W}$| has the Rohlin property where |$\mathcal{W}$| is the Razak–Jacelon algebra. Combing this result with the recent classification results and our previous result, we see that such actions are unique up to conjugacy. [ABSTRACT FROM AUTHOR]

In this paper we study derived equivalences for symplectic reflection algebras. We establish a version of the derived localization theorem between categories of modules over these algebras and categories of coherent sheaves over quantizations of |$\mathbb{Q}$| -factorial terminalizations of the symplectic quotient singularities. To do this we construct a Procesi sheaf on the terminalization and show that the quantizations of the terminalization are simple sheaves of algebras. We will also sketch some applications to the generalized Bernstein inequality and to perversity of wall crossing functors. [ABSTRACT FROM AUTHOR]

Let |$A$| be the path algebra of a Dynkin quiver over a finite field, and let |$C_1(\mathscr{P})$| be the category of 1-cyclic complexes of projective |$A$| -modules. In the present paper, we give a PBW-basis and a minimal set of generators for the Hall algebra |${\mathcal{H}}\,(C_1(\mathscr{P}))$| of |$C_1(\mathscr{P})$|⁠. Using this PBW-basis, we firstly prove the degenerate Hall algebra of |$C_1(\mathscr{P})$| is the universal enveloping algebra of the Lie algebra spanned by all indecomposable objects. Secondly, we calculate the relations among the generators in |${\mathcal{H}}\,(C_1(\mathscr{P}))$|⁠ , and obtain quantum Serre relations in a quotient of certain twisted version of |${\mathcal{H}}\,(C_1(\mathscr{P}))$|⁠. Moreover, we establish relations between the degenerate Hall algebra, twisted Hall algebra of |$A$| and those of |$C_1(\mathscr{P})$|⁠ , respectively. [ABSTRACT FROM AUTHOR]

A noncommutative deformation of a quadric surface is usually described by a three-dimensional cubic Artin–Schelter regular algebra. In this paper we show that for such an algebra its bounded derived category embeds into the bounded derived category of a commutative deformation of the Hilbert scheme of two points on the quadric. This is the second example in support of a conjecture by Orlov. Based on this example we formulate an infinitesimal version of the conjecture and provide some evidence in the case of smooth projective surfaces. [ABSTRACT FROM AUTHOR]

Let |$\textrm{coh}\ \mathbb{X}$| be the category of coherent sheaves over a weighted projective line |$\mathbb{X}$| and let |$D^b(\textrm{coh}\ \mathbb{X})$| be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in |$D^b(\textrm{coh}\ \mathbb{X})$| attached to certain line bundles. As applications, we first show that these mutation functors give rise to simple reflections for the Weyl group of the star-shaped quiver |${Q}$| associated with |$\mathbb{X}$|⁠. By further dealing with the Ringel–Hall algebra of |$\mathbb{X}$|⁠ , we show that these functors provide a realization for Tits' automorphisms of the Kac–Moody algebra |${\mathfrak g}_{Q}$| associated with |${Q}$|⁠ , as well as for Lusztig's symmetries of the quantum enveloping algebra of |${\mathfrak g}_{Q}$|⁠. [ABSTRACT FROM AUTHOR]

The goal of this paper is two-fold. First, we write down the semi-infinite Plücker relations, describing the Drinfeld–Plücker embedding of the (formal version of) semi-infinite flag varieties in type A. Second, we study the homogeneous coordinate ring, that is, the quotient by the ideal generated by the semi-infinite Plücker relations. We establish the isomorphism with the algebra of dual global Weyl modules and derive a new character formula. [ABSTRACT FROM AUTHOR]

Let |${\boldsymbol{\Sigma }}=(\Sigma ,\mathbb{M},\mathbb{O})$| be either an unpunctured surface with marked points and order-2 orbifold points or a once-punctured closed surface with order-2 orbifold points. For each pair |$(\tau ,\omega)$| consisting of a triangulation |$\tau $| of |${\boldsymbol{\Sigma }}$| and a function |$\omega :\mathbb{O}\rightarrow \{1,4\}$|⁠ , we define a chain complex |$C_\bullet (\tau , \omega)$| with coefficients in |$\mathbb{F}_2=\mathbb{Z}/2\mathbb{Z}$|⁠. Given |${\boldsymbol{\Sigma }}$| and |$\omega $|⁠ , we define a colored triangulation of |${\boldsymbol{\Sigma }_\omega }=(\Sigma ,\mathbb{M},\mathbb{O},\omega)$| to be a pair |$(\tau ,\xi)$| consisting of a triangulation of |${\boldsymbol{\Sigma }}$| and a 1-cocycle in the cochain complex that is dual to |$C_\bullet (\tau , \omega)$|⁠ ; the combinatorial notion of colored flip of colored triangulations is then defined as a refinement of the notion of flip of triangulations. Our main construction associates to each colored triangulation a species and a potential, and our main result shows that colored triangulations related by a flip have species with potentials (SPs) related by the corresponding SP-mutation as defined in [ 25 ]. We define the flip graph of |${\boldsymbol{\Sigma }_\omega }$| as the graph whose vertices are the pairs |$(\tau ,x)$| consisting of a triangulation |$\tau $| and a cohomology class |$x\in H^1(C^\bullet (\tau , \omega))$|⁠ , with an edge connecting two such pairs, |$(\tau ,x)$| and |$(\sigma ,z),$| if and only if there exist 1-cocycles |$\xi \in x$| and |$\zeta \in z$| such that |$(\tau ,\xi)$| and |$(\sigma ,\zeta)$| are colored triangulations related by a colored flip; then we prove that this flip graph is always disconnected provided the underlying surface |$\Sigma $| is not contractible. In the absence of punctures, we show that the Jacobian algebras of the SPs constructed are finite-dimensional and that whenever two colored triangulations have the same underlying triangulation, the Jacobian algebras of their associated SPs are isomorphic if and only if the underlying 1-cocycles have the same cohomology class. We also give a full classification of the nondegenerate SPs one can associate to any given pair |$(\tau ,\omega)$| over cyclic Galois extensions with certain roots of unity. The species constructed here are species realizations of the |$2^{|\mathbb{O}|}$| skew-symmetrizable matrices that Felikson–Shapiro–Tumarkin associated in [ 17 ] to any given triangulation of |${\boldsymbol{\Sigma }}$|⁠. In the prequel [ 25 ] of this paper we constructed a species realization of only one of these matrices, but therein we allowed the presence of arbitrarily many punctures. [ABSTRACT FROM AUTHOR]

We study the structure and representation theory of affine wreath product algebras and their cyclotomic quotients. These algebras, which appear naturally in Heisenberg categorification, simultaneously unify and generalize many important algebras appearing in the literature. In particular, special cases include degenerate affine Hecke algebras, affine Sergeev algebras (degenerate affine Hecke–Clifford algebras), and wreath Hecke algebras. In some cases, specializing the results of the current paper recovers known results, but with unified and simplified proofs. In other cases, we obtain new results, including proofs of two open conjectures of Kleshchev and Muth. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and study a large class of g-module algebras that we call factorizable by generalizing the Gauss factorization of square or rectangular matrices. This class includes coordinate algebras of corresponding reductive groups G , their parabolic subgroups, basic affine spaces, and many others. It turns out that products of factorizable algebras are also factorizable and it is easy to create a factorizable algebra out of virtually any g-module algebra. We also have quantum versions of all these constructions in the category of Uq(g)-module algebras. Quite surprisingly, our quantum factorizable algebras are naturally acted on by the quantized enveloping algebra Uq(g*) of the dual Lie bialgebra g* of g⁠. [ABSTRACT FROM AUTHOR]

Equivariant map algebras are Lie algebras of algebraic maps from a scheme (or algebraic variety) to a target finite-dimensional Lie algebra (in the case of the current paper, we assume the latter is a simple Lie algebra) that are equivariant with respect to the action of a finite group. In the first part of this paper, we define global Weyl modules for equivariant map algebras satisfying a mild assumption. We then identify a commutative algebra Aґλ that acts naturally on the global Weyl modules, which leads to a Weyl functor from the category of Aґλ -modules to the category of modules for the equivariant map algebra in question. These definitions extend the ones previously given for generalized current algebras (i.e., untwisted map algebras) and twisted loop algebras. In the second part of the paper, we restrict our attention to equivariant map algebras where the group involved is abelian, acts on the target Lie algebra by diagram automorphisms, and freely on (the set of rational points of) the scheme. Under these additional assumptions, we prove that Aґλ is finitely generated and the global Weyl module is a finitely generated Aґλ -module. We also define local Weyl modules via the Weyl functor and prove that these coincide with the local Weyl modules defined directly in [18]. Finally, we show that Aґλ is the algebra of coinvariants of the analogous algebra in the untwisted case. [ABSTRACT FROM AUTHOR]

In our recent paper Shoikhet [ 47 ], a version of the "generalized Deligne conjecture" for abelian $$n$$ -fold monoidal categories is proven. For $$n=1,$$ this result says that, given an abelian monoidal $$\Bbbk$$ -linear category $$\mathcal{A}$$ with unit $$e$$, $${\operatorname{char}}\Bbbk=0$$, $${\mathrm{RHom}}_{\mathcal{A}}(e,e)$$ is the first component of a Leinster 1-monoid in $${{\mathcal{A}lg}}(\Bbbk)$$ (provided a rather mild condition on the monoidal and the abelian structures in $$\mathcal{A}$$, called homotopy compatibility, is fulfilled). In this article, we introduce a new concept of a graded Leinster monoid. We show that the Leinster monoid in $${{\mathcal{A}lg}}(\Bbbk)$$, constructed by a monoidal $$\Bbbk$$ -linear abelian category in Shoikhet [ 47 ], is graded. We construct a functor, assigning an algebra over the chain operad $$C_{{{{\bullet}}}}(E_2,\Bbbk)$$, to a graded Leinster 1-monoid in $${{\mathcal{A}lg}}(\Bbbk)$$, which respects the weak equivalences. Consequently, this article together with Shoikhet [ 47 ] provides a complete proof of the "generalized Deligne conjecture" for 1-monoidal abelian categories, in the form most accessible for applications to deformation theory (such as Tamarkin's proof of the Kontsevich formality). [ABSTRACT FROM AUTHOR]

It is believed [ 13, 19, Section 4.1 ] that, among the coefficients entering Kontsevich's formality quasi-isomorphism [ 18 ], there are irrational (possibly even transcendental) numbers. In this article, we prove that a formality quasi-isomorphism for Hochschild cochains of a polynomial algebra over $${\mathbb Q}$$ can be constructed recursively. The proof that the proposed recursive algorithm works, is based on the existence of formality quasi-isomorphism over $${\mathbb R}$$. However, the algorithm requires no explicit knowledge of the coefficients entering Kontsevich's construction. Although this algorithm completely bypasses Tamarkin's approach [ 14, 23 ], the construction is inspired by Proposition 5.8 from classical paper [ 12 ] by V. Drinfeld. [ABSTRACT FROM AUTHOR]

In this paper, we prove the following differential analog of the Noether normalization lemma: for every d-dimensional differential algebraic variety over differentially closed field of zero characteristic there exists a surjective map on to the d-dimensional affine space. Equivalently, for every integral differential algebra A over differential field of zero characteristic there exist differentially independent b1, . . . , bd such that A is differentially algebraic over subalgebra B differentially generated by b1, . . . , bd, and whenever p ⊂ B is a prime differential ideal, there exists a prime differential ideal q ⊂ A such that p = B ∩ q. We also prove the analogous theorem for differential algebraic varieties over the ring of formal power series over an algebraically closed differential field and present some applications to differential equations. [ABSTRACT FROM AUTHOR]

Let G be a simple complex group of adjoint type. In his unpublished work, Z. Yun associated to each θ-group (G0, g1) and a vector X ∈ g1 a flat G-connection ∇X on the trivial G-bundle on P1−{0,∞}, generalizing the construction of Frenkel and Gross in [5]. In this paper, we study the local monodromies of the flat G-connection ∇X and compute the de Rham cohomology of ∇X with value in the adjoint representation of G. In particular, we show that in many cases the connection ∇X is cohomologically rigid. [ABSTRACT FROM AUTHOR]

Biological networks often change under different environmental and genetic conditions. In this paper, we model network change as the difference of two precision matrices and propose a novel loss function called the D-trace loss, which allows us to directly estimate the precision matrix difference without attempting to estimate the precision matrices themselves. Under a new irrepresentability condition, we show that the D-trace loss function with the lasso penalty can yield consistent estimators in high-dimensional settings if the difference network is sparse. A very efficient algorithm is developed based on the alternating direction method of multipliers to minimize the penalized loss function. Simulation studies and a real-data analysis show that the proposed method outperforms other methods. [ABSTRACT FROM AUTHOR]

The stack Mg,n of stable curves and its coarse moduli space Mg,n are defined over Z, and therefore over any field. Over an algebraically closed field of characteristic zero, in [11] Hacking showed that Mg,n is rigid (a conjecture of Kapranov), in [2, 23] Bruno and Mella for g = 0, and the second author for g = 1 showed that its automorphism group is the symmetric group Sn, permuting marked points unless (g,n) ∈ {(0, 4), (1, 1), (1, 2)}. The methods used in the papers above do not extend to positive characteristic. We show that in characteristic p > 0, the rigidity of Mg,n, with the same exceptions as over C, implies that its automorphism group is Sn. We prove that, over any field, M0,n is rigid and deduce that, over any field, Aut(M0,n) ≅ Sn for n = 5. Going back to characteristic zero, we prove that for g+n > 4, the coarse moduli space Mg,n is rigid, extending a result of acking who had proven it has no locally trivial deformations. Finally, we show that M1,2 is not rigid, although it does not admit locally trivial deformations, by explicitly computing his Kuranishi family. [ABSTRACT FROM AUTHOR]

In this paper, a class of closed queueing networks is modelled in the Markovian process algebra PEPA and solved using the classical mean value analysis. This approach is attractive as it negates the need to derive the entire state space, and so certain metrics from large models can be obtained with little computational effort. The class of models considered includes models that are not obviously classical closed queueing models. This paper extends previous results by deriving exact solutions for models with multiple copies of queue stations. Due to the cost of deriving an exact solution, a simplifying approximation is derived, which is shown to be highly accurate when the population is large. The approach is illustrated with three examples. [ABSTRACT FROM AUTHOR]

The π-calculus is a process algebra where agents interact by sending communication links to each other via noiseless communication channels. Taking into account the reality of noisy channels, an extension of the π-calculus, called the πN-calculus, has been introduced recently. This paper presents an early transitional semantics of the πN-calculus, which is not a directly translated version of the late semantics of πN, and then extends six kinds of behavioral equivalences consisting of reduction bisimilarity, barbed bisimilarity, barbed equivalence, barbed congruence, bisimilarity and full bisimilarity into the πN-calculus. Such behavioral equivalences are cast in a hierarchy, which is helpful to verify behavioral equivalence of two agents. In particular, this paper shows that due to the noisy nature of channels, the coincidence of bisimilarity and barbed equivalence, as well as the coincidence of full bisimilarity and barbed congruence, in the π-calculus does not hold in πN. [ABSTRACT FROM PUBLISHER]

In the papers [4–6] with Masaki Kashiwara, the author introduced the notion of symmetric crystals and presented the Lascoux–Leclerc–Thibon–Ariki-type conjectures for the affine Hecke algebras of type B. Namely, we conjectured that certain composition multiplicities and branching rules for the affine Hecke algebras of type B are described by using the lower global basis of symmetric crystals of . In the present paper, we prove the existence of crystal bases and global bases of for any symmetric quantized Kac–Moody algebra by using a geometry of quivers (with a Dynkin diagram involution). This is analogous to George Lusztig's geometric construction of U−v and its lower global basis. [ABSTRACT FROM PUBLISHER]

Typically, secondary and college algebra students attempt to utilize either completing the square or the quadratic formula as techniques to solve a quadratic equation only after frustration with factoring has arisen. While both completing the square and the quadratic formula are techniques which can determine solutions for all quadratic equations, this paper will demonstrate that the probability that a randomly selected quadratic with integer coefficients is factorable is extremely small. In addition to the ineffectiveness of factoring, this paper reveals some significant advantages associated with completing the square and the quadratic formula and that these techniques should be utilized more quickly, more often, and certainly before frustration mounts. [ABSTRACT FROM AUTHOR]