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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]

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]

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]

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]

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]

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]

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 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]

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]

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 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]

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 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]

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]

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 provide a new virtual description of the symmetric group action on the cohomology of ordered configuration space on |$SU_2$| up to translations. We use this formula to prove the Moseley–Proudfoot–Young conjecture. As a consequence we obtain the graded Frobenius character of the Orlik–Terao algebra of type |$A_n$|⁠. [ABSTRACT FROM AUTHOR]

We develop the representation theory of shifted quantum affine algebras |$\mathcal {U}_\mu (\hat {\mathfrak {g}})$| and of their truncations, which appeared in the study of quantized K-theoretic Coulomb branches of 3d |$N = 4$| SUSY quiver gauge theories. Our approach is based on novel techniques, which are new in the cases of shifted Yangians or ordinary quantum affine algebras as well: realization in terms of asymptotical subalgebras of the quantum affine algebra |$\mathcal {U}_q(\hat {\mathfrak {g}})$|⁠ , induction and restriction functors to the category |$\mathcal {O}$| of representations of the Borel subalgebra |$\mathcal {U}_q(\hat {\mathfrak {b}})$| of |$\mathcal {U}_q(\hat {\mathfrak {g}})$|⁠ , relations between truncations and Baxter polynomiality in quantum integrable models, and parametrization of simple modules via Langlands dual interpolation. We first introduce the category |$\mathcal {O}_\mu $| of representations of |$\mathcal {U}_\mu (\hat {\mathfrak {g}})$| and we classify its simple objects. Then we establish the existence of fusion products and we get a ring structure on the sum of the Grothendieck groups |$K_0(\mathcal {O}_\mu)$|⁠. We classify simple finite-dimensional representations of |$\mathcal {U}_\mu (\hat {\mathfrak {g}})$| and we obtain a cluster algebra structure on the Grothendieck ring of finite-dimensional representations. We prove a truncation has only a finite number of simple representations and we introduce a related partial ordering on simple modules. Eventually, we state a conjecture on the parametrization of simple modules of a non-simply-laced truncation in terms of the Langlands dual Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations. [ABSTRACT FROM AUTHOR]

Using Fock–Goncharov higher Teichmüller space variables we derive log-canonical coordinate representation for entries of general symplectic leaves of the |$\mathcal A_n$| groupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the reflection equation with the trigonometric |$R$| -matrix. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables for |$\mathcal A_3$| and |$\mathcal A_4$| in terms of geodesic functions for Riemann surfaces with holes. We realize braid-group transformations for |$\mathcal A_n$| via sequences of cluster mutations in the special |$\mathcal A_n$| -quiver. We prove the groupoid relations for normalized quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit. We prove the quantum algebraic relations of transport matrices for arbitrary (cyclic or acyclic) directed planar network. Dedicated to the memory of a great mathematician and person, Boris Dubrovin. [ABSTRACT FROM AUTHOR]

In operator algebra, the linearization trick is a technique that reduces the study of a non-commutative polynomial evaluated at elements of an algebra |${\mathcal {A}}$| to the study of a polynomial of degree one, evaluated on the enlarged algebra |${\mathcal {A}} \otimes M_r ({\mathbb {C}})$|⁠ , for some integer |$r$|⁠. We introduce a new instance of the linearization trick that is tailored to study a finitely supported random walk |$G$| by studying instead a nearest-neighbour coloured random walk on |$G \times \{1, \ldots , r \}$|⁠ , which is much simpler to analyze. As an application, we extend well-known results for nearest-neighbour walks on free groups and free products of finite groups to coloured random walks, thus showing how one can obtain new results for finitely supported random walks, namely an explicit description of the harmonic measure and formulas for the entropy and drift. [ABSTRACT FROM AUTHOR]

We prove a new quasiparticle sum expression for the character of the Ising model vertex algebra, related to the Jackson–Slater |$q$| -series identity of Rogers–Ramanujan type and to Nahm sums for the matrix $\left (\begin {smallmatrix}8&3\\3&2 \end {smallmatrix}\right) $ ⁠. We find, as consequences, an explicit monomial basis for the Ising model and a description of its singular support. We find that the ideal sheaf of the latter, defining it as a subscheme of the arc space of its associated scheme, is finitely generated as a differential ideal. We prove three new |$q$| -series identities of the Rogers–Ramanujan–Slater type associated with the three irreducible modules of the Virasoro Lie algebra of central charge |$1/2$|⁠. We give a combinatorial interpretation to the identity associated with the vacuum module. [ABSTRACT FROM AUTHOR]

We use the stabilization functors to study the combinatorial aspects of the $F$ -polynomial of a representation of any finite-dimensional basic algebra. We characterize the vertices of their Newton polytopes. We give an explicit formula for the $F$ -polynomial restricting to any face of its Newton polytope. For acyclic quivers, we give a complete description of all facets of the Newton polytope when the representation is general. We also prove that the support of the $F$ -polynomial is saturated for any rigid representation. We provide many examples and counterexamples and pose several conjectures. [ABSTRACT FROM AUTHOR]

Given a smooth curve |$C$|⁠ , we define and study analogues of KLR algebras and quiver Schur algebras, where quiver representations are replaced by torsion sheaves on |$C$|⁠. In particular, they provide a geometric realization for certain affinized symmetric algebras. When |$C={\mathbb{P}}^1$|⁠ , a version of curve Schur algebra turns out to be Morita equivalent to the imaginary semi-cuspidal category of the Kronecker quiver in any characteristic. As a consequence, we argue that one should not expect to have a reasonable theory of parity sheaves for affine quivers. [ABSTRACT FROM AUTHOR]

We give an interpretation of the |$(q,t)$| -deformed Cartan matrices of finite type and their inverses in terms of bigraded modules over the generalized preprojective algebras of Langlands dual type in the sense of Geiß–Leclerc–Schröer [ 33 ]. As an application, we compute the first extension groups between the generic kernels introduced by Hernandez–Leclerc [ 40 ] and propose a conjecture that their dimensions coincide with the pole orders of the normalized |$R$| -matrices between the corresponding Kirillov–Reshetikhin modules. [ABSTRACT FROM AUTHOR]

We show that Hilbert schemes of planar curve singularities and their parabolic variants can be interpreted as certain generalized affine Springer fibers for |$GL_n$|⁠ , as defined by Goresky–Kottwitz–MacPherson. Using a generalization of affine Springer theory for Braverman–Finkelberg–Nakajima's Coulomb branch algebras, we construct a rational Cherednik algebra action on the homology of the Hilbert schemes and compute it in examples. Along the way, we generalize to the parahoric setting the recent construction of Hilburn–Kamnitzer–Weekes, which may be of independent interest. In the spherical case, we make our computations explicit through a new general localization formula for Coulomb branches. Via results of Hogancamp–Mellit, we also show the rational Cherednik algebra acts on the HOMFLY-PT homologies of torus knots. This work was inspired in part by a construction in 3D |${\mathcal {N}}=4$| gauge theory. [ABSTRACT FROM AUTHOR]

We establish a bijection between |$d$| -simple-minded systems (⁠|$d$| -SMSs) of |$(-d)$| -Calabi–Yau cluster category |$\mathcal C_{-d}(H)$| and silting objects of |${\mathcal {D}}^{\mathrm {b}}(H)$| contained in |${\mathcal {D}}^{\le 0}\cap {\mathcal {D}}^{\ge 1-d}$| for hereditary algebra |$H$| of Dynkin type and |$d\ge 1$|⁠. We show that the number of |$d$| -SMSs in |$\mathcal C_{-d}(H)$| is the positive Fuss–Catalan number |$C_{d}^{+}(W)$| of the corresponding Weyl group |$W$|⁠ , by applying this bijection and Buan–Reiten–Thomas' and Zhu's results on Fomin–Reading's generalized cluster complexes. Our results are based on a refined version of silting- |$t$| -structure correspondence. [ABSTRACT FROM AUTHOR]

We investigate maximal tori in the Hochschild cohomology Lie algebra |${\operatorname {HH}}^1(A)$| of a finite dimensional algebra |$A$|⁠ , and their connection with the fundamental groups associated to presentations of |$A$|⁠. We prove that every maximal torus in |${\operatorname {HH}}^1(A)$| arises as the dual of some fundamental group of |$A$|⁠ , extending the work by Farkas, Green, and Marcos; de la Peña and Saorín; and Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental group of |$A$| is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Using this we prove that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras; hitherto this was known only for very restricted classes of monomial algebras. [ABSTRACT FROM AUTHOR]

We obtain a fractional Leibniz rule associated to bilinear Hermite pseudo-multipliers in the context of Hermite Besov and Hermite Triebel–Lizorkin spaces. As a byproduct, we show that the classes of bounded functions in these spaces (which include Hermite Sobolev and Hermite Hardy–Sobolev spaces) are algebras under pointwise multiplication. To obtain these results we develop appropriate decompositions for bilinear pseudo-multipliers and molecular estimates for certain families of functions in the Hermite setting. [ABSTRACT FROM AUTHOR]

Preprojective K-theoretic Hall algebras (KHAs), particular cases of KHAs of quivers with potential, are conjecturally positive halves of the Okounkov–Smirnov affine quantum algebras. It is thus natural to ask whether KHAs of quivers with potential are halves of a quantum group. For a symmetric quiver with potential satisfying a Künneth-type condition, we construct (positive and negative) extensions of its KHA, which are bialgebras. In particular, there are bialgebra extensions of preprojective KHAs and one can construct their Drinfeld double algebra. [ABSTRACT FROM AUTHOR]

We introduce the cohomological blowup of a graded Artinian Gorenstein algebra along a surjective map, which we term BUG (blowup Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blowup of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersections. We have included many examples throughout this manuscript. [ABSTRACT FROM AUTHOR]

We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let |$\Gamma $| be a finite index subgroup of |$\textrm {SL}(2,\mathbb Z)$| with an action on a complex simple Lie algebra |$\mathfrak g$|⁠ , which can be extended to |$\textrm {SL}(2,{\mathbb {C}})$|⁠. We show that the Lie algebra of the corresponding |$\mathfrak {g}$| -valued modular forms is isomorphic to the extension of |$\mathfrak {g}$| over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups |$\Gamma (N), \, N\leq 6$|⁠ , is considered in more detail in relation to the classical results of Klein and Fricke and the celebrated Markov Diophantine equation. We finish with a brief discussion of the extensions and representations of these Lie algebras. [ABSTRACT FROM AUTHOR]

We expand upon the notion of equivariant log concavity and make equivariant log concavity conjectures for Orlik–Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik–Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra |$\mathfrak{s}\mathfrak{l}_n$|⁠ , we exploit the theory of representation stability to give computer-assisted proofs of these conjectures in low degree. [ABSTRACT FROM AUTHOR]

We show how a novel construction of the sheaf of Cherednik algebras |$\mathscr {H}_{1, c, X, G}$| on a quotient orbifold |$Y:=X/G$| in author's prior work leads to results for |$\mathscr {H}_{1, c, X, G}$|⁠ , which until recently were viewed as intractable. First, for every orbit type stratum in |$X$|⁠ , we define a trace density map for the Hochschild chain complex of |$\mathscr {H}_{1, c, X, G}$|⁠ , which generalizes the standard Engeli–Felder's trace density construction for the sheaf of differential operators |$\mathscr {D}_X$|⁠. Second, by means of the newly obtained trace density maps, we prove an isomorphism in the derived category of complexes of |$\mathbb {C}_{Y}\llbracket \hbar \rrbracket $| -modules, which computes the hypercohomology of the Hochschild chain complex of the sheaf of formal Cherednik algebras |$\mathscr {H}_{1, \hbar , X, G}$|⁠. We show that this hypercohomology is isomorphic to the Chen–Ruan cohomology of the orbifold |$Y$| with values in the ring of formal power series |$\mathbb {C}\llbracket \hbar \rrbracket $|⁠. We infer that the Hochschild chain complex of the sheaf of skew group algebras |$\mathscr {H}_{1, 0, X, G}$| has a well-defined Euler characteristic that is equal to the orbifold Euler characteristic of |$Y$|⁠. Finally, we prove an algebraic index theorem. [ABSTRACT FROM AUTHOR]

Let |$\mathcal{B}_{\mathfrak{q}}$| be a finite-dimensional Nichols algebra of diagonal type with braiding matrix |$\mathfrak{q}$|⁠ , |$\mathcal{L}_{\mathfrak{q}}$| be the corresponding Lusztig algebra as in [ 4 ], and |$\operatorname{Fr}_{\mathfrak{q}}: \mathcal{L}_{\mathfrak{q}} \to U(\mathfrak{n}^{\mathfrak{q}})$| be the corresponding quantum Frobenius map as in [ 5 ]. We prove that the finite-dimensional Lie algebra |$\mathfrak{n}^{\mathfrak{q}}$| is either 0 or the positive part of a semisimple Lie algebra |$\mathfrak{g}^{\mathfrak{q}}$|⁠ , which is determined for each |$\mathfrak{q}$| in the list of [ 25 ]. [ABSTRACT FROM AUTHOR]

We introduce algebraic structures on the polyvector fields of an algebraic torus that serve to compute multiplicities in tropical and log Gromov–Witten theory while also connecting to the mirror symmetry dual deformation theory of complex structures. Most notably these structures include a tropical quantum field theory and an |$L_{\infty }$| -structure. The latter is an instance of Getzler's gravity algebra, and the |$l_2$| -bracket is a restriction of the Schouten–Nijenhuis bracket. We explain the relationship to string topology in the Appendix (thanks to Janko Latschev). [ABSTRACT FROM AUTHOR]

The |$\textbf{g}$| -vector fan of a finite-dimensional algebra is a fan whose rays are the |$\textbf{g}$| -vectors of its two-term presilting objects. We prove that the |$\textbf{g}$| -vector fan of a tame algebra is dense. We then apply this result to obtain a near classification of quivers for which the closure of the cluster |$\textbf{g}$| -vector fan is dense or is a half-space, using the additive categorification of cluster algebras by means of Jacobian algebras. As another application, we prove that for quivers with potentials arising from once-punctured closed surfaces, the stability and cluster scattering diagrams only differ by wall-crossing functions on the walls contained in a separating hyperplane. The appendix is devoted to the construction of truncated twist functors and their adjoints. [ABSTRACT FROM AUTHOR]

Let |$A$| be a finitely presented associative monomial algebra. We study the category |$\textsf{qgr}(A)$|⁠ , which is a quotient of the category of graded finitely presented |$A$| -modules by the finite-dimensional ones. As this category plays a role of the category of coherent sheaves on the corresponding noncommutative variety, we consider its bounded derived category |$\textbf{D}^b(\textsf{qgr}(A))$|⁠. We calculate the categorical entropy of the Serre twist functor on |$\textbf{D}^b(\textsf{qgr}(A))$| and show that it is equal to the (natural) logarithm of the entropy of the algebra |$A$| itself. Moreover, we relate these two kinds of entropy with the topological entropy of the Ufnarovski graph of |$A$| and the entropy of the path algebra of the graph. If |$A$| is a path algebra of some quiver, the categorical entropy is equal to the logarithm of the spectral radius of the quiver's adjacency matrix. [ABSTRACT FROM AUTHOR]

We show how to recover a discrete twist over an ample Hausdorff groupoid from a pair consisting of an algebra and what we call a quasi-Cartan subalgebra. We identify precisely which twists arise in this way (namely, those that satisfy the local bisection hypothesis), and we prove that the assignment of twisted Steinberg algebras to such twists and our construction of a twist from a quasi-Cartan pair are mutually inverse. We identify the algebraic pairs that correspond to effective groupoids and to principal groupoids. We also indicate the scope of our results by identifying large classes of twists for which the local bisection hypothesis holds automatically. [ABSTRACT FROM AUTHOR]

The notion of a Harish-Chandra bimodule, that is, finitely generated |$U(\mathfrak {g})$| -bimodule with locally finite adjoint action, was generalized to any filtered algebra in a work of Losev [ 9 ]. Similarly to the classical case we can define the notion of a unitarizable bimodule. We investigate a question when the regular bimodule, that is, the algebra itself, for a deformation of Kleinian singularity of type |$A$| is unitarizable. We obtain a partial classification of unitarizable regular bimodules. [ABSTRACT FROM AUTHOR]

We give a new realization of the prefundamental representations |$L^\pm _{r,a}$| introduced by Hernandez and Jimbo, when the quantum loop algebra |$U_q(\mathfrak {g})$| is of types |$A_n^{(1)}$| and |$D_n^{(1)}$| and the |$r$| -th fundamental weight |$\varpi _r$| for types |$A_n$| and |$D_n$| is minuscule. We define an action of the Borel subalgebra |$U_q(\mathfrak {b})$| of |$U_q(\mathfrak {g})$| on the unipotent quantum coordinate ring associated to the translation by |$-\varpi _r$| and show that it is isomorphic to |$L^\pm _{r,a}$|⁠. We then give a combinatorial realization of |$L^+_{r,a}$| in terms of the Lusztig data of the dual PBW vectors. [ABSTRACT FROM AUTHOR]

Consider a monoidal category that is at the same time abelian with enough projectives and such that projectives are flat on the right. We show that there is a |$B_{\infty }$| -algebra that is |$A_{\infty }$| -quasi-isomorphic to the derived endomorphism algebra of the tensor unit. This |$B_{\infty }$| -algebra is obtained as the co-Hochschild complex of a projective resolution of the tensor unit, endowed with a lifted |$A_{\infty }$| -coalgebra structure. We show that in the classical situation of the category of bimodules over an algebra, this newly defined |$B_{\infty }$| -algebra is isomorphic to the Hochschild complex of the algebra in the homotopy category of |$B_{\infty }$| -algebras. [ABSTRACT FROM AUTHOR]

We give a presentation of the Kauffman (BMW) skein algebra of the torus. This algebra is the "type |$BCD$| " analogue of the Homflypt skein algebra of torus, which was computed in earlier work of the 1st and 3rd authors [ 17 ]. This suggests the existence of a "type |$BCD$| " version of the Hall algebra of an elliptic curve [ 4 ]. In the appendix, we show this presentation is compatible with the Frohman–Gelca description of the Kauffman bracket (Temperley–Lieb) skein algebra of the torus [ 12 ]. [ABSTRACT FROM AUTHOR]

We construct a combinatorial generalization of the Leray models for hyperplane arrangement complements. Given a matroid and some combinatorial blow-up data, we give a presentation for a bigraded (commutative) differential graded algebra. If the matroid is realizable over |$\mathbb {C}$|⁠ , this is the familiar Morgan model for a hyperplane arrangement complement, embedded in a blowup of projective space. In general, we obtain a cdga that interpolates between the Chow ring of a matroid and the Orlik–Solomon algebra. Our construction can also be expressed in terms of sheaves on combinatorial blowups of geometric lattices. As a key technical device, we construct a monomial basis via a Gröbner basis for the ideal of relations. Combining these ingredients, we show that our algebra is quasi-isomorphic to the classical Orlik–Solomon algebra of the matroid. [ABSTRACT FROM AUTHOR]

We introduce a new approach to the study of finite-dimensional representations of the quantum group of the affine Lie superalgebra |$ \textrm {L}{\mathfrak {g}\mathfrak {l}}_{M|N}=\mathbb {C}[t,t^{-1}]\otimes \mathfrak {g}\mathfrak {l}_{M|N}$| (⁠|$M\neq N$|⁠). We explain how the representations of the quantum group of |$ \textrm {L}{\mathfrak {g}\mathfrak {l}}_{M|N}$| are directly related to those of the quantum affine algebra of type |$A$|⁠ , using an exact monoidal functor called truncation. This can be viewed as an affine analogue of super duality of type |$A$|⁠. [ABSTRACT FROM AUTHOR]

We study pre-Lie pairs, by which we mean a pair of a homotopy Lie algebra and a pre-Lie algebra with a compatible pre-Lie action. Such pairs provide a wealth of algebraic structure, which in particular can be used to analyze the homotopy Lie part of the pair. Our main application and the main motivation for this development are the dg Lie algebras of hairy graphs computing the rational homotopy groups of the mapping spaces of the little disks operads. We show that twisting with certain Maurer–Cartan elements trivializes their Lie algebra structure. The result can be used to understand the rational homotopy type of many connected components of these mapping spaces. [ABSTRACT FROM AUTHOR]