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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]

This note draws conclusions that arise by combining two recent papers, by Anuj Dawar, Erich Grädel and Wied Pakusa, published at ICALP 2019, and by Moritz Lichter, published at LICS 2021. In both papers, the main technical results rely on the combinatorial and algebraic analysis of the invertible-map equivalences |${\equiv ^{\text {IM}}_{k, Q}}$| on certain variants of Cai–Fürer–Immerman structures (CFI-structures for short). These |${\equiv ^{\text {IM}}_{k, Q}}$| -equivalences, for a natural number |$k$| and a set of primes |$Q$|⁠ , refine the well-known Weisfeiler–Leman equivalences used in algorithms for graph isomorphism. The intuition is that two graphs |$G{\equiv ^{\text {IM}}_{k, Q}}H$| cannot be distinguished by iterative refinements of equivalences on |$k$| -tuples defined via linear operators on vector spaces over fields of characteristic |$p \in Q$|⁠. In the first paper it has been shown, using considerable algebraic machinery, that for a prime |$q \notin Q$|⁠ , the |${\equiv ^{\text {IM}}_{k, Q}}$| equivalences are not strong enough to distinguish between non-isomorphic CFI-structures over the field |$\mathbb {F}_q$|⁠. In the second paper, a similar but not identical construction for CFI-structures over the rings |$\mathbb {Z}_{2^i}$| has, again by rather involved combinatorial and algebraic arguments, been shown to be indistinguishable with respect to |${\equiv ^{\text {IM}}_{k, \{2\}}}$|⁠. Together with an earlier work on rank logic, this second result suffices to separate rank logic from polynomial time. We show here that the two approaches can be unified to prove that CFI-structures over the rings |$\mathbb {Z}_{2^i}$| are in fact indistinguishable with respect to |${\equiv ^{\text {IM}}_{k, {\mathbb {P}}}}$|⁠ , for the set |${\mathbb {P}}$| of all primes. In particular, this implies the following two results. First, there is no fixed |$k$| such that the invertible-map equivalence |${\equiv ^{\text {IM}}_{k, {\mathbb {P}}}}$| coincides with isomorphism on all finite graphs. Second, no extension of fixed-point logic by linear-algebraic operators over fields can capture polynomial time. [ABSTRACT FROM AUTHOR]

This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson's constructive logic with strong negation, is the axiomatic extension of the substructural logic |$FL_{ew}$| (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. |$FL_{ew}$| -algebras) that includes both Heyting and Nelson algebras and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras , QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics. [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]

This paper focuses on order-preserving logics defined from varieties of distributive lattices with negation, and in particular on the problem of whether these can be axiomatized by means Hilbert-style calculi that are finite (i.e. that contain a finite number of finitary rule schemata). On the negative side, we provide a syntactic condition on the equational presentation of a variety that entails failure of finite axiomatizability for the corresponding logic. An application of this result is that the logic of all distributive lattices with negation is not finitely axiomatizable; we likewise establish that the order-preserving logic of the variety of all Ockham algebras is also not finitely axiomatizable. On the positive side, we show that an arbitrary subvariety of semi-De Morgan algebras is axiomatized by a finite number of equations if and only if the corresponding order-preserving logic is axiomatized by a finite Hilbert-style calculus. This equivalence also holds for every subvariety of a Berman variety of Ockham algebras. We obtain, as a corollary, a new proof that the implication-free fragment of intuitionistic logic is finitely axiomatizable, as well as a new corresponding Hilbert-style calculus. Our proofs are constructive in that they allow us to effectively convert an equational presentation of a variety of algebras into a Hilbert-style calculus for the corresponding order-preserving logic, and vice versa. We also consider the assertional logics associated to the above-mentioned varieties, showing in particular that the assertional logics of finitely axiomatizable subvarieties of semi-De Morgan algebras are finitely axiomatizable as well. [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 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]

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]

The KBc algebra is a subalgebra that has been used to construct classical solutions in Witten's open string field theory, such as the tachyon vacuum solution. The main purpose of this paper is to give various operator sets that satisfy the KBc algebra. In addition, since those sets can contain matter operators arbitrarily, we can reproduce the solution of Kiermaier, Okawa, and Soler, and that of Erler and Maccaferri. Starting with a single D-brane solution on the tachyon vacuum, we replace the original KBc in it with an appropriate set to generate each of the above solutions. Thus, it is expected that the KBc algebra, combined with the single D-brane solution, leads to a more unified description of classical solutions. [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]

The Post, axled and Łukasiewicz–Moisil algebras are important lattices studied in algebraic logic. In this paper, we investigate a useful interpretation between these algebras and some rings. We give a term equivalence between Post algebras of order |$p$| and |$p$| -rings, |$p$| prime and lift this result to the axled Łukasiewicz–Moisil algebra |$L \cong B_s \times P$| and the ring |$\prod ^s F_2 \times \prod ^l F_p$|⁠ , where |$B_s$| is a Boolean algebra of order |$2^s$|⁠ , |$P$| a |$p$| -valued Post algebra of order |$p^l$| and |$F_p$| is the prime field of order |$p$|⁠. [ABSTRACT FROM AUTHOR]

The non-commutative projective scheme |$\operatorname{\mathsf{Proj_{nc}}} S$| of a |$(\pm 1)$| -skew polynomial algebra S in n variables is considered to be a |$(\pm 1)$| -skew projective space of dimension n  − 1. In this paper, using combinatorial methods, we give a classification theorem for |$(\pm 1)$| -skew projective spaces. Specifically, among other equivalences, we prove that |$(\pm 1)$| -skew projective spaces |$\operatorname{\mathsf{Proj_{nc}}} S$| and |$\operatorname{\mathsf{Proj_{nc}}} S^{\prime}$| are isomorphic if and only if certain graphs associated with S and S ʹ are switching (or mutation) equivalent. We also discuss invariants of |$(\pm 1)$| -skew projective spaces from a combinatorial point of view. [ABSTRACT FROM AUTHOR]

Smooth and proper dg-algebras have an Euler class valued in the Hochschild homology of the algebra. This Euler class is worthy of this name since it satisfies many familiar properties including compatibility with the pairing on the Hochschild homology of the algebra and that of its opposite. This compatibility is the Riemann–Roch theorems of [ 21 , 14 ]. In this paper, we prove a broad generalization of these Riemann–Roch theorems. We generalize from the bicategory of dg-algebras and their bimodules to symmetric monoidal bicategories and from the Euler class to traces of non-identity maps. Our generalization also implies spectral Riemann–Roch theorems. We regard this result as an instantiation of a 2-dimensional generalized cobordism hypothesis. This perspective draws the result close to many others that generalize results about Euler characteristics and classes to bicategorical traces. [ABSTRACT FROM AUTHOR]

In the paper, we introduce multilattice versions of the basic congruent and monotonic modal logics. In the case of congruent and monotonic ones, we also study their extensions by Gödel's rule. We formulate these logics in the form of sequent calculi and prove syntactic embedding theorems (as a consequence, we obtain cut admissibility and decidability). Then we present them algebraically and semantically: via modal multilattices and via general and descriptive neighbourhood frames. We show the dual equivalency of the categories of modal multilattices and descriptive neighbourhood frames. Using Lindenbaum–Tarski algebras, we prove that the sequent calculi under consideration are sound and complete with respect to modal multilattices. [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 study a relationship between tilting modules with finite projective dimension and dominant dimension with respect to injective modules as a generalization of results of Crawley-Boevey–Sauter, Nguyen–Reiten–Todorov–Zhu and Pressland–Sauter. Moreover, we give characterizations of almost n -Auslander–Gorenstein algebras and almost n -Auslander algebras by the existence of tilting modules. As an application, we describe a sufficient condition for almost 1-Auslander algebras to be strongly quasi-hereditary by comparing such tilting modules and characteristic tilting modules. [ABSTRACT FROM AUTHOR]

In the theory of coalgebras, trace semantics can be defined in various distinct ways, including through algebraic logics, the Kleisli category of a monad or its Eilenberg–Moore category. This paper elaborates two new unifying ideas: (i) coalgebraic,draftrules trace semantics is naturally presented in terms of corecursive algebras, and (ii) all three approaches arise as instances of the same abstract setting. Our perspective puts the different approaches under a common roof and allows to derive conditions under which some of them coincide. [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]

Poisson–Lie T-duality/plurality was recently generalized to Jacobi–Lie T-plurality formulated in terms of double field theory and based on Leibniz algebras given by the structure coefficients fab c , fc ab , and Za , Z a . We investigate three- and four-dimensional sigma models corresponding to six-dimensional Leibniz algebras with fb ba ≠ 0, Z a = 0. We show that these algebras are plural one to another and, moreover, to an algebra with fb ba = 0, Z a = 0. These pluralities are used for construction of Jacobi–Lie models. It was conjectured that plural models should satisfy generalized supergravity equations. We have found examples of models satisfying "true" generalized supergravity equations where no trivialization to usual supergravity equations is possible. On the other hand, we show that there are also models corresponding to algebras with fb ba ≠ 0, Z a = 0 where the Killing vector appearing in generalized supergravity equations either vanishes or can be removed by suitable gauge transformation. Such models then satisfy usual supergravity equations, i.e. vanishing beta-function equations. [ABSTRACT FROM AUTHOR]

Let p be a prime number and let |${\mathbb{K}}$| be a field containing a root of 1 of order p. If the absolute Galois group |$G_{\mathbb{K}}$| satisfies |$\dim\, H^1(G_{\mathbb{K}},\mathbb{F}_p) \lt \infty$| and |$\dim\, H^{\,2}(G_{\mathbb{K}},\mathbb{F}_p)=1$|⁠ , we show that L. Positselski's and T. Weigel's Koszulity conjectures are true for |${\mathbb{K}}$|⁠. Also, under the above hypothesis, we show that the |$\mathbb{F}_p$| -cohomology algebra of |$G_{\mathbb{K}}$| is the quadratic dual of the graded algebra |${\rm gr}_\bullet\mathbb{F}_p[G_{\mathbb{K}}]$|⁠ , induced by the powers of the augmentation ideal of the group algebra |$\mathbb{F}_p[G_{\mathbb{K}}]$|⁠ , and these two algebras decompose as products of elementary quadratic algebras. Finally, we propose a refinement of the Koszulity conjectures, analogous to I. Efrat's elementary type conjecture. [ABSTRACT FROM AUTHOR]

Based on Mubarakzyanov's classification of four-dimensional real Lie algebras, we classify ten-dimensional Exceptional Drinfeld algebras (EDAs). The classification is restricted to EDAs whose maximal isotropic (geometric) subalgebras cannot be represented as a product of a 3D Lie algebra and a 1D abelian factor. We collect the obtained algebras into families depending on the dualities found between them. Despite algebras related by a generalized Yang–Baxter deformation we find two algebras related by a different Nambu–Lie U-duality transformation. We show that this duality relates two Type IIA backgrounds. [ABSTRACT FROM AUTHOR]

Quantifier variantists accept multiple alternative ontological languages in which quantifiers obey the usual inference rules despite having different meanings. But collapse arguments seem to show that these quantifiers would be provably equivalent to one another. Cian Dorr has pushed this discussion forward by formulating the collapse argument in terms of an algebra of meanings that are common amongst the languages. I attempt to show that quantifier variantists can respond. But an important distinction between types of quantifier variance emerges, between those in which quantifier meanings draw on a single objective backbone of "portions of reality," and those (such as the type that is arguably associated with neo-Fregeanism) in which they do not. [ABSTRACT FROM AUTHOR]

We introduce the notion of reflexivity for combinatory algebras. Reflexivity can be thought of as an equational counterpart of the Meyer–Scott axiom of combinatory models, which indeed allows us to characterize an equationally definable counterpart of combinatory models. This new structure, called strongly reflexive combinatory algebra, admits a finite axiomatization with seven closed equations, and the structure is shown to be exactly the retract of combinatory models. Lambda algebras can be characterized as strongly reflexive combinatory algebras that are stable. Moreover, there is a canonical construction of a lambda algebra from a strongly reflexive combinatory algebra. The resulting axiomatization of lambda algebras by the seven axioms for strong reflexivity together with those for stability is shown to correspond to the axiomatization of lambda algebras due to Selinger (2002, J. Funct. Program. 12, 549–566). [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 show that braided Cherednik algebras introduced by Bazlov and Berenstein are cocycle twists of rational Cherednik algebras of the imprimitive complex reflection groups |$G(m,p,n)$|⁠ , when m is even. This gives a new construction of mystic reflection groups which have Artin–Schelter regular rings of quantum polynomial invariants. As an application of this result, we show that a braided Cherednik algebra has a finite-dimensional representation if and only if its rational counterpart has one. [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]