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Over the years questions have arisen about T-ideals of (noncommutative) polynomials. But when evaluating a noncentral polynomial in subalgebras of matrices, one often has little control in determining the specific evaluations of the polynomial. One way of overcoming this difficulty in characteristic 0, is to reduce to multilinear polynomials and to utilize the representation theory of the symmetric group. But this technique is unavailable in characteristic p > 0. An alternative method, which succeeds, is the process of "hiking" a polynomial, in which one specializes its indeterminates in several stages, to obtain a polynomial in which Capelli polynomial is embedded, in order to get control on its evaluations. This method was utilized on homogeneous polynomials in the proof of Specht's conjecture for affine algebras over fields of positive characteristic. In this paper, we develop hiking further to nonhomogeneous polynomials, to apply to the "representability question." Kemer proved in 1988 that every affine relatively free PI algebra over an infinite field, is representable. In 2010, the first author of this paper proved more generally that every affine relatively free PI algebra over any commutative Noetherian unital ring is representable [A. Belov, Local finite basis property and local representability of varieties of associative rings, Izv. Russian Acad. Sci. (1) (2010) 3–134. English Translation Izv. Math. 74(1) (2010) 1–126]. We present a different, complete, proof, based on hiking nonhomogeneous polynomials, over finite fields. We then obtain the full result over a Noetherian commutative ring, using Noetherian induction on T-ideals. The bulk of the proof is for the case of a base field of positive characteristic. Here, whereas the usage of hiking is more direct than in proving Specht's conjecture, one must consider nonhomogeneous polynomials when the base ring is finite, which entails certain difficulties to be overcome. In Appendix A, we show how hiking can be adapted to prove the involutory versions, as well as various graded and nonassociative theorems. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if R is an affine algebra of dimension d ≥ 4 over ¯ p and 1 / (d − 1) ! ∈ R , then any unimodular row over R of length d can be mapped to a factorial row by elementary transformations. [ABSTRACT FROM AUTHOR]

Two presentations of quantum affine superalgebras were introduced by Yamane in [On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras, Publ. Res. Inst. Math. Sci. 35 (1999) 321–390], which were called Drinfeld–Jimbo realization and Drinfeld realization. Drinfeld realization contains infinite sequences of generators and relations. In this paper, we consider the Drinfeld realization of quantum affine superalgebra q ( ̂ m | n) associated to type m | n and define a simple algebra 0 ( ̂ m | n) generated by only a finite part of these sequences of quantum affine superalgebra q ( ̂ m | n). We show that the algebra 0 ( ̂ m | n) is isomorphic to the quantum affine superalgebra q ( ̂ m | n). Using the above isomorphism, we prove there exists an isomorphism between the two realizations. [ABSTRACT FROM AUTHOR]

In this paper, for an arbitrary Kac-Moody Lie algebra {\mathfrak g} and a diagram automorphism \mu of {\mathfrak g} satisfying certain natural linking conditions, we introduce and study a \mu-twisted quantum affinization algebra {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) of {\mathfrak g}. When {\mathfrak g} is of finite type, {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) is Drinfeld's current algebra realization of the twisted quantum affine algebra. When \mu =\mathrm {id} and {\mathfrak g} in affine type, {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) is the quantum toroidal algebra introduced by Ginzburg, Kapranov and Vasserot. As the main results of this paper, we first prove a triangular decomposition for {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right). Second, we give a simple characterization of the affine quantum Serre relations on restricted {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right)-modules in terms of "normal order products". Third, we prove that the category of restricted {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right)-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right). Last, we study the classical limit of {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) and abridge it to the quantization theory of extended affine Lie algebras. In particular, based on a classification result of Allison-Berman-Pianzola, we obtain the \hbar-deformation of all nullity 2 extended affine Lie algebras. [ABSTRACT FROM AUTHOR]

The paper written many years ago by Resco and Small, constructed an example in prime characteristic of a finitely generated Noetherian K$K$‐algebra that does not necessarily remain Noetherian under field extensions. After this work was completed, Yuri Medvedev, then at the University of Ottawa, visited Resco at the University of Oklahoma, read the paper, and suggested an approach using Jordan algebras that would yield a similar example in characteristic 0. This was announced in the paper by Resco and Small, but to our knowledge the details were never published nor even entirely verified. We take the opportunity to do this now, but stress that the key sequence {an∣n=1,2,3,...}$\lbrace a_n\mid n=1,2,3,\ldots \rbrace$ and its relevant properties are due to Medvedev. Also, Resco suggested avoiding Jordan algebras and working directly and ring theoretically with the exterior algebra. This is the approach we take. [ABSTRACT FROM AUTHOR]

A way to construct the natural representation of the quantized affine algebra U v ( s l ̂ l) is via the deformed Fock space by Misra and Miwa. This relates the classes of Weyl modules for U q (s l N) were q is a root of unity to the action of U v ( s l ̂ l) as N tends toward infinity. In this paper we investigate the situation outside of type A. In classical types, we construct embeddings of the Grothendieck group of finite dimensional U q (g) -modules into Fock spaces of different charges and define an action of an affine quantum symmetric pair that plays the role of the quantized affine algebra. We describe how the action is related to the linkage principal for quantum groups at a root of unity and tensor product multiplicities. [ABSTRACT FROM AUTHOR]

For a ∈ C ⁎ there are two natural evaluation maps ev a and ev a from the affine Hecke algebra H ▵ (r) C to the Hecke algebra H (r) C. The maps ev a and ev a induce evaluation maps ev ˜ a and ev ˜ a from the affine quantum Schur algebra S ▵ (n , r) C to the quantum Schur algebra S (n , r) C , respectively. In this paper we prove that the evaluation map ev ˜ a (resp. ev ˜ a) is compatible with the evaluation map Ev a (resp. Ev (− 1) n a q n ) for quantum affine sl n. Furthermore we compute the Drinfeld polynomials associated with the simple S ▵ (n , r) C -modules which come from the simple S (n , r) C -modules via the evaluation maps ev ˜ a. Then we characterize finite-dimensional irreducible S ▵ (n , r) C -modules which are irreducible as S (n , r) C -modules for n > r. As an application, we characterize finite-dimensional irreducible modules for the affine Hecke algebra H ▵ (r) C which are irreducible as modules for the Hecke algebra H (r) C. [ABSTRACT FROM AUTHOR]

Let F$F$ be a field of characteristic zero and G$G$ a finite abelian group. In this paper, we prove that an affine variety of G$G$‐graded PI‐algebras is minimal if and only if it is generated by a graded algebra UT(A1,⋯,Am;γ)$UT(A_1,\dots,A_m;\gamma)$ of upper block triangular matrices where A1,⋯,Am$A_1,\dots,A_m$ are finite‐dimensional G$G$‐simple algebras. [ABSTRACT FROM AUTHOR]

This paper develops a new approach to small time local attainability of smooth manifolds of any dimension, possibly with boundary and to prove Hölder continuity of the minimum time function. We give explicit pointwise conditions of any order by using higher order hamiltonians which combine derivatives of the controlled vector field and the functions that locally define the target. For the controllability of a point our sufficient conditions extend some classically known results for symmetric or control affine systems, using the Lie algebra instead, but for targets of higher dimension our approach and results are new. We find our sufficient higher order conditions explicit and easy to compute for targets with curvature and general control systems. Some cases of nonsmooth targets are also included. [ABSTRACT FROM AUTHOR]

In this paper, a sponge in Rd$\mathbb {R}^d$ is the attractor of an iterated function system consisting of finitely many strictly contracting affine maps whose linear part is a diagonal matrix. A suitable separation condition is introduced under which a variational formula is proved for the Lq$L^q$ spectrum of any self‐affine measure defined on a sponge for all q∈R$q\in \mathbb {R}$. Apart from some special cases, even the existence of their box dimension was not proved before. Under certain conditions, the formula has a closed form which in general is an upper bound. The Frostman and box dimension of these measures is also determined. The approach unifies several existing results and extends them to arbitrary dimensions. The key ingredient is the introduction of a novel pressure function which aims to capture the growth rate of box counting quantities on sponges. We show that this pressure satisfies a variational principle which resembles the Ledrappier–Young formula for Hausdorff dimension. [ABSTRACT FROM AUTHOR]

The notion of shifted quantum groups has recently played an important role in algebraic geometry. This subtle modification of the original definition brings more flexibility in the representation theory of quantum groups. The first part of this paper presents new mathematical results for the shifted quantum toroidal gl (1) and quantum affine sl (2) algebras (resp. denoted U ¨ q 1 , q 2 μ (gl (1)) and U ˙ q μ (sl (2)) ). It defines several new representations, including finite dimensional highest ℓ -weight representations for the toroidal algebra, and a vertex representation of U ˙ q μ (sl (2)) acting on Hall–Littlewood polynomials. It also explores the relations between representations of U ˙ q μ (sl (2)) and U ¨ q 1 , q 2 μ (gl (1)) in the limit q 1 → ∞ ( q 2 fixed), and present the construction of several new intertwiners. These results are used in the second part to construct BPS observables for 5d N = 1 and 3d N = 2 gauge theories. In particular, it is shown that 5d hypermultiplets and 3d chiral multiplets can be introduced in the algebraic engineering framework using shifted representations, and the Higgsing procedure is revisited from this perspective. [ABSTRACT FROM AUTHOR]

According to a theorem of Cantat [Ann. of Math. (2) 174 (2011), pp. 299–340] and Urech [J. Reine Angew. Math. 770 (2021), pp. 27–57], an analog of the classical Tits alternative holds for the group of birational automorphisms of a compact complex Kähler surface. We established in Arzhantsev and Zaidenberg [Int. Math. Res. Not. IMRN 2022 (2022), pp. 8162–8195] the following Tits-type alternative: if X is a toric affine variety and G\subset \operatorname {Aut}(X) is a subgroup generated by a finite set of unipotent subgroups normalized by the acting torus then either G contains a nonabelian free subgroup or G is a unipotent affine algebraic group. In the present paper we extend the latter result to any group G of automorphisms of a complex affine surface generated by a finite collection of unipotent algebraic subgroups. It occurs that either G contains a nonabelian free subgroup or G is a metabelian unipotent algebraic group. [ABSTRACT FROM AUTHOR]

This paper is the first in a series where we attempt to define defects in critical lattice models that give rise to conformal field theory topological defects in the continuum limit. We focus mostly on models based on the Temperley–Lieb algebra, with future applications to restricted solid-on-solid (also called anyonic chains) models, as well as non-unitary models like percolation or self-avoiding walks. Our approach is essentially algebraic and focusses on the defects from two points of view: the "crossed channel" where the defect is seen as an operator acting on the Hilbert space of the models, and the "direct channel" where it corresponds to a modification of the basic Hamiltonian with some sort of impurity. Algebraic characterizations and constructions are proposed in both points of view. In the crossed channel, this leads us to new results about the center of the affine Temperley–Lieb algebra; in particular we find there a special basis with non-negative integer structure constants that are interpreted as fusion rules of defects. In the direct channel, meanwhile, this leads to the introduction of fusion products and fusion quotients, with interesting algebraic properties that allow to describe representations content of the lattice model with a defect, and to describe its spectrum. [ABSTRACT FROM AUTHOR]

Generalized structurable algebras contain the class of Lie algebras, alternative algebras, Poisson algebras, and a class of nonassociative algebras appeared in mathematical physics and they are therefore significant for applications. In this paper we investigate the construction of affine Lie algebras from generalized structurable algebras. [ABSTRACT FROM AUTHOR]

Let X be an irreducible affine algebraic variety that is spherical with respect to an action of a connected reductive group G. In this paper, we provide sufficient conditions, formulated in terms of weight combinatorics, for the existence of one-parameter additive actions on normalized by a Borel subgroup . As an application, we prove that every G-stable prime divisor in X can be connected with an open G-orbit by means of a suitable B-normalized one-parameter additive action. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic 0, let S be a complete local ring with coefficient field K , let K [ [ x 1 , ... , x n ] ] be the ring of formal power series in variables x 1 , ... , x n with coefficients from K , let K [ [ x 1 , ... , x n ] ] → S be a K -algebra surjection and let E • • , • be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of S. Nicholas Switala [12] proved that this spectral sequence is independent of the surjection beginning with the E 2 page, and the groups E 2 p , q are all finite-dimensional over K. In this paper we extend this result to affine varieties. Namely, let Y be an affine variety over K , let X be a non-singular affine variety over K , let Y ⊂ X be an embedding over K and let E • • , • be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of Y. Then this spectral sequence is independent of the embedding beginning with the E 2 page, and the groups E 2 p , q are all finite-dimensional over K. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the affine ageing algebra a g e ̂ (1) , which is a central extension of the loop algebra of the one-spatial ageing algebra a g e (1). Certain Verma-type modules including Verma modules and imaginary Verma modules of a g e ̂ (1) are studied. Particularly, the simplicity of these modules are characterized and their irreducible quotient modules are determined. We also study the restricted modules of a g e ̂ (1) which are also the modules of the affine vertex algebra arising from the one-spatial ageing algebra a g e (1). We present certain constructions of simple restricted a g e ̂ (1) -modules and an explicit such example of simple restricted module via the Whittaker module of a g e ̂ (1) is given. [ABSTRACT FROM AUTHOR]

In this paper, we give an RTT presentation of the twisted quantum affine algebra of type A2n−1(2) and show that it is isomorphic to the Drinfeld new realization via the Gauss decomposition of the L-operators. This provides the first such presentation for twisted quantum affine algebras with nontrivial central elements. [ABSTRACT FROM AUTHOR]

The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant weight, it is proved by Lenart– Lubovsky that the quantum alcove model does not depend on the choice of a reduced alcove path, which is a shortest path of alcoves from the fundamental one to its translation by the given dominant weight. This is established through quantum Yang–Baxter moves, which biject the objects of the models associated to two such alcove paths, and can be viewed as a generalization of jeu de taquin slides to arbitrary root systems. The purpose of this paper is to give a generalization of quantum Yang–Baxter moves to the quantum alcove model corresponding to an arbitrary weight, which was used to express a general Chevalley formula for the equivariant K-group of semi-infinite flag manifolds. The generalized quantum Yang–Baxter moves give rise to a “sijection” (bijection between signed sets), and are shown to preserve certain important statistics, including weights and heights. As an application, we prove that the generating function of these statistics does not depend on the choice of a reduced alcove path. Also, we obtain an identity for the graded characters of Demazure submodules of level-zero extremal weight modules over a quantum affine algebra, which can be thought of as a representation-theoretic analogue of the mentioned Chevalley formula. [ABSTRACT FROM AUTHOR]

In this paper, we complete the characterization of tame irreducible extended affine root supersystems. We give a complete description of tame irreducible extended affine root supersystems of type X = C(1, 1),C(1, 2), C(2, 2) and BC(1, 1) and determine isomorphic classes. [ABSTRACT FROM AUTHOR]

Zigzag strip bundles are combinatorial models realizing the crystals B (∞) and the highest weight crystals B (λ) over quantum affine algebras U q (g) , and they are closely related with Nakajima monomials and Kashiwara embeddings of specific type. In this paper, we introduce new zigzag strip bundles for U q (g) , which we call reverse zigzag strip bundles in order to distinguish from the previous ones, and we give explicit 1-1 correspondences with the set of Nakajima monomials and the image of Kashiwara embedding which are different from those appeared in the study of the zigzag strip bundles. Another main result is to give different points of view about zigzag strip bundles and reverse zigzag strip bundles. Indeed, the previous 1-1 correspondences between the sets of zigzag strip bundles or reverse zigzag strip bundles, and the sets of Nakajima monomials or images of Kashiwara embeddings were determined based on the number of blocks in the columns of these strip bundles. In our work, we use the number of blocks in the rows of zigzag strip bundles and reverse zigzag strip bundles to provide characterizations of Nakajima monomials and images of Kashiwara embeddings that differ from the previous ones. That is, we give characterizations of Nakajima monomials and Kashiwara embeddings of 4 types. Further, we discuss the connection between zigzag strip bundles and reverse zigzag strip bundles. [ABSTRACT FROM AUTHOR]

The Miura transformation plays a crucial role in the study of integrable systems. There have been various extensions of the Miura transformation, which have been used to relate different kinds of integrable equations and to classify the bi‐Hamiltonian structures. In this paper, we are mainly concerned with the geometric aspects of the Miura transformation. The generalized Miura transformations from the mKdV‐type hierarchies to the KdV‐type hierarchies are constructed under both algebraic and geometric settings. It is shown that the Miura transformations not only relate integrable curve flows in different geometries but also induce the transition between different moving frames. Moreover, the Miura transformation gives the factorization of generating operators of constraint Gelfand–Dickey hierarchy. Other geometric formulations are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we study a family of infinite-dimensional Lie algebras X ˆ S , where X stands for the type: A , B , C , D , and S is an abelian group, which generalize the A , B , C , D series of trigonometric Lie algebras. Among the main results, we identify X ˆ S with what are called the covariant algebras of the affine Lie algebra L S ˆ with respect to some automorphism groups, where L S is an explicitly defined associative algebra viewed as a Lie algebra. We then show that restricted X ˆ S -modules of level ℓ naturally correspond to equivariant quasi modules for affine vertex algebras related to L S. Furthermore, for any finite cyclic group S , we completely determine the structures of these four families of Lie algebras, showing that they are essentially affine Kac-Moody Lie algebras of certain types. [ABSTRACT FROM AUTHOR]

In this paper, a new categorical crystal structure for the quantum affine algebras is presented. We introduce the notion of extended crystals B ^ 픤 ⁢ (∞) for an arbitrary quantum group U q ⁢ (픤) , which is the product of infinite copies of the crystal B ⁢ (∞) . For a complete duality datum 풟 in the Hernandez–Leclerc category 풞 픤 0 of a quantum affine algebra U q ′ ⁢ (픤) , we prove that the set ℬ 풟 ⁢ (픤) of the isomorphism classes of simple modules in 풞 픤 0 has an extended crystal structure isomorphic to B ^ 픤 fin ⁢ (∞) . An explicit combinatorial description of the extended crystal ℬ 풟 ⁢ (픤) for affine type A n (1) is given in terms of affine highest weights. [ABSTRACT FROM AUTHOR]

In this paper, we construct various simple vertex superalgebras which are extensions of affine vertex algebras, by using abelian cocycle twists of representation categories of quantum groups. This solves the Creutzig and Gaiotto conjectures (Creutzig and Gaiotto in Comm Math Phys 379:785–845, 2020, Conjecture 1.1 and 1.4) in the case of type ABC. If the twist is trivial, the resulting algebras correspond to chiral differential operators in the chiral case, and to WZW models in the non-chiral case. [ABSTRACT FROM AUTHOR]

This is the second of a series of papers devoted to the study of relaxed highest-weight modules over affine vertex algebras and W-algebras. The first [K. Kawasetsu and D. Ridout, Relaxed highest-weight modules I: Rank 1 cases, Commun. Math. Phys. 368 (2019) 627–663, arXiv:1803.01989 [math.RT]] studied the simple "rank- 1 " affine vertex superalgebras L k ( 2) and L k ( (1 | 2)) , with the main results including the first complete proofs of certain conjectured character formulae (as well as some entirely new ones). Here, we turn to the question of classifying relaxed highest-weight modules for simple affine vertex algebras of arbitrary rank. The key point is that this can be reduced to the classification of highest-weight modules by generalizing Olivier Mathieu's coherent families [O. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier  (Grenoble)  50 (2000) 537–592]. We formulate this algorithmically and illustrate its practical implementation with several detailed examples. We also show how to use coherent family technology to establish the non-semisimplicity of category in one of these examples. [ABSTRACT FROM AUTHOR]

In this paper, we examine maximal dimensions of Lie algebras of infinitesimal affine transformations of tangent bundles with a complete lift connection over spaces with a semisymmetric connection. [ABSTRACT FROM AUTHOR]

In this paper, we answer a question asked by Kaiming Zhao at the mini course "Simple representations of the Virasoro algebra" at Xiamen mathematical center in the fall term of 2021. More precisely, we prove that if one special element in the Lie algebra g (including the Virasoro algebra, the Heisenberg-Virasoro algebra and affine Virasoro algebras) acts locally nilpotently on an irreducible module V, then V is a restricted g -module. [ABSTRACT FROM AUTHOR]

In [12] , the authors classified a class of simple modules over untwised affine Kac-Moody Lie algebras, on these modules each weight vector of the positive parts of affine Kac-Moody Lie algebras acts locally finitely. In this paper, for all affine-Virasoro algebras we also classify all simple modules with the similar property. We determine that there are precisely three classes of simple modules on which each weight vector of the positive part of any affine-Virasoro algebra acts locally finitely: simple highest weight or Whittaker Virasoro algebra modules, simple highest weight or Whittaker affine Lie algebra modules, and simple highest weight or Whittaker affine-Virasoro algebra modules which are neither simple Virasoro algebra modules nor simple affine Lie algebra modules. We also obtain three equivalent conditions to characterize such simple modules over affine-Virasoro algebras. [ABSTRACT FROM AUTHOR]

For a split semisimple algebraic group H with its split maximal torus S, let f:CH(BH)→CH(BS)W$f: \mathop {\mathrm{CH}}\nolimits (\mathcal {B}H)\rightarrow \mathop {\mathrm{CH}}\nolimits (\mathcal {B}S)^W$ be the restriction homomorphism of the Chow rings CH$\mathop {\mathrm{CH}}\nolimits$ of the classifying spaces B$\mathcal {B}$ of H and S, where W is the Weyl group. A constraint on the image of f, given by the Steenrod operations, has been applied to the spin groups in a previous paper. Here, we describe and apply to the spin groups another constraint, which is given by the reductive envelopes of H. We also recover in this way some older results on orthogonal groups. [ABSTRACT FROM AUTHOR]

In this paper we show that evolution algebras over any given field \Bbbk are universally finite. In other words, given any finite group G, there exist infinitely many regular evolution algebras X such that Aut(X)\cong G. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme \mathbf {G} over \Bbbk is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra. [ABSTRACT FROM AUTHOR]

Let $U_q'({\mathfrak {g}})$ be a quantum affine algebra with an indeterminate $q$ , and let $\mathscr {C}_{\mathfrak {g}}$ be the category of finite-dimensional integrable $U_q'({\mathfrak {g}})$ -modules. We write $\mathscr {C}_{\mathfrak {g}}^0$ for the monoidal subcategory of $\mathscr {C}_{\mathfrak {g}}$ introduced by Hernandez and Leclerc. In this paper, we associate a simply laced finite-type root system to each quantum affine algebra $U_q'({\mathfrak {g}})$ in a natural way and show that the block decompositions of $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$ are parameterized by the lattices associated with the root system. We first define a certain abelian group $\mathcal {W}$ (respectively $\mathcal {W} _0$) arising from simple modules of $\mathscr {C}_{\mathfrak {g}}$ (respectively $\mathscr {C}_{\mathfrak {g}}^0$) by using the invariant $\Lambda ^\infty$ introduced in previous work by the authors. The groups $\mathcal {W}$ and $\mathcal {W} _0$ have subsets $\Delta$ and $\Delta _0$ determined by the fundamental representations in $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$ , respectively. We prove that the pair $(\mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ is an irreducible simply laced root system of finite type and that the pair $(\mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} , \Delta)$ is isomorphic to the direct sum of infinite copies of $(\mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ as a root system. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and to investigate the basic properties of qconvex, q-affine and q-concave sequences and to establish their surprising connection to Chebyshev polynomials of the first and of the second kind. One of the main results shows that qconcave sequences are the pointwise minima of q-affine sequences. As an application, we consider a nonlinear selfmap of the n-dimensional space and prove that it has a unique fixed point. For the proof of this result, we introduce a new norm on the space in terms of a q-concave sequence and show that the nonlinear operator becomes a contraction with respect to this norm, and hence, the Banach Fixed Point theorem can be applied. [ABSTRACT FROM AUTHOR]

Let p be a prime, k be a p-closed field of characteristic different from p, and 1 → T → G → F → 1 be an exact sequence of algebraic groups over k, where T is a torus and F is a finite p-group. In this paper, we study the essential dimension ed(G; p) of G at p. R. Lötscher, M. MacDonald, A. Meyer, and the first author showed that min dim(V) - dim(G) 6 ed(G; p) 6 min dim(W) - dim(G); where V and W range over the p-faithful and p-generically free k-representations of G, respectively. In the special case where G = F, one recovers the formula for ed(F; p) proved earlier by N. Karpenko and A. Merkurjev. In the case where F = T, one recovers the formula for ed(T; p) proved earlier by R. Lötscher et al. In both of these cases, the upper and lower bounds on ed(G; p) given above coincide. In general, there is a gap between them. Lötscher et al. conjectured that the upper bound is, in fact, sharp; that is, ed(G; p) = min dim(W) - dim(G), where W ranges over the p-generically free representations. We prove this conjecture in the case where F is diagonalizable. [ABSTRACT FROM AUTHOR]

In this paper, we used a method based on mixing formalism to improve theoretical predictions for energy spectra and quadrupole transition rates of 1 1 2 Cd nucleus. To this aim, the transitional Hamiltonian in the affine SU(1,1) algebra has been extended by adding the O(6) Casimir operator and mixing Hamiltonian to improve the description of 4 2 + and 2 3 + intruder levels of this nucleus. We also considered the wave functions of both regular and intruder states as a combination in the N and N +2 bosons spaces. The results confirm the advantages of using such mixing approaches and describe the energy and transition rates with high accuracy. [ABSTRACT FROM AUTHOR]

Learning and synthesizing stabilizing controllers for unknown nonlinear control systems is a challenging problem for real-world and industrial applications. Koopman operator theory allows one to analyze nonlinear systems through the lens of linear systems and nonlinear control systems through the lens of bilinear control systems. The key idea of these methods lies in the transformation of the coordinates of the nonlinear system into the Koopman observables, which are coordinates that allow the representation of the original system (control system) as a higher dimensional linear (bilinear control) system. However, for nonlinear control systems, the bilinear control model obtained by applying Koopman operator based learning methods is not necessarily stabilizable. Simultaneous identification of stabilizable lifted bilinear control systems as well as the associated Koopman observables is still an open problem. In this paper, we propose a framework to construct such stabilizable bilinear models and identify their associated observables from data by simultaneously learning a bilinear Koopman embedding for the underlying unknown control affine nonlinear system as well as a Control Lyapunov Function (CLF) for the Koopman based bilinear model using a learner and falsifier. Our proposed approach thereby provides provable guarantees of asymptotic stability for the Koopman based representation of the unknown control affine nonlinear control system as a bilinear system. Numerical simulations are provided to validate the efficacy of our proposed class of stabilizing feedback controllers for unknown control-affine nonlinear systems. [ABSTRACT FROM AUTHOR]

In this paper, we give a counterexample to Zariski's cancellation problem for principal ${\mathbb{G}}_{a}$ G a -bundles over certain open subvarieties of non- $\mathbb{A}^{1}$ A 1 -uniruled affine varieties. [ABSTRACT FROM AUTHOR]

Generalized double affine Hecke algebras (GDAHA) are flat deformations of the group algebras of 2-dimensional crystallographic groups associated to star-shaped simply laced affine Dynkin diagrams. In this paper, we first construct a functor that sends representations of the D ˜ 4 -type GDAHA to representations of the E ˜ 6 -type one for specialised parameters. Then, under no restrictions on the parameters, we construct embeddings of both GDAHAs of type D ˜ 4 and E ˜ 6 into matrix algebras over quantum cluster X -varieties, thus linking to the theory of higher Teichmüller spaces. For E ˜ 6 , the two explicit representations we provide over distinct quantum tori are shown to be related by quiver reductions and mutations. [ABSTRACT FROM AUTHOR]

We study solutions of the reflection equation related to the quantum affine algebra U q ( s l n ^) . First, we explain how to construct a family of stochastic integrable vertex models with fixed boundary conditions. Then, we construct upper- and lower-triangular solutions of the reflection equation related to symmetric tensor representations of U q ( s l n ^) with arbitrary spin. We also prove the star–star relation for the Boltzmann weights of the Ising-type model, conjectured by Bazhanov and Sergeev, and use it to verify certain properties of the solutions obtained. [ABSTRACT FROM AUTHOR]

Given an affine algebraic variety X, we prove that if the neutral component \mathrm {Aut}^\circ (X) of the automorphism group consists of algebraic elements, then it is nested, i.e., is a direct limit of algebraic subgroups. This improves our earlier result (see Perepechko and Regeta [Transform. Groups 28 (2023), pp. 401–412]). To prove it, we obtain the following fact. If a connected ind-group G contains a closed connected nested ind-subgroup H\subset G, and for any g\in G some positive power of g belongs to H, then G=H. [ABSTRACT FROM AUTHOR]

We produce counterexamples to the birational Torelli theorem for Calabi–Yau manifolds in arbitrarily high dimension: this is done by exhibiting a series of non-birational pairs of Calabi–Yau (n^2-1)-folds which, for n \geq 2 even, admit an isometry between their middle cohomologies. These varieties also satisfy an \mathbb {L}-equivalence relation in the Grothendieck ring of varieties, i.e. the difference of their classes annihilates a power of the class of the affine line. We state this last property for a broader class of Calabi–Yau pairs, namely all those which are realized as pushforwards of a general (1,1)-section on a homogeneous roof in the sense of Kanemitsu, along its two extremal contractions. [ABSTRACT FROM AUTHOR]

Let G be an affine algebraic group defined over a field k of characteristic 0. We study the derived moduli space of G-local systems on a pointed connected CW complex X trivialized at the basepoint of X. This derived moduli space is represented by an affine DG scheme RLocG(X,*): we call the (co)homology of the structure sheaf of RLocG(X,*) the representation homology of X in G and denote it by HR*(X,G). The 0-dimensional homology, HR0(X,G), is isomorphic to the coordinate ring of the G-representation variety RepG[-1(X)] of the fundamental group of X -- a well-known algebro-geometric invariant that plays a role in many areas of topology. The higher representation homology is much less studied. In particular, when X is simply connected, HR0(X,G) is trivial but HR*(X,G) is still an interesting rational invariant of X that depends on the Lie algebra of G. In this paper, we use Quillen's rational homotopy theory to compute the representation homology of an arbitrary simply connected space (of finite rational type) in terms of its Lie and Sullivan algebraic models. When G is reductive, we also compute HR*(X,G)G, the G-invariant part of representation homology, and study the question when HR*(X,G)G is free of locally finite type as a graded commutative algebra. This question turns out to be related to the so-called Strong Macdonald Conjecture, a celebrated result in representation theory proposed (as a conjecture) by Feigin and Hanlon in the 1980s and proved by Fishel, Grojnowski and Teleman in 2008. Reformulating the Strong Macdonald Conjecture in topological terms, we give a simple characterization of spaces X for which HR*(X,G)G is a graded symmetric algebra for any complex reductive group G. 1980s and proved by Fishel, Grojnowski and Teleman in 2008. Reformulating the Strong Macdonald Conjecture in topological terms, we give a simple characterization of spaces X forwhich HR*(X,G)G is a graded symmetric algebra for any complex reductive group G. [ABSTRACT FROM AUTHOR]

Motivated by Maulik-Okounkov stable maps associated to quiver varieties, we define and construct algebraic stable maps on tensor products of representations in the category \mathcal {O} of the Borel subalgebra of an arbitrary untwisted quantum affine algebra. Our representation-theoretical construction is based on the study of the action of Cartan-Drinfeld subalgebras. We prove the algebraic stable maps are invertible and depend rationally on the spectral parameter. As an application, we obtain new R-matrices in the category \mathcal {O} and we establish that a large family of simple modules, including the prefundamental representations associated to Q-operators, generically commute as representations of the Cartan-Drinfeld subalgebra. We also establish categorified QQ^*-systems in terms of the R-matrices we construct. [ABSTRACT FROM AUTHOR]

It is essential to guarantee product quality by detecting defects in industrial printed labels. Defect detection based on reference comparison is a common method to achieve this task. However, this method yields poor performance under large deformations of printed labels, due to the lack of accurate registration ability between testing images and reference images. Therefore, it is an urgent task to achieve accurate image registration for printed label defect detection. In this paper, a coarse-to-fine end-to-end registration network, named APPR-Net, is proposed to deal with the large deformation. First, we adopt a strategy of image patch splitting and stitching to improve the scalability of image resolution. Second, we design a four-stream affine transformation module followed by a multi-scale pyramid registration network, where a coarse registration is obtained by the former module and then gradually refined by the later network in a coarse-to-fine manner. Third, we introduce a distortion loss function to constrain the text distortion of the warped image after image registration. Finally, to simulate printed labels with defects and large deformation, we build a synthetic database based on real-world industrial printed labels for performance comparison. The results demonstrate that the proposed APPR-Net significantly outperforms other compared methods. • Propose a coarse-to-fine registration network to deal with large deformation. • Take advantages of image patch splitting and stitching strategy. • Design a four-stream affine transformation module followed by a pyramid network. • Introduce a distortion loss function to constrain text distortions. • Build a synthetic database based on real-world printed labels. [ABSTRACT FROM AUTHOR]

In this paper we study the integral form of the lattice vertex algebra V L. We show that divided powers of general vertex operators preserve the integral lattice spanned by Schur functions indexed by partition-valued functions. We also show that the Garland operators, counterparts of divided powers of Heisenberg elements in affine Lie algebras, also preserve the integral form. These construe analogs of the Kostant Z -forms for the enveloping algebras of simple Lie algebras and the algebraic affine Lie groups in the situation of the lattice vertex algebras. [ABSTRACT FROM AUTHOR]

Motivated by the potential connection between metric-affine gravity and linear generalized uncertainty principle (GUP) in the phase space, we develop a covariant form of linear GUP and an associated modified Poincaré algebra, which exhibits distinctive behavior, nearing nullity at the minimal length scale proposed by linear GUP. We use three-torus geometry to visually represent linear GUP within a covariant framework. The three-torus area provides an exact geometric representation of Bekenstein's universal bound. We depart from Bousso's approach, which adapts Bekenstein's bound by substituting the Schwarzschild radius ( r s ) with the radius (R) of the smallest sphere enclosing the physical system, thereby basing the covariant entropy bound on the sphere's area. Instead, our revised covariant entropy bound is described by the area of a three-torus, determined by both the inner radius r s and outer radius R where r s ⩽ R due to gravitational stability. This approach results in a more precise geometric representation of Bekenstein's bound, notably for larger systems where Bousso's bound is typically much larger than Bekensetin's universal bound. Furthermore, we derive an equation that turns the standard uncertainty inequality into an equation when considering the contribution of the three-torus covariant entropy bound, suggesting a new avenue of quantum gravity. [ABSTRACT FROM AUTHOR]

The SL (2 , Z) -symmetry of Cherednik's spherical double affine Hecke algebras in Macdonald theory includes a distinguished generator which acts as a discrete time evolution of Macdonald operators, which can also be interpreted as a torus Dehn twist in type A. We prove for all twisted and untwisted affine algebras of type ABCD that the time-evolved q-difference Macdonald operators, in the t → ∞ q-Whittaker limit, form a representation of the associated discrete integrable quantum Q-systems, which are obtained, in all but one case, via the canonical quantization of suitable cluster algebras. The proof relies strongly on the duality property of Macdonald and Koornwinder polynomials, which allows, in the q-Whittaker limit, for a unified description of the quantum Q-system variables and the conserved quantities as limits of the time-evolved Macdonald operators and the Pieri operators, respectively. The latter are identified with relativistic q-difference Toda Hamiltonians. A crucial ingredient in the proof is the use of the "Fourier transformed" picture, in which we compute time-translation operators and prove that they commute with the Pieri operators or Hamiltonians. We also discuss the universal solutions of Koornwinder-Macdonald eigenvalue and Pieri equations, for which we prove a duality relation, which simplifies the proofs further. [ABSTRACT FROM AUTHOR]