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For any complete C -algebraic variety X and its associated compact C -analytic space X , it follows from the well known GAGA principle that the algebraic Picard group P i c (X) and the analytic Picard group P i c (X) are isomorphic. Our main purpose here is to investigate an analogous situation for non complete situation, namely C -algebraic affine hypersurfaces. [ABSTRACT FROM AUTHOR]

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus T q , the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of T q ; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals. [ABSTRACT FROM AUTHOR]

We consider reduced imaginary Verma modules for the untwisted quantum affine algebras U q (g ˆ) and define a crystal-like base which we call imaginary crystal base using the Kashiwara algebra K q constructed in earlier work by Ben Cox and two of the authors. We prove the existence of the imaginary crystal base for any object in a suitable category O r e d , i m q containing the reduced imaginary Verma modules for U q (g ˆ). [ABSTRACT FROM AUTHOR]

This article, published in the Journal of Mathematical Sciences, explores the computation of the component group of an arbitrary real algebraic group. The author demonstrates that the component group is always an elementary Abelian 2-group. The computation is based on structure results on algebraic groups and Galois cohomology methods. The paper provides necessary material on algebraic groups and Galois cohomology, and includes an example of the real locus of an elliptic curve. This document discusses the computation of the group of connected components of the real locus of a connected algebraic group defined over R. It introduces the concepts of cocycles, exact sequences of groups, and Galois cohomology. The main result of the document is a formula for computing the group of connected components, which involves lattices and coroot lattices. The proof of the formula is similar to a previous proof in another paper. The given text discusses the component groups of certain mathematical structures, specifically linear algebraic groups and Abelian varieties. It presents theorems and examples related to these component groups, providing mathematical proofs and explanations. The text explores different cases and scenarios, such as when the period lattice is preserved by complex multiplication or when it is multiplied by certain values. The examples given include elliptic curves and their connected components. The text is written in a technical and mathematical language, and it may be useful for researchers studying these specific topics. [Extracted from the article]

We establish a Schur-Weyl duality between a shifted quantum affine algebra and an Ariki-Koike algebra. Then, we realize a cyclotomic q -Schur algebra in the context of the Schur-Weyl duality. [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]

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]

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]

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]

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]

One of the features of Baxter's Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine sl 2 and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras. [ABSTRACT FROM AUTHOR]

We introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them which generalize the T-systems among KR-modules and unipotent quantum minors in the quantum unipotent coordinate algebras simultaneously. We develop new combinatorial tools: admissible chains of i -boxes which produce commuting families of affine determinantial modules, and box moves which describe the T-system in a combinatorial way. Using these results, we prove that various module categories over the quantum affine algebras provide monoidal categorifications of cluster algebras. As special cases, Hernandez-Leclerc categories C g 0 and C g − provide monoidal categorifications of the cluster algebras for an arbitrary quantum affine algebra. [ABSTRACT FROM AUTHOR]

Inspired by Sugawara operators, we introduce a new basis for the universal non-degenerate Whittaker module of level κ over the classical affine Kac-Moody algebra of type B N (1) , C N (1) or D N (1). This basis is different from the one derived from the PBW basis and would help us to understand the structure of Whittaker modules. As a result, we classify simple non-degenerate Whittaker modules for the classical affine Kac-Moody algebras whether at the critical or noncritical level. [ABSTRACT FROM AUTHOR]

In pursuit of a noncommutative spectrum functor, we argue that the Heyneman-Sweedler finite dual coalgebra can be viewed as a quantization of the maximal spectrum of a commutative affine algebra, integrating prior perspectives of Takeuchi, Batchelor, Kontsevich-Soibelman, and Le Bruyn. We introduce fully residually finite-dimensional algebras A as those with enough finite-dimensional representations to let A ∘ act as an appropriate depiction of the noncommutative maximal spectrum of A ; importantly, this class includes affine noetherian PI algebras. In the case of prime affine algebras that are module-finite over their center, we describe how the Azumaya locus is represented in the finite dual. This is used to describe the finite dual of quantum planes at roots of unity as an endeavor to visualize the noncommutative space on which these algebras act as functions. Finally, we discuss how a similar analysis can be carried out for other maximal orders over surfaces. [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]

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]

We generalize the theory of linked partition ideals due to Andrews using finite automata in formal language theory and apply it to prove three Rogers-Ramanujan type identities for modulus 14 that were posed by Nandi through a vertex operator theoretic construction of the level 4 standard modules of the affine Lie algebra A(2) 2 . [ABSTRACT FROM AUTHOR]

An algebraic variety X is called a homogeneous variety if the automorphism group Aut (X) acts on X transitively, and a homogeneous space if there exists a transitive action of an algebraic group on X. We prove a criterion of smoothness of a suspension to construct a wide class of homogeneous varieties. As an application, we give criteria for a Danielewski surface to be a homogeneous variety and a homogeneous space. Also, we construct affine suspensions of arbitrary dimension that are homogeneous varieties but not homogeneous spaces. [ABSTRACT FROM AUTHOR]

Recently, Blasiak–Morse–Seelinger introduced symmetric func- tions called Katalan functions, and proved that the K-theoretic k-Schur functions due to Lam–Schilling–Shimozono form a subfamily of the Katalan functions. They conjectured that another subfamily of Katalan functions called closed k-Schur Katalan functions is identified with the Schubert structure sheaves in the K-homology of the affine Grassmannian. Our main result is a proof of this conjecture. We also study a K-theoretic Peterson isomorphism that Ikeda, Iwao, and Maeno constructed, in a nongeometric manner, based on the unipotent solution of the relativistic Toda lattice of Ruijsenaars. We prove that the map sends a Schubert class of the quantum K-theory ring of the flag variety to a closed K-k-Schur Katalan function up to an explicit factor related to a translation element with respect to an antidominant coroot. In fact, we prove this map coincides with a map whose existence was conjectured by Lam, Li, Mihalcea, Shimozono, and proved by Kato, and more recently by Chow and Leung. [ABSTRACT FROM AUTHOR]

This is a review article on the quantum toroidal algebras, focusing on their roles in various solvable structures of 2d conformal field theory, supersymmetric gauge theory, and string theory. Using W -algebras as our starting point, we elucidate the interconnection of affine Yangians, quantum toroidal algebras, and double affine Hecke algebras. Our exploration delves into the representation theory of the quantum toroidal algebra of gl 1 in full detail, highlighting its connections to partitions, W -algebras, Macdonald functions, and the notion of intertwiners. Further, we also discuss integrable models constructed on Fock spaces and associated R -matrices, both for the affine Yangian and the quantum toroidal algebra of gl 1. The article then demonstrates how quantum toroidal algebras serve as a unifying algebraic framework that bridges different areas in physics. Notably, we cover topological string theory and supersymmetric gauge theories with eight supercharges, incorporating the AGT duality. Drawing upon the representation theory of the quantum toroidal algebra of gl 1 , we provide a rather detailed review of its role in the algebraic formulations of topological vertex and q q -characters. Additionally, we briefly touch upon the corner vertex operator algebras and quiver quantum toroidal algebras. [ABSTRACT FROM AUTHOR]

We prove that the nonvarying strata of abelian and quadratic differentials in low genus have trivial tautological rings and are affine varieties. We also prove that strata of k -differentials of infinite area are affine varieties for all k. Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of the Hodge bundle for abelian and quadratic differentials of finite area is extremal in the sense that merging two zeros in each stratum leads to an extremal effective divisor in the boundary. A common feature throughout these results is a relation of divisor classes in strata of differentials as well as its incarnation in Teichmüller dynamics. [ABSTRACT FROM AUTHOR]

We construct the basic representation of the double affine Hecke algebra at critical level q = 1 associated to an irreducible reduced affine root system R with a reduced gradient root system. For R of untwisted type such a representation was studied by Oblomkov [A. Oblomkov, Double affine Hecke algebras and Calogero–Moser spaces, Represent. Theory 8(2004) 243–266] and further detailed by Gehles [K. E. Gehles, Properties of Cherednik algebras and graded Hecke algebras, PhD thesis, University of Glasgow (2006)] in the presence of minuscule weights. [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]

We establish an explicit embedding of a quantum affine s l n into a quantum affine s l n + 1 . This embedding serves as a common generalization of two natural, but seemingly unrelated, embeddings, one on the quantum affine Schur algebra level and the other on the non-quantum level. The embedding on the quantum affine Schur algebras is used extensively in the analysis of canonical bases of quantum affine s l n and g l n . The embedding on the non-quantum level is used crucially in a work of Riche and Williamson on the study of modular representation theory of general linear groups over a finite field. The same embedding is also used in a work of Maksimau on the categorical representations of affine general linear algebras. We further provide a more natural compatibility statement of the embedding on the idempotent version with that on the quantum affine Schur algebra level. A g l ^ n -variant of the embedding is also established. [ABSTRACT FROM AUTHOR]

We prove analogs of A. Selberg's result for finitely generated subgroups of \operatorname {Aut}(A) and of Engel's theorem for subalgebras of \operatorname {Der}(A) for a finitely generated associative commutative algebra A over an associative commutative ring. We prove also an analog of the theorem of W. Burnside and I. Schur about local finiteness of torsion subgroups of \operatorname {Aut}(A). [ABSTRACT FROM AUTHOR]

Considering a Monge-Ampère equation with prescribed affine Gauss curvature, we first show the completeness of centroaffine metric on the convex domain and derive a gradient estimate of the convex solution and then give different orders of two eigenvalues of the Hessian with respect to the distance function. We also show that the curvature of level sets of the convex solution is uniformly bounded, and show that there exist a class of Euclidean-complete hyperbolic surfaces with prescribed affine Gauss curvature and with bounded affine principal curvatures. [ABSTRACT FROM AUTHOR]

Using compactifications in the logarithmic cotangent bundle, we obtain a formula for the Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement. This generalizes earlier results of Aluffi and Wu‐Zhou. The first application of our formula is a geometric description of Chern‐Mather classes of an arbitrary very affine variety, generalizing earlier results of Huh which held under the smooth and schön assumptions. As the second application, we prove an involution formula relating sectional maximum likelihood (ML) degrees and ML bidegrees, which was conjectured by Huh and Sturmfels in 2013. [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]

We consider certain infinite dimensional modules of level 1 for the double Yangian DY (gl 2) which are based on the Iohara–Kohno realization. We show that they possess topological bases of Feigin–Stoyanovsky-type, i.e. the bases expressed in terms of semi-infinite monomials of certain integrable operators which stabilize and satisfy the difference two condition. Finally, we give some applications of these bases to the representation theory of the corresponding quantum affine vertex algebra. [ABSTRACT FROM AUTHOR]

We study the category Osh of representations over a shifted Yangian. This category has a tensor product structure and contains distinguished modules, the positive prefundamental modules and the negative prefundamental modules. Motivated by the representation theory of the Borel subalgebra of a quantum affine algebra and by the relevance of quantum integrable systems in this context, we prove that tensor products of prefundamental modules with irreducible modules are either cyclic or cocyclic. This implies the existence and uniqueness of morphisms, the R-matrices, for such tensor products. We prove the R-matrices are polynomial in the spectral parameter, and we establish functional relations for the R-matrices. As applications, we prove the Jordan-Holder property in the category Osh. We also obtain a proof, uniform for any finite type, that any irreducible module factorizes through a truncated shifted Yangian. [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]

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, 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]

This article concerns a question asked by M. V. Nori on homotopy of sections of projective modules defined on the polynomial algebra over a smooth affine domain R. While this question has an affirmative answer, it is known that the assertion does not hold if: (1) dim (R) = 2 ; or (2) d ≥ 3 but R is not smooth. We first prove that an affirmative answer can be given for dim (R) = 2 when R is an F ‾ p -algebra. Next, for d ≥ 3 we find the precise obstruction for the failure in the singular case. Further, we improve a result of Mandal (related to Nori's question) in the case when the ring A is an affine F ‾ p -algebra of dimension d. We apply this improvement to define the n -th Euler class group E n (A) , where 2 n ≥ d + 2. Moreover, if A is smooth, we associate to a unimodular row v of length n + 1 its Euler class e (v) ∈ E n (A) and show that the corresponding stably free module, say, P (v) has a unimodular element if and only if e (v) vanishes in E n (A). [ABSTRACT FROM AUTHOR]

In finite-dimensional simple Lie algebras and affine Kac-Moody Lie algebras, Chevalley involutions are crucial ingredients of the modular theory. Towards establishing the modular theory for extended affine Lie algebras, we investigate the existence of "Chevalley involutions" for Lie tori and extended affine Lie algebras. We first discuss how to lift a Chevalley involution from the centerless core which is characterized to be a centerless Lie torus to the core and then to the entire extended affine Lie algebra. We then prove by a type-dependent argument the existence of Chevalley involutions for centerless Lie tori. [ABSTRACT FROM AUTHOR]

We construct a class of modules over the affine Lie algebra sl ˆ 2 from the modules over the degree-2 Weyl algebra K 2 and modules over the 2-dimensional solvable Lie algebra b. We determine the irreducibility and isomorphisms of these modules with examples given by Bavula's construction of modules for generalized Weyl algebras in [5]. Finally, we show these irreducible sl ˆ 2 -modules are generally new. [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]

Subregular W-algebras are an interesting and increasingly important class of quantum hamiltonian reductions of affine vertex algebras. Here, we show that the n + 1 subregular W-algebra can be realized in terms of the n + 1 regular W-algebra and the half lattice vertex algebra Π. This generalizes the realizations found for n = 1 and 2 in [D. Adamović, Realizations of simple affine vertex algebras and their modules: The cases s l (2) ̂ and o s p (1 , 2) ̂ , Comm. Math. Phys. 366 (2019) 1025–1067, arXiv:1711.11342 [math.QA]; D. Adamović, K. Kawasetsu and D. Ridout, A realization of the Bershadsky–Polyakov algebras and their relaxed modules, Lett. Math. Phys., 111 (2021) 1–30, arXiv:2007.00396 [math.QA]] and can be interpreted as an inverse quantum hamiltonian reduction in the sense of Adamović. We use this realization to explore the representation theory of n + 1 subregular W-algebras. Much of the structure encountered for 2 and 3 is also present for n + 1 . Particularly, the simple n + 1 subregular W-algebra at nondegenerate admissible levels can be realized purely in terms of the W n + 1 minimal model vertex algebra and Π. [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]

Let g be an algebra over K with a bilinear operation [ ⋅ , ⋅ ] : g × g → g not necessarily associative. For A ⊆ g , let A k be the set of elements of g written combining k elements of A via + and [ ⋅ , ⋅ ]. We show a "sum-bracket theorem" for simple Lie algebras over K of the form g = sl n , so n , sp 2 n , e 6 , e 7 , e 8 , f 4 , g 2 : if char (K) is not too small, we have growth of the form | A k | ≥ | A | 1 + ε for all generating symmetric sets A away from subfields of K. Over F p in particular, we have a diameter bound matching the best analogous bounds for groups of Lie type [2]. As an independent intermediate result, we prove also an estimate of the form | A ∩ V | ≤ | A k | dim ⁡ (V) / dim ⁡ (g) for linear affine subspaces V of g. This estimate is valid for all simple algebras, and k is especially small for a large class of them including associative, Lie, and Mal'cev algebras, and Lie superalgebras. [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 simple, simply connected complex affine algebraic group 퐺, we prove the existence of a flat projective connection on the bundle of nonabelian theta functions on the moduli spaces of semistable parabolic 퐺-bundles for families of smooth projective curves with marked points. [ABSTRACT FROM AUTHOR]

We consider arc spaces for the compositions of Plücker and Veronese embeddings of the flag varieties for simple Lie groups of types ADE. The arc spaces are not reduced and we consider the homogeneous coordinate rings of the corresponding reduced schemes. We show that each graded component of a homogeneous coordinate ring is a cocyclic module over the current algebra and is acted upon by the algebra of symmetric polynomials. We show that the action of the polynomial algebra is free and that the fiber at the special point of a graded component is isomorphic to an affine Demazure module whose level is the degree of the Veronese embedding. In type A1 (which corresponds to the Veronese curve), we give the precise list of generators of the reduced arc space. In general type, we introduce the notion of global higher level Demazure modules, which generalizes the standard notion of the global Weyl modules, and identify the graded components of the homogeneous coordinate rings with these modules. [ABSTRACT FROM AUTHOR]

This work deals with the stability analysis of nonlinear sampled-data systems under nonuniform sampling. It establishes novel relationships between specific stability properties of the exact discrete-time model and stability properties of the continuous-time system that is obtained when the maximum admissible sampling period tends to zero. These results can be used to infer stability properties for the sampled-data system by direct inspection of the stability of the mentioned continuous-time system, a task which is typically easier than the analysis of the closed-loop sampled-data system. Compared to the literature, our results allow to prove stronger (asymptotic) sampled-data stability properties for nonlinear systems in cases for which existing results only guarantee practical stability. • Novel links between continuous- and discrete-time models' stability properties. • Simpler stability proofs for nonlinear sampled-data systems under nonuniform sampling. • Present results guarantee stronger (asymptotic) stability properties. • Mild conditions allow for non-globally Lipschitz and non-input-affine systems. [ABSTRACT FROM AUTHOR]