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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]

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]

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]

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]

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]

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]

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]

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]

We define two realizations of the affine super-Yangian for a special linear Kac–Moody superalgebra and an arbitrary system of simple roots: in terms of a "minimalist" system of generators and in terms of the new system of Drinfeld generators. We construct an isomorphism between these two realizations of the super-Yangian in the case of a fixed system of simple roots. We consider the Weyl groupoid, define its quantum analogue, and its action on the super Yangians defined by the systems of simple roots. We show that the action of the quantum Weyl groupoid induces isomorphisms between super-Yangians defined by different simple root systems. [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]

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]

Let U ħ (g ^) be the untwisted affinization of a symmetrizable quantum Kac-Moody algebra U ħ (g) . For ℓ ∈ C , we construct an ħ -adic quantum vertex algebra V g ^ , ħ (ℓ , 0) , and establish a one-to-one correspondence between ϕ -coordinated V g ^ , ħ (ℓ , 0) -modules and restricted U ħ (g ^) -modules of level ℓ . Suppose that ℓ is a positive integer. We construct a quotient ħ -adic quantum vertex algebra L g ^ , ħ (ℓ , 0) of V g ^ , ħ (ℓ , 0) , and establish a one-to-one correspondence between certain ϕ -coordinated L g ^ , ħ (ℓ , 0) -modules and restricted integrable U ħ (g ^) -modules of level ℓ . Suppose further that g is of finite type. We prove that L g ^ , ħ (ℓ , 0) / ħ L g ^ , ħ (ℓ , 0) is isomorphic to the simple affine vertex algebra L g ^ (ℓ , 0) . [ABSTRACT FROM AUTHOR]

A family of vertex algebras whose universal Verma modules coincide with the cohomology of affine Laumon spaces is found. This result is based on an explicit expression for the generating function of Poincaré polynomials of these spaces. There is a variant of quantum Hamiltonian reduction that realizes vertex algebras which we call iterated W -algebras and our main conjecture is that the vertex algebras associated to the affine Laumon spaces are subalgebras of iterated W -algebras. [ABSTRACT FROM AUTHOR]

Subspace clustering (SC) exploits the potential capacity of self-expressive modeling of unsupervised learning frameworks, representing each data point as a linear combination of the other related data points. Advanced self-expressive approaches construct an affinity matrix from the representation coefficients by imposing an additional regularization, reflecting the prior data distribution. An affine constraint is widely used for regularization in subspace clustering studies according on the grounds that, in real-world applications, data points usually lie in a union of multiple affine subspaces rather than linear subspaces. However, a strict affine constraint is not flexible enough to handle the real-world cases, as the observed data points are always corrupted by noise and outliers. To address this issue, we introduce the concept of scalable affine constraint to the SC formulation. Specifically, each coefficient vector is constrained to sum up to a soft scalar s rather than 1. The proposed method can estimate the most appropriate value of scalar s in the optimization stage, adaptively enhancing the clustering performance. Besides, as clustering benefits from multiple representations, we extend the scalable affine constraint to a multi-view clustering framework designed to achieve collaboration among the different representations adopted. An efficient optimization approach based on ADMM is developed to minimize the proposed objective functions. The experimental results on several datasets demonstrate the effectiveness of the proposed clustering approach constrained by scalable affine regularisation, with superior performance compared to the state-of-the-art. [ABSTRACT FROM AUTHOR]

We analyze the boundaries of the moduli spaces of compactifications of the heterotic string on Td, making particular emphasis on d = 2 and its F-theory dual. We compute the OPE algebras as we approach all the infinite distance limits that correspond to (possibly partial) decompactification limits in some dual frame. When decompactifying k directions, we find infinite towers of states becoming light that enhance the algebra arising at a given point in the moduli space of the Td−k compactification to its k-loop version, where the central extensions are given by the k KK vectors. For T2 compactifications, we reproduce all the affine algebras that arise in the F-theory dual, and show all the towers explicitly, including some that are not manifest in the F-theory counterparts. Furthermore, we construct the affine SO(32) algebra arising in the full decompactification limit, both in the heterotic and in the F-theory sides, showing that not only affine algebras of exceptional type arise in the latter. [ABSTRACT FROM AUTHOR]

We establish the Q Q ~ -systems for the twisted quantum affine algebras that were conjectured in Frenkel and Hernandez (Commun Math Phys 362(2):361–414, 2018). We develop the representation theory of Borel subalgebra of twisted quantum affine algebras and we construct their prefundamental representations. We also propose a general conjecture on the relations between twisted and non-twisted types. We prove this conjecture for some particular classes of representations, including prefundamental representations. [ABSTRACT FROM AUTHOR]

Let g be a simple Lie algebra. We study 1/2-BPS Wilson loops of supersymmetric 5d g -type quiver gauge theories on a circle, in a non-trivial instanton background. The Wilson loops are codimension 4 defects of the gauge theory, and their interaction with self-dual instantons is captured by a modified 1d ADHM quantum mechanics. We compute the partition function as its Witten index. This index is a "qq-character" of a finite-dimensional irreducible representation of the quantum affine algebra U q (g ^) . Using gauge/vortex duality, we can understand the 5d physics in 3d gauge theory terms. Namely, we reinterpret the 5d theory with vortex flux from the point of view of the vortices themselves. This vortex perspective has an advantage: it has yet another dual description in terms of deformed g -type Toda Theory on a cylinder, in free field formalism. We show that the gauge theory partition function is equal to a chiral correlator of the deformed Toda Theory, with stress tensor and higher spin operator insertions. We derive all the above results from type IIB string theory, compactified on a resolved ADE singularity times a cylinder with punctures, with various branes wrapping the blown-up 2-cycles. [ABSTRACT FROM AUTHOR]