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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]

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]

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]

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 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 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]

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]

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]

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

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]

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]

Schur duality is an equivalence, for d ≤ n, between the category of finite-dimensional representations over C of the symmetric group Sd on d letters, and the category of finite-dimensional representations over C of GL(n,C) whose irreducible subquotients are subquotients of E⊗d, E = Cn. The latter are called polynomial representations homogeneous of degree d. It is based on decomposing E⊗d as a C[Sd] × GL(n,C)-bimodule. It was used by Schur to conclude the semisimplicity of the category of finite-dimensional complex GL(n,C)-modules from the corresponding result for Sd that had been obtained by Young. Here we extend this duality to the affine super case by constructing a functor F: M 7→ M ⊗C[Sd] E⊗d, E now being the super vector space Cm|n, from the category of finite-dimensional C[Sd x Zd]-modules, or representations of the affine Weyl, or symmetric, group Sa d = Sd x Zd, to the category of finite-dimensional representations of the universal enveloping algebra of the affine Lie superalgebra U(bsl(m|n)) that are E⊗d-compatible, namely the subquotients of whose restriction to U(sl(m|n)) are constituents of E⊗d. Both categories are not semisimple. When d < m+n the functor defines an equivalence of categories. As an application we conclude that the irreducible finite-dimensional E⊗d-compatible representations of the affine superalgebra bsl(m|n) are tensor products of evaluation representations at distinct points of C×. [ABSTRACT FROM AUTHOR]

The quantum Sugawara operators associated with a simple Lie algebra g are elements of the center of a completion of the quantum affine algebra U q (g ˆ) at the critical level. By the foundational work of Reshetikhin and Semenov-Tian-Shansky (1990), such operators occur as coefficients of a formal Laurent series ℓ V (z) associated with every finite-dimensional representation V of the quantum affine algebra. As demonstrated by Ding and Etingof (1994), the quantum Sugawara operators generate all singular vectors in generic Verma modules over U q (g ˆ) at the critical level and give rise to a commuting family of transfer matrices. Furthermore, as observed by E. Frenkel and Reshetikhin (1999), the operators are closely related with the q -characters and q -deformed W -algebras via the Harish-Chandra homomorphism. We produce explicit quantum Sugawara operators for the quantum affine algebra of type A which are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This opens a way to understand all the related objects via their explicit constructions. We consider one application by calculating the Harish-Chandra images of the quantum Sugawara operators. The operators act by scalar multiplication in the q -deformed Wakimoto modules and we calculate the eigenvalues by identifying them with the Harish-Chandra images. [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]

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]

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]

In this paper, we explore a canonical connection between the algebra of q -difference operators V ˜ q , affine Lie algebras and affine vertex algebras associated to certain subalgebra A of the Lie algebra gl ∞. We also introduce and study a category R of V ˜ q -modules. More precisely, we obtain a realization of V ˜ q as a covariant algebra of the affine Lie algebra A ⁎ ˆ , where A ⁎ is a 1-dimensional central extension of A. We prove that restricted V q ˜ -modules of level ℓ 12 correspond to Z -equivariant ϕ -coordinated quasi-modules for the vertex algebra V A ˜ (ℓ 12 , 0) , where A ˜ is a generalized affine Lie algebra of A. In the end, we show that objects in the category R are restricted V q ˜ -modules, and we classify simple modules in the category R. [ABSTRACT FROM AUTHOR]

For a complex finite-dimensional simple Lie algebra g , we introduce the notion of Q-datum, which generalizes the notion of a Dynkin quiver with a height function from the viewpoint of Weyl group combinatorics. Using this notion, we develop a unified theory describing the twisted Auslander–Reiten quivers and the twisted adapted classes introduced in Oh and Suh (J Algebra 535(1):53–132, 2019) with an appropriate notion of the generalized Coxeter elements. As a consequence, we obtain a combinatorial formula expressing the inverse of the quantum Cartan matrix of g , which generalizes the result of Hernandez and Leclerc (J Reine Angew Math 701:77–126, 2015) in the simply-laced case. We also find several applications of our combinatorial theory of Q-data to the finite-dimensional representation theory of the untwisted quantum affine algebra of g . In particular, in terms of Q-data and the inverse of the quantum Cartan matrix, (i) we give an alternative description of the block decomposition results due to Chari and Moura (Int Math Res Not 5:257–298, 2005) and Kashiwara et al. (Block decomposition for quantum affine algebras by the associated simply-laced root system, 2020. arXiv:2003.03265), (ii) we present a unified (partially conjectural) formula of the denominators of the normalized R-matrices between all the Kirillov–Reshetikhin modules, and (iii) we compute the invariants Λ (V , W) and Λ ∞ (V , W) introduced in Kashiwara et al. (Compos Math 156(5):1039–1077, 2020) for each pair of simple modules V and W. [ABSTRACT FROM AUTHOR]

We define a family of symmetric polynomials G ν , λ (z 1 , ... , z n + 1 , q) indexed by a pair of dominant integral weights for a root system of type A n. The polynomial G ν , 0 (z , q) is the specialized Macdonald polynomial P ν (z , q , 0) and is known to be the graded character of a level one Demazure module associated to the affine Lie algebra sl ˆ n + 1. We prove that G 0 , λ (z , q) is the graded character of a level two Demazure module for sl ˆ n + 1. Under suitable conditions on (ν , λ) (which apply to the pairs (ν , 0) and (0 , λ)) we prove that G ν , λ (z , q) is Schur positive, i.e., it can be written as a linear combination of Schur polynomials with coefficients in Z + [ q ]. We further prove that P ν (z , q , 0) is a linear combination of elements G 0 , λ (z , q) with the coefficients being essentially products of q -binomials. Together with a result of K. Naoi, a consequence of our result is an explicit formula for the specialized Macdonald polynomial associated to a non-simply laced Lie algebra as a linear combination of the level one Demazure characters of the non-simply laced algebra. [ABSTRACT FROM AUTHOR]

Let g0 be a simple Lie algebra of type ADE and let U′q(g) be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group B(g0) on the quantum Grothendieck ring Kt(g) of Hernandez-Leclerc's category C0g. Focused on the case of type AN-1, we construct a family of monoidal autofunctors {fi}i∈Z on a localization TN of the category of finite-dimensional graded modules over the quiver Hecke algebra of type A∞. Under an isomorphism between the Grothendieck ring K(TN) of TN and the quantum Grothendieck ring Kt(A(1)N-1), the functors {fi}1≤i≤N-1 recover the action of the braid group B(AN-1). We investigate further properties of these functors. [ABSTRACT FROM AUTHOR]

Let a reductive group G act on a smooth affine complex algebraic variety X. Let g be the Lie algebra of G and µ : T ∗(X) → g ∗ be the moment map. If the moment map is flat, and for a generic character χ : g → C, the action of G on µ −1 (χ) is free, then we show that for very generic characters χ the corresponding quantum Hamiltonian reduction of the ring of differential operators D(X) is simple. [ABSTRACT FROM AUTHOR]

For a smooth manifold X equipped with a volume form, let 𝓛0 (X) be the Lie algebra of volume preserving smooth vector fields on X. Lichnerowicz proved that the abelianization of 𝓛0 (X) is a finite-dimensional vector space, and that its dimension depends only on the topology of X. In this paper we provide analogous results for some classical examples of non-singular complex affine algebraic varieties with trivial canonical bundle, which include certain algebraic surfaces and linear algebraic groups. The proofs are based on a remarkable result of Grothendieck on the cohomology of affine varieties, and some techniques that were introduced with the purpose of extending the Andersén–Lempert theory, which was originally developed for the complex spaces ℂn, to the larger class of Stein manifolds that satisfy the density property. [ABSTRACT FROM AUTHOR]

We investigate a class of Kac–Moody algebras previously not considered. We refer to them as n-extended Lorentzian Kac–Moody algebras defined by their Dynkin diagrams through the connection of an A n Dynkin diagram to the node corresponding to the affine root. The cases n = 1 and n = 2 correspond to the well-studied over- and very-extended Kac–Moody algebras, respectively, of which the particular examples of E 10 and E 11 play a prominent role in string and M-theory. We construct closed generic expressions for their associated roots, fundamental weights and Weyl vectors. We use these quantities to calculate specific constants from which the nodes can be determined that when deleted decompose the n-extended Lorentzian Kac–Moody algebras into simple Lie algebras and Lorentzian Kac–Moody algebra. The signature of these constants also serves to establish whether the algebras possess SO(1, 2) and/or SO(3)-principal subalgebras. [ABSTRACT FROM AUTHOR]

We construct a differential graded Lie algebra g controlling the Poisson deformations of an affine Poisson variety. We analyze g in the case of affine Gorenstein toric Poisson varieties. Moreover, explicit description of the second and third Hochschild cohomology groups is given for three-dimensional affine Gorenstein toric varieties. [ABSTRACT FROM AUTHOR]

The Whittaker module M ψ and its quotient Whittaker module L ψ , ξ for the twisted affine Nappi-Witten Lie algebra H ˆ 4 [ τ ] are studied. For nonsingular type, it is proved that if ξ ≠ 0 , then L ψ , ξ is irreducible and any irreducible Whittaker H ˆ 4 [ τ ] -module of type ψ with k acting as a non-zero scalar ξ is isomorphic to L ψ , ξ. Furthermore, for ξ = 0 , all Whittaker vectors of L ψ , 0 are completely determined. For singular type, the Whittaker vectors of L ψ , ξ with ξ ≠ 0 are fully characterized. [ABSTRACT FROM AUTHOR]

Let g be a complex simple Lie algebra and U q L g the corresponding quantum affine algebra. We construct a functor F θ between finite-dimensional modules over a quantum symmetric pair subalgebra of affine type U q k ⊂ U q L g and an orientifold KLR algebra arising from a framed quiver with a contravariant involution, providing a boundary analogue of the Kang-Kashiwara-Kim-Oh generalized Schur-Weyl duality. With respect to their construction, our combinatorial model is further enriched with the poles of a trigonometric K-matrix intertwining the action of U q k on finite-dimensional U q L g -modules. By construction, F θ is naturally compatible with the Kang-Kashiwara-Kim-Oh functor in that, while the latter is a functor of monoidal categories, F θ is a functor of module categories. Relying on a suitable isomorphism à la Brundan-Kleshchev-Rouquier, we prove that F θ recovers the Schur-Weyl dualities due to Fan-Lai-Li-Luo-Wang-Watanabe in quasi-split type AIII. [ABSTRACT FROM AUTHOR]

For a simple Lie algebra g we consider an analogue of the affine algebra g ˆ k with n singularities, defined starting from the ring of functions on the n -pointed disk. We study the center of its completed enveloping algebra and prove an analogue of the Feigin-Frenkel theorem in this setting. In particular, we first give an algebraic description of the center by providing explicit topological generators; we then characterize the center geometrically as the ring of functions on the space of G L -Opers over the n -pointed disk. Finally, we prove some factorization properties of our isomorphism, thus establishing a relation between our isomorphism and the usual isomorphism of Feigin-Frenkel. [ABSTRACT FROM AUTHOR]

In this paper we construct a family of exact functors from the category of Whittaker modules of the simple complex Lie algebra of type A n to the category of finite-dimensional modules of the graded affine Hecke algebra of type A ℓ. Using results of Backelin [2] and of Arakawa-Suzuki [1] , we prove that these functors map standard modules to standard modules (or zero) and simple modules to simple modules (or zero). Moreover, we show that each simple module of the graded affine Hecke algebra appears as the image of a simple Whittaker module. Since the Whittaker category contains the BGG category O as a full subcategory, our results generalize results of Arakawa-Suzuki [1] , which in turn generalize Schur-Weyl duality between finite-dimensional representations of SL n (C) and representations of the symmetric group S n. [ABSTRACT FROM AUTHOR]

In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra g. The first predicts the maximal dimension of simple g-modules and in this paper we apply the Lefschetz Principle and classical techniques from Lie theory to prove this conjecture for all restricted Lie subalgebras of gln(k) whenever k is an algebraically closed field of sufficiently large characteristic p (depending on n). As a consequence we deduce that the conjecture holds for the Lie algebra of an affine algebraic group scheme over any commutative ring, after specialising to an algebraically closed field of almost any characteristic. In the appendix to this paper, written by Akaki Tikaradze, an alternative, short proof of the first Kac&-Weisfeiler conjecture is given for the Lie algebra of a group scheme over a finitely generated ring R ⊆ C, after base change to a field of large positive characteristic [ABSTRACT FROM AUTHOR]

• System identification techniques aim at estimating unmeasured states, parameters, and inputs. • They can only be successful if the estimated quantities are observable. • A method to investigate observability in presence of unmeasured inputs is presented. • It applies to nonlinear systems whose state and measurement equations are affine in all inputs. This paper presents a geometric algorithm to investigate the theoretical observability of nonlinear systems with partially measured inputs and outputs. The algorithm is based on Lie algebra and applies to systems whose state and measurement equations are analytical and affine in all inputs. It investigates whether the system satisfies a necessary observability condition that is named the Observability Rank Condition for systems with Direct Feedthrough (ORC-DF). The presented algorithm allows to assess the observability of the dynamic system states, the identifiability of constant-in-time parameters, and the ability to track unmeasured inputs, which is referred to as system invertibility. It is also shown how the developed methodology can be extended to investigate the observability of non-smooth systems that can be broken into different smooth branches, often encountered in mechanical applications related to sliding and damage. Possible applications are illustrated with several examples from structural engineering. [ABSTRACT FROM AUTHOR]

We conjecture that quantum Gaudin models in affine types admit families of higher Hamiltonians, labelled by the (countably infinite set of) exponents, whose eigenvalues are given by functions on a space of meromorphic opers associated with the Langlands dual Lie algebra. This is in direct analogy with the situation in finite types. However, in stark contrast to finite types, we prove that in affine types such functions take the form of hypergeometric integrals, over cycles of a twisted homology defined by the levels of the modules at the marked points. That result prompts the further conjecture that the Hamiltonians themselves are naturally expressed as such integrals. We go on to describe the space of meromorphic affine opers on an arbitrary Riemann surface. We prove that it fibres over the space of meromorphic connections on the canonical line bundle Ω. Each fibre is isomorphic to the direct product of the space of sections of the square of Ω with the direct product, over the exponents j not equal to 1, of the twisted cohomology of the j th tensor power of Ω. [ABSTRACT FROM AUTHOR]

The direct sum of irreducible level one integrable representations of affine Kac-Moody Lie algebra of (affine) type ADE carries a structure of P/Q-graded vertex operator algebra. There exists a filtration on this direct sum studied by Kato and Loktev such that the corresponding graded vector space is a direct sum of global Weyl modules. The associated graded space with respect to the dual filtration is isomorphic to the homogenous coordinate ring of semi-infinite flag variety. We describe the ring structure in terms of vertex operators and endow the homogenous coordinate ring with a structure of P/Q-graded vertex operator algebra. We use the vertex algebra approach to derive semi-infinite Plücker-type relations in the homogeneous coordinate ring. [ABSTRACT FROM AUTHOR]

We consider control affine systems, as well as cost-extended control systems, on the (four-dimensional) Engel group. Specifically, we classify the full-rank left-invariant control affine systems (under both detached feedback equivalence and strongly detached feedback equivalence). The cost-extended control systems with quadratic cost are then classified (under cost equivalence), as are their associated Hamilton-Poisson systems (up to affine isomorphism). In all cases, we exhibit a complete list of equivalence class representatives. [ABSTRACT FROM AUTHOR]

We construct certain Steinberg groups associated to extended afine Lie algebras and their root systems. Then by the integration methods of Kac and Peterson for integrable Lie algebras, we associate a group to every tame extended afine Lie algebra. Afterwards, we show that the extended afine Weyl group of the ground Lie algebra can be recovered as a quotient group of two subgroups of the group associated to the underlying algebra similar to Kac-Moody groups. [ABSTRACT FROM AUTHOR]

Abstract Let g be any untwisted affine Kac–Moody algebra, μ any fixed complex number, and g ˜ (μ) the corresponding toroidal extended affine Lie algebra of nullity two. For any k -tuple λ = (λ 1 , ⋯ , λ k) of weights of g , and k -tuple a = (a 1 , ⋯ , a k) of distinct non-zero complex numbers, we construct a class of modules V ˜ (λ , a) for the extended affine Lie algebra g ˜ (μ). We prove that the g ˜ (μ) -module V ˜ (λ , a) is completely reducible. We also prove that the g ˜ (μ) -module V ˜ (λ , a) is integrable when all weights λ i in λ are dominant. Thus, we obtain a new class of irreducible integrable weight modules for the toroidal extended affine Lie algebra g ˜ (μ). [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study Virasoro vertex algebras and affine vertex algebras over a general field of characteristic p > 2. More specifically, we study certain quotients of the universal Virasoro and affine vertex algebras by ideals related to the p -centers of the Virasoro algebra and affine Lie algebras. Among the main results, we classify their irreducible N -graded modules by explicitly determining their Zhu algebras and show that these vertex algebras have only finitely many irreducible N -graded modules and they are C 2 -cofinite. [ABSTRACT FROM AUTHOR]

The localized equivariant homology of the quiver variety of type A(1) N-1 can be identified with the level one Fock space by assigning a normalized torus fixed point basis to certain symmetric functions: Jack(glN) symmetric functions introduced by Uglov. We show that this correspondence is compatible with actions of two algebras, the Yangian for slN and the affine Lie algebra s^lN, on both sides. Consequently we obtain the affine Yangian action on the Fock space. [ABSTRACT FROM AUTHOR]

Kirillov–Reshetikhin (KR) crystals are colored directed graphs encoding the structure of certain finite-dimensional representations of affine Lie algebras. A tensor product of column shape Kirillov–Reshetikhin crystals has recently been realized in a uniform way, for all untwisted affine types, in terms of the so-called quantum alcove model. We enhance this model by using it to give a uniform realization of the corresponding combinatorial R -matrix, i.e., the unique affine crystal isomorphism permuting factors in a tensor product of column shape KR crystals. In other words, we are generalizing to all Lie types Schützenberger's sliding game (jeu de taquin) for Young tableaux, which realizes the combinatorial R -matrix in type A . Our construction is in terms of certain combinatorial moves, called quantum Yang–Baxter moves, which are explicitly described by reduction to the rank 2 root systems. We also show that the quantum alcove model does not depend on the choice of a sequence of alcoves joining the fundamental one to a translation of it. [ABSTRACT FROM AUTHOR]

We reprove the results of Jordan [18] and Siebert [30] and show that the Lie algebra of polynomial vector fields on an irreducible affine variety X is simple if and only if X is a smooth variety. Given proof is self-contained and does not depend on papers mentioned above. Besides, the structure of the module of polynomial functions on an irreducible smooth affine variety over the Lie algebra of vector fields is studied. Examples of Lie algebras of polynomial vector fields on an N-dimensional sphere, non-singular hyperelliptic curves and linear algebraic groups are considered. [ABSTRACT FROM AUTHOR]

We complete the classification of conformal embeddings of a maximally reductive subalgebra k into a simple Lie algebra g at non-integrable non-critical levels k by dealing with the case when k has rank less than that of g. We describe some remarkable instances of decomposition of the vertex algebra Vk(g) as a module for the vertex subalgebra generated by k. We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. In particular, we study an example of conformal embeddings A1×A1↪C3 at level k=-1/2, and obtain explicit branching rules by applying certain q-series identity. In the analysis of conformal embedding A1×D4↪C8 at level k=-1/2 we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs. [ABSTRACT FROM AUTHOR]

Let be a complex affine algebraic group, and let and be commuting anti-holomorphic involutions of . We construct an algebraic family of algebraic groups over the complex projective line and a real structure on the family that interpolates between the real forms and . [ABSTRACT FROM AUTHOR]

Dixmier and Moeglin gave an algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the universal enveloping algebra of a finite-dimensional complex Lie algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the universal enveloping algebra of a finite-dimensional complex Lie algebra satisfies the Dixmier–Moeglin equivalence . We define quantities which measure how “close” an arbitrary prime ideal of a noetherian algebra is to being primitive, rational, and locally closed; if every prime ideal is equally “close” to satisfying each of these three properties, then we say that the algebra satisfies the strong Dixmier–Moeglin equivalence . Using the example of the universal enveloping algebra of sl 2 ( C ) , we show that the strong Dixmier–Moeglin equivalence is strictly stronger than the Dixmier–Moeglin equivalence. For a simple complex Lie algebra g , a non-root of unity q ≠ 0 in an infinite field K , and an element w of the Weyl group of g , De Concini, Kac, and Procesi have constructed a subalgebra U q [ w ] of the quantised enveloping K -algebra U q ( g ) . These quantum Schubert cells are known to satisfy the Dixmier–Moeglin equivalence and we show that they in fact satisfy the strong Dixmier–Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the strong Dixmier–Moeglin equivalence. [ABSTRACT FROM AUTHOR]

A classical construction of Atiyah assigns to any (real or complex) Lie group G , manifold M and principal homogeneous G -space P over M , a Lie algebroid over M ( [1] ). The spirit behind our work is to put this work within an algebraic context, replace M by a scheme X and G by a “simple” reductive group scheme G over X (in the sense of Demazure–Grothendieck) that arise naturally with an attached torsor (which plays the role of P ) in the study of Extended Affine Lie Algebras (see [9] for an overview). Lie algebroids in an algebraic sense were also considered by Beilinson and Bernstein. We will explain how the present work relates to theirs. [ABSTRACT FROM AUTHOR]

In this work, we study minimal realization problem for an affine control system on a connected Lie group . We construct a minimal realization by using a canonical projection and by characterizing indistinguishable points of the system. [ABSTRACT FROM AUTHOR]