Displaying similar documents to “Quantum Affine Algebras and Deformations of the Virasoro and 𝒲 -Algebras”

Multiloop algebras, iterated loop algebras and extended affine Lie algebras of nullity 2

Bruce Allison, Stephen Berman, Arturo Pianzola (2014)

Journal of the European Mathematical Society

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Let 𝕄 n be the class of all multiloop algebras of finite dimensional simple Lie algebras relative to n -tuples of commuting finite order automorphisms. It is a classical result that 𝕄 1 is the class of all derived algebras modulo their centres of affine Kac-Moody Lie algebras. This combined with the Peterson-Kac conjugacy theorem for affine algebras results in a classification of the algebras in 𝕄 1 . In this paper, we classify the algebras in 𝕄 2 , and further determine the relationship between...

The structure of split regular Hom-Poisson algebras

María J. Aragón Periñán, Antonio J. Calderón Martín (2016)

Colloquium Mathematicae

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We introduce the class of split regular Hom-Poisson algebras formed by those Hom-Poisson algebras whose underlying Hom-Lie algebras are split and regular. This class is the natural extension of the ones of split Hom-Lie algebras and of split Poisson algebras. We show that the structure theorems for split Poisson algebras can be extended to the more general setting of split regular Hom-Poisson algebras. That is, we prove that an arbitrary split regular Hom-Poisson algebra is of the form...

Smallness problem for quantum affine algebras and quiver varieties

David Hernandez (2008)

Annales scientifiques de l'École Normale Supérieure

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The geometric small property (Borho-MacPherson [2]) of projective morphisms implies a description of their singularities in terms of intersection homology. In this paper we solve the smallness problem raised by Nakajima [37, 35] for certain resolutions of quiver varieties [37] (analogs of the Springer resolution): for Kirillov-Reshetikhin modules of simply-laced quantum affine algebras, we characterize explicitly the Drinfeld polynomials corresponding to the small resolutions. We use...

Kontsevich Deformation Quantization on Lie Algebras

Nabiha Ben Amar, Mouna Chaabouni, Mabrouka Hfaiedh (2007)

Bollettino dell'Unione Matematica Italiana

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We consider Kontsevich star product on the dual 𝔤 * of a general Lie algebra g equipped with the linear Poisson bracket. We show that this star product provides a deformation quantization by partial embeddings in the direction of the Poisson bracket.

The geometric reductivity of the quantum group S L q ( 2 )

Michał Kępa, Andrzej Tyc (2011)

Colloquium Mathematicae

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We introduce the concept of geometrically reductive quantum group which is a generalization of the Mumford definition of geometrically reductive algebraic group. We prove that if G is a geometrically reductive quantum group and acts rationally on a commutative and finitely generated algebra A, then the algebra of invariants A G is finitely generated. We also prove that in characteristic 0 a quantum group G is geometrically reductive if and only if every rational G-module is semisimple,...

The Dixmier-Moeglin equivalence and a Gel’fand-Kirillov problem for Poisson polynomial algebras

K. R. Goodearl, S. Launois (2011)

Bulletin de la Société Mathématique de France

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The structure of Poisson polynomial algebras of the type obtained as semiclassical limits of quantized coordinate rings is investigated. Sufficient conditions for a rational Poisson action of a torus on such an algebra to leave only finitely many Poisson prime ideals invariant are obtained. Combined with previous work of the first-named author, this establishes the Poisson Dixmier-Moeglin equivalence for large classes of Poisson polynomial rings, including semiclassical limits of quantum...

Relating quantum and braided Lie algebras

X. Gomez, S. Majid (2003)

Banach Center Publications

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We outline our recent results on bicovariant differential calculi on co-quasitriangular Hopf algebras, in particular that if Γ is a quantum tangent space (quantum Lie algebra) for a CQT Hopf algebra A, then the space k ⊕ Γ is a braided Lie algebra in the category of A-comodules. An important consequence of this is that the universal enveloping algebra U ( Γ ) is a bialgebra in the category of A-comodules.

Quantization of semisimple real Lie groups

Kenny De Commer (2024)

Archivum Mathematicum

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We provide a novel construction of quantized universal enveloping * -algebras of real semisimple Lie algebras, based on Letzter’s theory of quantum symmetric pairs. We show that these structures can be ‘integrated’, leading to a quantization of the group C * -algebra of an arbitrary semisimple algebraic real Lie group.

On a Construction of ModularGMS-algebras

Abd El-Mohsen Badawy (2015)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this paper we investigate the class of all modular GMS-algebras which contains the class of MS-algebras. We construct modular GMS-algebras from the variety 𝐊 ̲ 2 by means of K ̲ 2 -quadruples. We also characterize isomorphisms of these algebras by means of K ̲ 2 -quadruples.

Quantization of Drinfeld Zastava in type A

Michael Finkelberg, Leonid Rybnikov (2014)

Journal of the European Mathematical Society

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Drinfeld Zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of the affine Lie algebra 𝔰𝔩 ^ n . We introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the Zastava space bijectively at the level of complex points. The natural Poisson structure on the Zastava space can be described on Z in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) Lie algebra. The...

Symmetries in connected graded algebras and their PBW-deformations

Yongjun Xu, Xin Zhang (2023)

Czechoslovak Mathematical Journal

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We focus on connected graded algebras and their PBW-deformations endowed with additional symmetric structures. Many well-known algebras such as negative parts of Drinfeld-Jimbo’s quantum groups, cubic Artin-Schelter algebras and three-dimensional Sklyanin algebras appear in our research framework. As an application, we investigate a 𝒦 2 algebra 𝒜 which was introduced to compute the cohomology ring of the Fomin-Kirillov algebra ℱ𝒦 3 , and explicitly construct all the (self-)symmetric and sign-(self-)symmetric...

Free dynamical quantum groups and the dynamical quantum group S U Q d y n ( 2 )

Thomas Timmermann (2012)

Banach Center Publications

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We introduce dynamical analogues of the free orthogonal and free unitary quantum groups, which are no longer Hopf algebras but Hopf algebroids or quantum groupoids. These objects are constructed on the purely algebraic level and on the level of universal C*-algebras. As an example, we recover the dynamical S U q ( 2 ) studied by Koelink and Rosengren, and construct a refinement that includes several interesting limit cases.

Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras

Peng Shan (2011)

Annales scientifiques de l'École Normale Supérieure

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We define the i -restriction and i -induction functors on the category 𝒪 of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.

A geometric approach to full Colombeau algebras

R. Steinbauer (2010)

Banach Center Publications

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We present a geometric approach to diffeomorphism invariant full Colombeau algebras which allows a particularly clear view of the construction of the intrinsically defined algebra ^ ( M ) on the manifold M given in [gksv].

Generalized Post algebras and their application to some infinitary many-valued logics

Cat-Ho Nguyen

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CONTENTSIntroduction............................................................................................................................................................................... 5Part I. A generalization of Post algebras............................................................................................................................. 7   1. Definition and characterization of generalized Post algebras............................................. 7   2. Post...

Standardly stratified split and lower triangular algebras

Eduardo do N. Marcos, Hector A. Merklen, Corina Sáenz (2002)

Colloquium Mathematicae

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In the first part, we study algebras A such that A = R ⨿ I, where R is a subalgebra and I a two-sided nilpotent ideal. Under certain conditions on I, we show that A is standardly stratified if and only if R is standardly stratified. Next, for A = U 0 M V , we show that A is standardly stratified if and only if the algebra R = U × V is standardly stratified and V M is a good V-module.

Division algebras that generalize Dickson semifields

Daniel Thompson (2020)

Communications in Mathematics

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We generalize Knuth’s construction of Case I semifields quadratic over a weak nucleus, also known as generalized Dickson semifields, by doubling of central simple algebras. We thus obtain division algebras of dimension 2 s 2 by doubling central division algebras of degree s . Results on isomorphisms and automorphisms of these algebras are obtained in certain cases.

The affineness criterion for quantum Hom-Yetter-Drinfel'd modules

Shuangjian Guo, Shengxiang Wang (2016)

Colloquium Mathematicae

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Quantum integrals associated to quantum Hom-Yetter-Drinfel’d modules are defined, and the affineness criterion for quantum Hom-Yetter-Drinfel’d modules is proved in the following form. Let (H,α) be a monoidal Hom-Hopf algebra, (A,β) an (H,α)-Hom-bicomodule algebra and B = A c o H . Under the assumption that there exists a total quantum integral γ: H → Hom(H,A) and the canonical map β : A ⊗ B A → A ⊗ H , a ⊗ B b ↦ S - 1 ( b [ 1 ] ) α ( b [ 0 ] [ - 1 ] ) ⊗ β - 1 ( a ) β ( b [ 0 ] [ 0 ] ) , is surjective, we prove that the induction functor A ⊗ B - : ̃ ( k ) B → A H is an equivalence of categories.

Quantised 𝔰𝔩 2 -differential algebras

Andrey Krutov, Pavle Pandžić (2024)

Archivum Mathematicum

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We propose a definition of a quantised 𝔰𝔩 2 -differential algebra and show that the quantised exterior algebra (defined by Berenstein and Zwicknagl) and the quantised Clifford algebra (defined by the authors) of  𝔰𝔩 2 are natural examples of such algebras.

Braided coproduct, antipode and adjoint action for U q ( s l 2 )

Pavle Pandžić, Petr Somberg (2024)

Archivum Mathematicum

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Motivated by our attempts to construct an analogue of the Dirac operator in the setting of U q ( 𝔰𝔩 n ) , we write down explicitly the braided coproduct, antipode, and adjoint action for quantum algebra U q ( 𝔰𝔩 2 ) . The braided adjoint action is seen to coincide with the ordinary quantum adjoint action, which also follows from the general results of S. Majid.

Solvable Leibniz algebras with NF n ⊕ [...] F m 1 F m 1 nilradical

L.M. Camacho, B.A. Omirov, K.K. Masutova, I.M. Rikhsiboev (2017)

Open Mathematics

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All finite-dimensional solvable Leibniz algebras L, having N = NFn⊕ [...] Fm1 F m 1 as the nilradical and the dimension of L equal to n+m+3 (the maximal dimension) are described. NFn and [...] Fm1 F m 1 are the null-filiform and naturally graded filiform Leibniz algebras of dimensions n and m, respectively. Moreover, we show that these algebras are rigid.