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A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namelyA 2,B 2, andG 2. Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger.The author was supported by an NSF Postdoctoral Fellowship, grant #DMS-9107908.  相似文献   

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In this paper we apply a previously published method [1] to the construction of boson realizations for Lie algebraG 2. These realizations are expressed by means of certain recurrent formulae in terms of five Bose pairs and generators of the subalgebrasl(2, R).  相似文献   

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Given a strict Lie 2-algebra, we can integrate it to a strict Lie 2-group by integrating the corresponding Lie algebra crossed module. On the other hand, the integration procedure of Getzler and Henriques will also produce a 2-group. In this paper, we show that these two integration results are Morita equivalent. As an application, we integrate a non-strict morphism between Lie algebra crossed modules to a generalized morphism between their corresponding Lie group crossed modules.  相似文献   

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Bialgebra structures compatible with Lie algebrae(2) ⊕u(1) and two-dimensional Weyl algebra are classified. Most of them are noncoboundary. The corresponding Poisson brackets are also calculated. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.  相似文献   

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Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer–Cartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of Kosmann–Schwarzbach. This approach provides a definition of an L -(quasi)bialgebra (strongly homotopy Lie (quasi)bialgebra). We recover an L -algebra structure as a particular case of our construction. The formal geometry interpretation leads to a definition of an L (quasi)bialgebra structure on V as a differential operator Q on V, self-commuting with respect to the big bracket. Finally, we establish an L -version of a Manin (quasi) triple and get a correspondence theorem with L -(quasi)bialgebras. This paper is dedicated to Jean-Louis Loday on the occasion of his 60th birthday with admiration and gratitude.  相似文献   

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We construct the moduli spaces associated to the solutions of equations of motion (modulo gauge transformations) of the Poisson sigma model with target being an integrable Poisson manifold. The construction can be easily extended to a case of a generic integrable Lie algebroid. Indeed for any Lie algebroid one can associate a BF-like topological field theory which localizes on the space of algebroid morphisms, that can be seen as a generalization of flat connections to the groupoid case. We discuss the finite gauge transformations and discuss the corresponding moduli spaces. We consider the theories both without and with boundaries.  相似文献   

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Due to the Cappelli-Itzykson-Zuber classification, the minimal conformally invariant quantum field theories withSU(2) currents are classified by the ADE Lie algebras. Here I give a conceptual proof of the empirically valid relation between their partition functions and the Lie algebra exponents.  相似文献   

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In this Letter, we discuss a series of linearization problems – for Poisson structures, Lie algebroids, and Lie groupoids. The last problem involves a conjecture on the structure of proper groupoids. Attempting to prove this by the method of averaging leads to problems concerning almost actions of compact groups and almost invariant submanifolds for compact group actions. The Letter ends with a discussion of possible extensions of the convexity theorems for momentum maps of hamiltonian actions of compact groups.  相似文献   

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We consider a vector space VV over K=RK=R or CC, equipped with a skew symmetric bracket [⋅,⋅]:V×V→V[,]:V×VV and a 2-form ω:V×V→Kω:V×VK. A simple change of the Jacobi identity to the form [A,[B,C]]+[C,[A,B]]+[B,[C,A]]=ω(B,C)A+ω(A,B)C+ω(C,A)B[A,[B,C]]+[C,[A,B]]+[B,[C,A]]=ω(B,C)A+ω(A,B)C+ω(C,A)B opens up new possibilities, which shed new light on the Bianchi classification of three-dimensional Lie algebras.  相似文献   

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The external algebra over holomorphic first order differential forms on a complex Lie groupG is endowed with the structure of a graded Poisson Lie algebra. This structure is introduced via graded bicovariant brackets that are shown to be in one to one correspondence withG-invariant tensors of special symmetry. Complete classification of graded Poisson Lie structures defined by homogeneous brackets is obtained for the case of classical complex Lie groups.  相似文献   

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By using a six-dimensional matrix Lie algebra [Y.F. Zhang and Y. Wang, Phys. Lett. A 360 (2006) 92], three induced Lie algebras are constructed. One of them is obtained by extending Lie bracket, the others are higher-dimensional complex Lie algebras constructed by using linear transformations. The equivalent Lie algebras of the later two with multi-component forms are obtained as well. As their applications, we derive an integrable coupling and quasi-Hamiltonian structure of the modified TC hierarchy of soliton equations.  相似文献   

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Abstract

The Lie point symmetries associated with a number of (2 + 1)-dimensional generalizations of soliton equations are investigated. These include the Niznik – Novikov – Veselov equation and the breaking soliton equation, which are symmetric and asymmetric generalizations respectively of the KDV equation, the (2+1)-dimensional generalization of the nonlinear Schrödinger equation by Fokas as well as the (2+1)-dimensional generalized sine-Gordon equation of Konopelchenko and Rogers. We show that in all these cases the Lie symmetry algebra is infinite-dimensional; however, in the case of the breaking soliton equation they do not possess a centerless Virasorotype subalgebra as in the case of other typical integrable (2+1)-dimensional evolution equations. We work out the similarity variables and special similarity reductions and investigate them.  相似文献   

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By using a six-dimensional matrix Lie algebra [Y.F. Zhang and Y. Wang, Phys. Lett. A 360 (2006) 92], three induced Lie algebras are constructed. One of them is obtained by extending Lie bracket, the others are higher-dimensional complex Lie algebras constructed by using linear transformations. The equivalent Lie algebras of the later two with multi-component forms are obtained as well. As their applications, we derive an integrable coupling and quasi-Hamiltonian structure of the modified TC hierarchy of soliton equations.  相似文献   

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The purpose of this work is to join Lie field structures with certain infinite-dimensional Lie algebras with locally convex topology. These topological Lie algebras allow topological groups which are a generalization of the connected nilpotent Lie groups. We showed the existence of the continuous unitary representations of the gained groups and then we proved the analogue of Gårding theorem. Using this theorem we established the existence of representations of Lie field structures into Lie algebras of skew-symmetric operators on Hilbert spaces.Work supported by National Science Foundation.On leave of absence from the Institute Rudjer Bokovi, Zagreb.  相似文献   

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We prove that every non-Abelian, finite-dimensional Lie algebra admits an exact bialgebra structure.  相似文献   

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The nonstandard and so(2, 2) Lie bialgebras are generalized to the so(3, 2) case in two natural ways by considering this algebra as the conformal algebra of the 2+1 Minkowskian spacetime. Lie bialgebra contractions are analyzed providing conformal bialgebras of the 2+1 Galilean and Carroll spacetimes. The corresponding quantum Hopf so(3, 2) algebras are presented and contractions are performed at the quantum level.  相似文献   

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