In this paper,we first give a direct sum decomposition of Lie comodules,and then accord- ing to the Lie comodule theory,construct some(triangular)Lie bialgebras through Lie coalgebras.
In this paper, Lie bialgebra structures on generalized Virasoro-like algebras are studied. It is proved that all such Lie bialgebras are triangular coboundary.
We first prove that,for any generalized Hamiltonian type Lie algebra H,the first co- homology group H^1(H,H(?)H) is trivial.We then show that all Lie bialgebra structures on H are triangular.
The first cohomology group of generalized loop Virasoro algebras with coefficients in the tensor product of its adjoint module is shown to be trivial. The result is used to prove that Lie bialgebra structures on gener...The first cohomology group of generalized loop Virasoro algebras with coefficients in the tensor product of its adjoint module is shown to be trivial. The result is used to prove that Lie bialgebra structures on generalized loop Virasoro algebras are coboundary triangular. The authors generalize the results to generalized map Virasoro algebras.展开更多
A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket. We construct a bialgebra theory of compatible Lie Mgebras as an analogue of a piiLie bialge...A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket. We construct a bialgebra theory of compatible Lie Mgebras as an analogue of a piiLie bialgebra. They can also be regarded as a "compatible version" of Lie bialgebras, that is, a pair of Lie biaJgebras such that any linear combination of the two Lie bialgebras is still a Lie bialgebra. Many properties of compatible Lie bialgebras as the "compatible version" of the corresponding properties of Lie biaJgebras are presented. In particular, there is a coboundary compatible Lie bialgebra theory with a construction from the classical Yang-Baxter equation in compatible Lie algebras as a combination of two classical Yang-Baxter equations in lAe algebras. FUrthermore, a notion of compatible pre-Lie Mgebra is introduced with an interpretation of its close relation with the classical Yang-Baxter equation in compatible Lie a/gebras which leads to a construction of the solutions of the latter. As a byproduct, the compatible Lie bialgebras fit into the framework to construct non-constant solutions of the classical Yang-Baxter equation given by Golubchik and Sokolov.展开更多
In a recent article by Liu,Pei,and Zhu,Lie bialgebra structures on the twisted Heisenberg-Virasoro algebra were determined. By disposing the indexing set, the generalized Heisenberg-Virasoro algebra was considered. It...In a recent article by Liu,Pei,and Zhu,Lie bialgebra structures on the twisted Heisenberg-Virasoro algebra were determined. By disposing the indexing set, the generalized Heisenberg-Virasoro algebra was considered. It is proved that all Lie bialgebra structures on centerless generalized Heisenberg-Virasoro algebra L are coboundary triangular by proving that the first cohomology group H1 (L,V) =0.展开更多
In this article, we use the general method of quantization by Drinfeld’s twist to quantize explicitly the Lie bialgebra structures on Lie algebras of Block type.
In this paper, the extended affine Lie algebra sl2(Cq) is quantized from three different points of view, which produces three non-commutative and non-cocommutative Hopf algebra structures, and yields other three qua...In this paper, the extended affine Lie algebra sl2(Cq) is quantized from three different points of view, which produces three non-commutative and non-cocommutative Hopf algebra structures, and yields other three quantizations by an isomorphism of sl2 (Cq) correspondingly. Moreover, two of these quantizations can be restricted to the extended affine Lie algebra sl2(Cq).展开更多
Let M be an exact symplectic manifold with contact type boundary such that cl(M) = O. Motivated by noncommutative symplectic geometry and string topology, we show that the cyclic cohomology of the Fukaya category of...Let M be an exact symplectic manifold with contact type boundary such that cl(M) = O. Motivated by noncommutative symplectic geometry and string topology, we show that the cyclic cohomology of the Fukaya category of M has an involutive Lie bialgebra structure.展开更多
We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the ...We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first eohomology group H1 (W, W × W) is trivial,展开更多
We give a classification of Lie bialgebra structures on generalized loop Schr?dinger-Virasoro algebras st). Then we find out that not all Lie bialgebra structures on generalized loop Schr?dinger-Virasoro algebras st) ...We give a classification of Lie bialgebra structures on generalized loop Schr?dinger-Virasoro algebras st). Then we find out that not all Lie bialgebra structures on generalized loop Schr?dinger-Virasoro algebras st) are triangular coboundary.展开更多
We quantize the W-algebra W(2,2), whose Verma modules, Harish-Chandra modules, irreducible weight modules and Lie bialgebra structures have been investigated and determined in a series of papers recently.
基金the Educational Ministry Key Foundation of China(Grant No.108154)the National Natural Science Foundation of China(Grant No.10571153)
文摘In this paper,we first give a direct sum decomposition of Lie comodules,and then accord- ing to the Lie comodule theory,construct some(triangular)Lie bialgebras through Lie coalgebras.
基金Supported by an NSF Grant 10471096 of China,"One Hundred Talents Program"from University of Science and Technology of China and"Trans-Century Training Programme Foundation for the Talents"from National Education Ministry of China
文摘In this paper, Lie bialgebra structures on generalized Virasoro-like algebras are studied. It is proved that all such Lie bialgebras are triangular coboundary.
基金the National Natural Science Foundation of China (Nos. 10471091, 10671027)the One Hundred Talents Program from University of Science and Technology of China
文摘Lie bialgebra structures on a family of Lie algebras of Block type are shown to be triangular coboundary.
基金This work was supported by the National Natural Science Foundation of China (Grant No.10471091)"One Hundred Talents Program"from University of Science and Technology of China
文摘We first prove that,for any generalized Hamiltonian type Lie algebra H,the first co- homology group H^1(H,H(?)H) is trivial.We then show that all Lie bialgebra structures on H are triangular.
基金supported by the National Natural Science Foundation of China(Nos.10825101,11431010,11271284,11101269)the Scientific Research Starting Foundation for Doctors,Shanghai Ocean University(No.A-0209-13-0105380)the Youth Scholars of Shanghai Higher Education Institutions(No.ZZHY14026)
文摘The first cohomology group of generalized loop Virasoro algebras with coefficients in the tensor product of its adjoint module is shown to be trivial. The result is used to prove that Lie bialgebra structures on generalized loop Virasoro algebras are coboundary triangular. The authors generalize the results to generalized map Virasoro algebras.
基金Supported by National Natural Science Foundation of China under Grant Nos.11271202,11221091,11425104Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20120031110022
文摘A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket. We construct a bialgebra theory of compatible Lie Mgebras as an analogue of a piiLie bialgebra. They can also be regarded as a "compatible version" of Lie bialgebras, that is, a pair of Lie biaJgebras such that any linear combination of the two Lie bialgebras is still a Lie bialgebra. Many properties of compatible Lie bialgebras as the "compatible version" of the corresponding properties of Lie biaJgebras are presented. In particular, there is a coboundary compatible Lie bialgebra theory with a construction from the classical Yang-Baxter equation in compatible Lie algebras as a combination of two classical Yang-Baxter equations in lAe algebras. FUrthermore, a notion of compatible pre-Lie Mgebra is introduced with an interpretation of its close relation with the classical Yang-Baxter equation in compatible Lie a/gebras which leads to a construction of the solutions of the latter. As a byproduct, the compatible Lie bialgebras fit into the framework to construct non-constant solutions of the classical Yang-Baxter equation given by Golubchik and Sokolov.
基金National Natural Science Foundations of China(No.11001046,No.11201305)the Fundamental Research Funds for the Central Universities+1 种基金Foundation of Outstanding Young Teachers of Donghua University,ChinaInnovation Project of Shanghai Education Committee,China(No.12YZ081)
文摘In a recent article by Liu,Pei,and Zhu,Lie bialgebra structures on the twisted Heisenberg-Virasoro algebra were determined. By disposing the indexing set, the generalized Heisenberg-Virasoro algebra was considered. It is proved that all Lie bialgebra structures on centerless generalized Heisenberg-Virasoro algebra L are coboundary triangular by proving that the first cohomology group H1 (L,V) =0.
基金supported by the National Science Foundation of China (10825101)"One Hundred Talents Program" from University of Science and Technology of Chinathe China Postdoctoral Science Foundation (20090450810)
文摘In this article, we use the general method of quantization by Drinfeld’s twist to quantize explicitly the Lie bialgebra structures on Lie algebras of Block type.
文摘In this paper, the extended affine Lie algebra sl2(Cq) is quantized from three different points of view, which produces three non-commutative and non-cocommutative Hopf algebra structures, and yields other three quantizations by an isomorphism of sl2 (Cq) correspondingly. Moreover, two of these quantizations can be restricted to the extended affine Lie algebra sl2(Cq).
基金The authors would like to thank the referees for pointing out several errors and hnprecisions and Professors Yiming Long, Yongbin Ruan and Gang Tian for their helpful communications and supports during these years. H. -L. Her was partially supported by the National Natural Science Foundation of China (Grant No. 10901084) S. Sun was partially supported by the National Natural Science Foundation of China (Grant Nos. 10731080, 11131004), PHR201106118, and the Institute of Mathematics and Interdisciplinary Science at Capital Normal University all authors are also partially supported by the National Natural Science Foundation of China (Grant No. 11271269).
文摘Let M be an exact symplectic manifold with contact type boundary such that cl(M) = O. Motivated by noncommutative symplectic geometry and string topology, we show that the cyclic cohomology of the Fukaya category of M has an involutive Lie bialgebra structure.
文摘We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first eohomology group H1 (W, W × W) is trivial,
基金the National Natural Science Foundation of China(Grant Nos.11801369,11771279,11801363,11431010)the Natural Science Foundation of Shanghai(Grant No.16ZR1415000).
文摘We give a classification of Lie bialgebra structures on generalized loop Schr?dinger-Virasoro algebras st). Then we find out that not all Lie bialgebra structures on generalized loop Schr?dinger-Virasoro algebras st) are triangular coboundary.
基金Supported by NSF'of China (Grant Nos. 10825101, 10926166), Special Grade of the Financial Support from China Postdoctoral Science Foundation (Grant No. 201003326) and the Natural Science Research Project for Higher Institutions of Jiangsu Province (Grant No. 09KJB110001)
文摘We quantize the W-algebra W(2,2), whose Verma modules, Harish-Chandra modules, irreducible weight modules and Lie bialgebra structures have been investigated and determined in a series of papers recently.