arXiv subjects
Publications and source records attributed to Guang'ai Song.
In this paper, by studying the maximal good subspaces, we determine the dual Lie coalgebras of the centerless twisted Heisenberg-Virasoro algebra. Based on this, we construct the dual Lie bialgebras structures of the twisted Heisenberg-Virasoro type. As by-products, four new infinite dimensional Lie algebras are obtained.
Let $A=F[x,y]$ be the polynomial algebra on two variables $x,y$ over an algebraically closed field $F$ of characteristic zero. Under the Poisson bracket, $A$ is equipped with a natural Lie algebra structure. It is proven that the maximal good subspace of $A^*$ induced from the multiplication of the associative commutative algebra $A$ coincides with the maximal good subspace of $A^*$ induced from the Poisson bracket of the Poisson Lie algebra $A$. Based on this, structures of dual Lie bialgebras of the Poisson type are investigated. As by-products, five classes of new infinite dimensional Lie algebras are obtained.
For any factorization domain $\cal A$ and an algebra endomorphism $σ$ of $\cal A$, there exists a non-associative algebra $({\cal A},σ,[\cdot,\cdot])$ with multiplication satisfying skew-symmetry and generalized (twisted) Jacobi identities, called a $σ$-deformed Witt algebra. In this paper, we obtain the necessary and sufficient conditions for the algebra $({\cal A},σ,[\cdot,\cdot])$ to be simple.
Structures of dual Lie bialgebras on the one sided Witt algebra, the Witt algebra and the Virasoro algebra are investigated. As a result, we obtain some infinite dimensional Lie algebras.
In a recent paper by the authors, Lie bialgebras structures of generalized Virasoro-like type were considered. In this paper, the explicit formula of the quantization of generalized Virasoro-like algebras is presented.
In two recent papers by the authors, all Lie bialgebra structures on Lie algebras of generalized Witt type are classified. In this paper all Lie bialgebra structures on generalized Virasoro-like algebras are determined. It is proved that all such Lie bialgebras are triangular coboundary.
We first prove that, for any generalized Hamiltonian type Lie algebra $L$, the first cohomology group $H^1(L,L \otimes L)$ is trivial. We then show that all Lie bialgebra structures on $L$ are triangular.
In a paper by Michaelis a class of infinite-dimensional Lie bialgebras containing the Virasoro algebra was presented. This type of Lie bialgebras was classified by Ng and Taft. In this paper, all Lie bialgebra structures on the Lie algebras of generalized Witt type are classified. It is proved that, for any Lie algebra $W$ of generalized Witt type, all Lie bialgebras on $W$ are coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group $H^1(W,W \otimes W)$ is trivial.