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Kaiming Zhao

Publications and source records attributed to Kaiming Zhao.

At least 73 records · Page 4Linked to original sources
arXiv2012

A class of simple weight Virasoro modules

For a simple module $M$ over the positive part of the Virasoro algebra (actually for any simple module over some finite dimensional solvable Lie algebras $\mathfrak{a}_r$) and any $α\in\C$, a class of weight modules $\mathcal {N}(M, α)$ over the Virasoro algebra are constructed. The necessary and sufficient condition for $\mathcal {N}(M, \a)$ to be simple is obtained. We also determine the necessary and sufficient conditions for two such irreducible Virasoro modules to be isomorphic. Many examples for such irreducible Virasoro modules with different features are provided. In particular the irreducible weight Virasoro modules $Γ(α_1, α_2, λ_1, λ_2)$ are defined on the polynomial algebra $\C[x]\otimes \C[t, t^{-1}]$ for any $α_1, α_2, λ_1, λ_2\in\C$ with $λ_1$ or $λ_2$ nonzero. By twisting the weight modules $\mathcal {N}(M, α)$ we also obtain nonweight simple Virasoro modules $\mathcal {N}(M, β)$ for any $β\in\C[t,t^{-1}]$.

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arXiv2012

Simple Virasoro modules which are locally finite over a positive part

We propose a very general construction of simple Virasoro modules generalizing and including both highest weight and Whittaker modules. This reduces the problem of classification of simple Virasoro modules which are locally finite over a positive part to classification of simple modules over a family of finite dimensional solvable Lie algebras. For one of these algebras all simple modules are classified by R. Block and we extend this classification to the next member of the family. As a result we recover many known but also construct a lot of new simple Virasoro modules. We also propose a revision of the setup for study of Whittaker modules.

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arXiv2011

Characterization of simple highest weight modules

We prove that for simple complex finite dimensional Lie algebras, affine Kac-Moody Lie algebras, the Virasoro algebra and the Heisenberg-Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro and for Heisenberg algebras.

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arXiv2011

A new method of finding all Roots of simple quaternionic polynomials

In this paper, we provide a new method to find all zeros of polynomials with quaternionic coefficients located on only one side of the powers of the variable (these polynomials are called simple polynomials). This method is much more efficient and much simpler than the known one in [D. Janovska and G. Opfer, A note on the computation of all zeros of simple quaternionic polynomials, SIAM J. Numer. Anal., 48(1)(2010), pp. 244-256]. We recover several known results, and deduce several interesting consequences concerning solving equations with all real coefficients or complex coefficients which do not seem to be deduced easily from the results in [D. Janovska and G. Opfer, A note on the computation of all zeros of simple quaternionic polynomials, SIAM J. Numer. Anal., 48(1)(2010), pp. 244-256]. We also give a necessary and sufficient condition for a simple quaternionic polynomials to have finitely many solutions (only isolated solutions).

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arXiv2009

Supports of weight modules over Witt algebras

In this paper, as the first step towards classification of simple weight modules with finite dimensional weight spaces over Witt algebras $W_n$, we explicitly describe supports of such modules. We also obtain some descriptions on the support of an arbitrary simple weight module over a $\Z^n$-graded Lie algebra $\mathfrak{g}$ having a root space decomposition $\oplus_{α\in\Z^n}\mathfrak {g}_α$ with respect to the abelian subalgebra $\mathfrak {g}_0$, with the property $[\mathfrak{g}_α,\mathfrak {g}_β]= \mathfrak {g}_{α+β}$ for all $α,β\in\Z^n$, $α\neq β$ (this class contains the algebra $W_n$).

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arXiv2008

Classification of irreducible Harish-Chandra modules over the loop-Virasoro algebra

The loop-Virasoro algebra is the Lie algebra of the tensor product of the Virasoro algebra and the Laurent polynomial algebra. This paper classifies irreducible Harish-Chandra modules over the loop-Virasoro algebra, which turn out to be highest weight modules, lowest weight modules and evaluation modules of the intermediate series (all wight spaces are 1-dimensional). As a by-product, we obtain a classification of irreducible Harish-Chandra modules over truncated Virasoro algebras. We also determine the necessary and sufficient conditions for highest weigh irreducible modules over the loop-Virasoro algebra to have all finite dimensional weight spaces, as well as the necessary and sufficient conditions for highest weigh Verma modules to be irreducible.

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arXiv2008

Automorphisms and derivations of Borel subalgebras and their nilradicals in Kac-Moody algebras

In this paper, we determine derivations of Borel subalgebras and their derived subalgebras called nilradicals, in Kac-Moody algebras (and contragredient Lie algebras) over any field of characteristic 0; and we also determine automorphisms of those subalgebras in symmetrizable Kac-Moody algebras. The results solve a conjecture posed by R. V. Moody about 30 years ago which generalizes a result by B. Kostant and which is discussed by A. Fialowski using Lie algebra cohomology in case of affine type.

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arXiv2006

Normal forms for orthogonal similarity classes of skew-symmetric matrices

Let F be an algebraically closed field of characteristic different from 2. We show that every nonsingular skew-symmetric n by n matrix X over F is orthogonally similar to a bidiagonal skew-symmetric matrix. In the singular case one has to allow some 4-diagonal blocks as well. If further the characteristic is 0, we construct the normal form for O_n(F)-similarity classes of skew-symmetric matrices. In this case the known normal forms (as presented in the well known book by Gantmacher) are quite different. Finally we study some related varieties of matrices. We prove that the variety of normalized nilpotent n by n bidiagonal matrices for n=2s+1 is irreducible of dimension s. As a consequence the skew-symmetric nilpotent bidiagonal n by n matrices are shown to form a variety of pure dimension s.

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arXiv2006

Classification of irreducible Harish-Chandra modules over generalized Virasoro algebras

Let $G$ be an arbitrary additive subgroup of $C$ and $Vir[G]$ the corresponding generalized Virasoro algebra. In the present paper, irreducible weight modules with finite dimensional weight spaces over $Vir[G]$ are completely determined. The classification strongly depends on the index group $G$. If $G$ does not have a direct summand $Z$, then such irreducible modules over $Vir[G]$ are only modules of intermediate series whose weight spaces are all 1-dimensional. Otherwise, there is one more class of modules which are constructed by using intermediate series modules over a generalized Virasoro subalgebra $Vir[G_0]$ of $Vir[G]$ for a direct summand $G_0$ of $G$ with corank 1.

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arXiv2005

Classification of irreducible weight modules over the twisted Heisenberg-Virasoro algebra

In this paper, all irreducible weight modules with finite dimensional weight spaces over the twisted Heisenberg-Virasoro algebra are determined. There are two different classes of them. One class is formed by simple modules of intermediate series, whose weight spaces are all 1-dimensional; the other class consists of the irreducible highest weight modules and lowest weight modules.

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arXiv2005

Classification of irreducible weight modules over higher rank Virasoro algebras

Let $G$ be a rank $n$ additive subgroup of $\bC$ and $\Vir[G]$ the corresponding Virasoro algebra of rank $n$. In the present paper, irreducible weight modules with finite dimensional weight spaces over $\Vir[G]$ are completely determined. There are two different classes of them. One class consists of simple modules of intermediate series whose weight spaces are all 1-dimensional. The other is constructed by using intermediate series modules over a Virasoro subalgebra of rank $n-1$. The classification of such modules over the classical Virasoro algebra was obtained by O. Mathieu in 1992 using a completely different approach.

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arXiv2005

Classification of simple weight Virasoro modules with a finite-dimensional weight space

We show that the support of a simple weight module over the Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all non-trivial weight spaces of such module are infinite dimensional. As a corollary we obtain that every simple weight module over the Virasoro algebra, having a non-trivial finite-dimensional weight space, is a Harish-Chandra module (and hence is either a simple highest or lowest weight module or a simple module from the intermediate series). This implies positive answers to two conjectures about simple pointed and simple mixed modules over the Virasoro algebra.

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arXiv2004

Structure of Algebras of Weyl Type

In a paper by the authors, the associative and the Lie algebras of Weyl type $A[D]=A\otimes F[D]$ were introduced, where $A$ is a commutative associative algebra with an identity element over a field $F$ of any characteristic, and $F[D]$ is the polynomial algebra of a commutative derivation subalgebra $D$ of $A$. In the present paper, a class of the above associative and Lie algebras $A[D]$ with $F$ being a field of characteristic 0 and $D$ consisting of locally finite derivations of $A$, is studied. The isomorphism classes of these associative and Lie algebras are determined. The structure of these algebras is described explicitly.

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arXiv2004

Classification of derivation-simple color algebras related to locally finite derivations

We classify the pairs $(A,D)$ consisting of an $(ε,Γ)$-olor-commutative associative algebra $A$ with an identity element over an algebraically closed field $F$ of characteristic zero and a finite dimensional subspace $D$ of $(ε,Γ)$-color-commutative locally finite color-derivations of $A$ such that $A$ is $Γ$-graded $D$-simple and the eigenspaces for elements of $D$ are $Γ$-graded. Such pairs are the important ingredients in constructing some simple Lie color algebras which are in general not finitely-graded. As some applications, using such pairs, we construct new explicit simple Lie color algebras of generalized Witt type, Weyl type.

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arXiv2003

Weight modules over exp-polynomial Lie algebras

In this paper, we generalize a result by Berman and Billig on weight modules over Lie algebras with polynomial multiplication. More precisely, we show that a highest weight module with an exp-polynomial ``highest weight'' has finite dimensional weight spaces. We also get a class of irreducible weight modules with finite dimensional weight spaces over generalized Virasoro algebras which do not occur over the classical Virasoro algebra.

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arXiv2003

Simple Lie Color Algebras of Weyl Type

For an $(ε,G)$-color-commutative associative algebra $A$ with an identity element over a field $F$ of characteristic not 2, and for a color-commutative subalgebra $D$ of color-derivations of $A$, denote by $A[D]$ the associative subalgebra of ${\rm End}(A)$ generated by $A$ (regarding as operators on $A$ via left multiplication) and $D$. It is easily proved that, as an associative algebra, $A[D]$ is $G$-graded simple if and only if $A$ is $\G$-graded $D$-simple. Suppose $A$ is $\G$-graded $D$-simple. Then, (a) $A[D]$ is a free left $A$-module; (b) as a Lie color algebra, the subquotient $[A[D],A[D]]/Z(A[D])\cap[A[D],A[D]]$ is simple (except one minor case), where $Z(A[D])$ is the color center of $A[D]$. The structure of this subquotient is explicitly described.

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arXiv2003

Z x Z-graded Lie algebras generated by the Virasoro algebra and $sl_2$

The present paper is another step toward the classification of $\Z\times\Z$-graded Lie algebras: we classify $\Z\times\Z$-graded Lie algebras $A=\oplus_{i,j\in\Z}A_{i,j}$ over a field $F$ of characteristic 0 with dim $A_{i,j}\le1$ for $i,j\in\Z$, satisfying: (1) $Å_0=\oplus_{i\in \Z}Å_{i,0}$ is isomorphic to the centerless Virasoro algebra, (2) $A_{0,-1},A_{0,0},A_{0,1}$ span the 3-dimensional simple Lie algebra $sl_2$, (3) $A$ is generated by the centerless Virasoro algebra and $sl_2$. These algebras include some rank 2 centerless Virasoro algebras, and some rank 2 Block algebras and their extensions.

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