arXiv subjects
Publications and source records attributed to Yucai Su.
We determine the derivation algebras and the isomorphism classes of a family of the simple Lie algebras introduced recently by Xu [Manuscripta Math 100 (1999), 489-518]. The structure space of these algebras is given explicitly.
A class of nongraded Hamiltonian Lie algebras was earlier introduced by Xu. These Lie algebras have a Poisson bracket structure. In this paper, the isomorphism classes of these Lie algebras are determined by employing a ``sandwich'' method and by studying some features of these Lie algebras. It is obtained that two Hamiltonian Lie algebras are isomorphic if and only if their corresponding Poisson algebras are isomorphic. Furthermore, the derivation algebras and the second cohomology groups are determined.
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.
For a nondegenerate additive subgroup $G$ of the $n$-dimensional vector space $F^n$ over an algebraically closed field $F$ of characteristic zero, there is an associative algebra and a Lie algebra of Weyl type $W(G,n)$ spanned by all differential operators $u D_1^{m_1}... D_n^{m_n}$ for $u\in F[G]$ (the group algebra), and $m_1,...,m_n \ge 0$, where $D_1, ...,D_n$ are degree operators. In this paper, it is proved that an irreducible quasifinite $W(\Z,1)$-module is either a highest or lowest weight module or else a module of the intermediate series; furthermore, a classification of uniformly bounded $W(\Z,1)$-modules is completely given. It is also proved that an irreducible quasifinite $W(G,n)$-module is a module of the intermediate series and a complete classification of quasifinite $W(G,n)$-modules is also given, if $G$ is not isomorphic to $\Z$.
In a paper by Xu, some simple Lie algebras of generalized Cartan type were constructed, using the mixtures of grading operators and down-grading operators. Among them, are the simple Lie algebras of generalized Witt type, which are in general nongraded and have no torus. In this paper, some representations of these simple Lie algebras of generalized Witt type are presented.
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.
Generalized Virasoro algebras (defined as the universal central extension of some generalized Witt algebras) and super-Virasoro algebras and modules of the intermediate series are studied and discussed.
It is proved that uniformly bounded simple modules over higher rank super-Virasoro algebras are modules of the intermediate series, and that simple modules with finite dimensional weight spaces are either modules of the intermediate series or generalized highest weight modules.
Classical contact Lie algebras are the fundamental algebraic structures on the manifolds of contact elements of configuration spaces in classical mechanics. In this paper, we determine the structure of the currently largest known category of contact simple Lie algebras introduced earlier by the second author. These algebras are in general not finitely-graded.
In one of our recent papers, the associative and the Lie algebras of Weyl type $A[D]=A\otimes F[D]$ were defined and studied, 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, $D$ consisting of locally finite but not locally nilpotent derivations of $A$, are studied. The isomorphism classes and automorphism groups of these associative and Lie algebras are determined.
In a recent paper by Zhao and the author, the Lie algebras $A[D]=A\otimes F[D]$ of Weyl type were defined and studied, 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, the 2-cocycles of a class of the above Lie algebras $A[D]$ (which are called the Lie algebras of generalized differential operators in the present paper), with $F$ being a field of characteristic 0, are determined. Among all the 2-cocycles, there is a special one which seems interesting. Using this 2-cocycle, the central extension of the Lie algebra is defined.
We give a classification of infinite dimensional indecomposable weight modules over the Lie superalgebra $sl(2/1)$.
It is proved that an indecomposable Harish-Chandra module over the Virasoro algebra must be (i) a uniformly bounded module, or (ii) a module in Category $\cal O$, or (iii) a module in Category ${\cal O}^-$, or (iv) a module which contains the trivial module as one of its composition factors.
Over a field $F$ of any characteristic, for a commutative associative algebra $A$ with an identity element and for the polynomial algebra $F[D]$ of a commutative derivation subalgebra $D$ of $A$, the associative and the Lie algebras of Weyl type on the same vector space $A[D]=A\otimes F[D]$ are defined. It is proved that $A[D]$, as a Lie algebra (modular its center) or as an associative algebra, is simple if and only if $A$ is $D$-simple and $A[D]$ acts faithfully on $A$. Thus a lot of simple algebras are obtained.
In this paper, we determine the isomorphism classes of the central simple Poisson algebras introduced earlier by the second author. The Lie algebra structures of these Poisson algebras are in general not finitely-graded.
One of the four well-known series of simple Lie algebras of Cartan type is the series of Lie algebras of Special type, which are divergence-free Lie algebras associated with polynomial algebras and the operators of taking partial derivatives, connected with volume-preserving diffeomorphisms. In this paper, we determine the structure space of the divergence-free Lie algebras associated with pairs of a commutative associative algebra with an identity element and its finite-dimensional commutative locally-finite derivation subalgebra such that the commutative associative algebra is derivation-simple with respect to the derivation subalgebra.
We classify all the pairs of a commutative associative algebra with an identity element and its finite-dimensional commutative locally-finite derivation subalgebra such that the commutative associative algebra is derivation-simple with respect to the derivation subalgebra over an algebraically closed field with characteristic 0. Such pairs are the fundamental ingredients for constructing generalized simple Lie algebras of Cartan type. Moreover, we determine the isomorphic classes of the generalized simple Lie algebras of Witt Type. The structure space of these algebras is given explicitly.
To study finite-dimensional modules of the Lie superalgebras, Kac introduced the Kac-modules and divided them into typical or atypical modules according as they are simple or not. For Lambda being atypical, Hughes et al have an algorithm to determine all the composition factors of a Kac-module; they conjectured that there exists a bijection between the composition factors of a Kac-module and the so-called permissible codes. The aim of this paper is to prove this conjecture. We give a partial proof here, i.e., to any unlinked code, by constructing explicitly the primitive vector, we prove that there corresponds a composition factor of the Kac-module. We will give a full proof of the conjecture in another paper.