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Yucai Su

Publications and source records attributed to Yucai Su.

At least 91 records · Page 5Linked to original sources
arXiv2005

2-Cocycles of the Lie superalgebras of Weyl type

In a paper by Su and Zhao, the Lie algebra $A[D]=A\otimes F[D]$ of Weyl type was defined and studied, where $A$ is a commutative associative algebra with an identity element over a field $F$ of arbitrary characteristic, and $F[D]$ is the polynomial algebra of a commutative derivation subalgebra $D$ of $A$. The 2-cocycles of a class of $A[D]$ were determined by Su. In the present paper, we determine the 2-cocycles of a class of Lie superalgebras of Weyl type over a field $F$ of characteristic 0.

math.QA↗
arXiv2005

Derivations and 2-Cocycles of Contact Lie Algebras Related to Locally-Finite Derivations

Classical contact Lie algebras are the fundamental algebraic structures on the manifolds of contact elements of configuration spaces in classical mechanics. Xu introduced a large category of contact simple Lie algebras which are related to locally finite derivations and are in general not finitely graded. The isomorphism classes of these Lie algebras were determined in a previous paper by Xu and Su. In this paper, the derivation algebras and the 2-cohomology groups of these Lie algebras are determined, and it is obtained that the 2-cohomology groups of these Lie algebras are all trivial.

math.QA↗
arXiv2005

Classification of quasifinite $W_\infty$-modules

It is proved that an irreducible quasifinite $W_\infty$-module is a highest or lowest weight module or a module of the intermediate series; a uniformly bounded indecomposable weight $W_\infty$-module is a module of the intermediate series. For a nondegenerate additive subgroup $G$ of $F^n$, where $F$ is a field of characteristic zero, there is a simple Lie or associative algebra $W(G,n)^{(1)}$ spanned by differential operators $uD_1^{m_1}... D_n^{m_n}$ for $u\in F[G]$ (the group algebra), and $m_i\ge0$ with $\sum_{i=1}^n m_i\ge1$, where $D_i$ are degree operators. It is also proved that an indecomposable quasifinite weight $W(G,n)^{(1)}$-module is a module of the intermediate series if $G$ is not isomorphic to $Z$.

math.RT↗
arXiv2005

Verma modules over the generalized Heisenberg-Virasoro algebra

For any additive subgroup $G$ of an arbitrary field $F$ of characteristic zero, there corresponds a generalized Heisenberg-Virasoro algebra $L[G]$. Given a total order of $G$ compatible with its group structure, and any $h,h_I,c,c_I,c_{LI}\in F$, a Verma module $M(h,h_I,c,c_I,c_{LI})$ over $L[G]$ is defined. In the this note, the irreducibility of Verma modules $M(h,h_I,c,c_I,c_{LI})$ is completely determined.

math.QA↗
arXiv2005

Derivations for the even part of the odd Hamiltonian superalgebra in positive characteristic

In this paper we mainly study the derivations for even part of the finite-dimensional odd Hamiltonian superalgebra $HO$ over a field of prime characteristic. We first give the generator set of $HO_{\bar{0}}.$ Then we determine the homogeneous derivationsfrom $HO_{\bar{0}}$ into $W_{\bar{0}},$ the even part of the generalized Witt superalgebra $W$. Finally, wedetermine the derivation algebra and outer derivation algebra of $HO_{\bar{0}}.$

math.RA↗
arXiv2005

Derivations for the even part of the Hamiltonian superalgebra in positive characteristic

In this paper we consider the derivations for even part of the finite-dimensional Hamiltonian superalgebra $H$ over a field of prime characteristic. We first introduce an ideal $\frak{N}$ of $H_{\bar{0}}$ and show that the derivation space from $H_{\bar{0}}$ into $W_{\bar{0}}$ can be obtained by the derivation space from $\frak{N}$ into $W_{\bar{0}},$ the even part of the generalized Witt superalgebra $W$. For further application we also give the generating set of the ideal $\frak{N}$. Then we describe three series of exceptional derivations from $ {H}_{\bar{0}}$ into $W_{\bar{0}}.$ Finally, we determine all the derivations vanishing on the non-positive $\mathbb{Z}$-graded part of $ H_{\bar{0}}$, the odd $\mathbb{Z}$-homogeneous derivations, and negative $\mathbb{Z}$-homogeneous derivations from ${H}_{\bar{0}}$ into $W_{\bar{0}}.$

math.RA↗
arXiv2005

Lie bialgebras of generalized Witt type

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.

math.QA↗
arXiv2004

Character and dimension formulae for general linear superalgebra

The generalized Kazhdan-Lusztig polynomials for the finite dimensional irreducible representations of the general linear superalgebra are computed explicitly. Using the result we establish a one to one correspondence between the set of composition factors of an arbitrary $r$-fold atypical $gl_{m|n}$-Kac-module and the set of composition factors of some $r$-fold atypical $gl_{r|r}$-Kac-module. The result of Kazhdan-Lusztig polynomials is also applied to prove a conjectural character formula put forward by van der Jeugt et al in the late 80s. We simplify this character formula to cast it into the Kac-Weyl form, and derive from it a closed formula for the dimension of any finite dimensional irreducible representation of the general linear superalgebra.

math.QA↗
arXiv2004

Cohomology of Lie superalgebras $sl_{m|n}$ and $osp_{2|2n}$

We explicitly compute the first and second cohomology groups of the classical Lie superalgebras $sl_{m|n}$ and $osp_{2|2n}$ with coefficients in the finite dimensional irreducible modules and the Kac modules. We also show that the second cohomology groups of these Lie superalgebras with coefficients in the respective universal enveloping algebras (under the adjoint action) vanish. The latter result in particular implies that the universal enveloping algebras $U(sl_{m|n})$ and $U(osp_{2|2n})$ do not admit any non-trivial formal deformations of Gerstenhaber type.

math.QA↗
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.

math.QA↗
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.

math.QA↗
arXiv2004

Low dimensional cohomology of general conformal algebras $gc_N$

We compute the low dimensional cohomologies $\tilde H^q(gc_N,C)$, $H^q(gc_N,\C)$ of the infinite rank general Lie conformal algebras $gc_N$ with trivial coefficients for $q\le3, N=1$ or $q\le2, N\ge2$. We also prove that the cohomology of $gc_N$ with coefficients in its natural module is trivial, i.e., $H^*(gc_N,\C[\ptl]^N)=0$; thus partially solve an open problem of Bakalov-Kac-Voronov in [{\it Comm. Math. Phys.,} {\bf200} (1999), 561-598].

math.QA↗
arXiv2003

Structure of the Lie algebras related to those of block

We determine the isomorphism classes of the first family of infinite dimensional simple Lie algebras recently introduced by Xu. The structure space of these algebras is given explicitly. The derivations of these algebras are also determined.

math.QA↗