Lie Algebras Of Block Type Research Articles

Biderivations of a Lie algebra of Block type and applications

Let be a Lie algebra of Block type with basis and relations . In the present paper, the biderivations of are explicitly described. In particular, it is shown that any biderivation of is inner. As applications, the linear commuting maps and commutative post-Lie algebra structures on this algebra are given.

Partial classification of cuspidal simple modules for Virasoro-like algebra

Let P be the Lie algebra of Hamiltonian vector fields on the torus, which is also known as the Virasoro-like algebra, a special kind of the so-called Block type Lie algebra. And let A be the Laurent polynomial algebra in two variables. In this paper, by following S.E. Rao's strategy of “backward induction”, we prove that any quasi-finite simple (A,P)-module has to come from Larsson–Shen's construction.

Structure of a class of Lie algebras of Block type, II

Let q be a nonzero complex number. In this paper, we completely determine the derivation algebras and the automorphism groups of the Block type Lie algebras ℬ(q).

Classification of ℤ2-graded modules of intermediate series over a Block-type Lie algebra

Let [Formula: see text] be the Lie algebra of Block type with basis {Li,j | i,j ∈ ℤ} and relations [Li,j, Lk,l] = ((j + 1)k - (l + 1)i)Li+k,j+l. Since [Formula: see text] is ℤ2-graded, it is natural to study ℤ2-graded modules over [Formula: see text]. In this paper, ℤ2-graded [Formula: see text]-modules of the intermediate series are classified.

The Lie Conformal Algebra of a Block Type Lie Algebra

Let L be a Lie algebra of Block type over ℂ with basis {Lα,i | α,i ∈ ℤ} and brackets [Lα,i,Lβ,j]=(β(i+1)-α(j+1)) Lα+β,i+j. In this paper, we first construct a formal distribution Lie algebra of L. Then we decide its conformal algebra B with ℂ[∂]-basis {Lα(w) | α ∈ ℤ} and λ-brackets [Lα(w)λ Lβ(w)]= (α∂+(α+β)λ) Lα+β(w). Finally, we give a classification of free intermediate series B-modules.

Block (or Hamiltonian) Lie Symmetry of Dispersionless D-Type Drinfeld–Sokolov Hierarchy

In this paper, the dispersionless D-type Drinfeld–Sokolov hierarchy, i.e. a reduction of the dispersionless two-component BKP hierarchy, is studied. The additional symmetry flows of this hierarchy are presented. These flows form an infinite-dimensional Lie algebra of Block type as well as a Lie algebra of Hamiltonian type.

Irreducible quasifinite modules over a class of Lie algebras of block type

For any nonzero complex number $q$, there is a Lie algebra of Block type, denoted by $\mathcal{B}(q)$. In this paper, a complete classification of irreducible quasifinite modules is given. More precisely, an irreducible quasifinite module is a highest weight or lowest weight module, or a module of intermediate series. As a consequence, a classification for uniformly bounded modules over another class of Lie algebras, the semi-direct product of the Virasoro algebra and a module of intermediate series, is also obtained. Our method is conceptional, instead of computational.

Irreducible Quasi-Finite Representations of a Block Type Lie Algebra

Let B be the Block type Lie algebra over ℂ with basis {L α, i , C 1, C 2 | (α, i) ∈ ℤ × ℤ \\ {(0, − 2)}} and Lie bracket [L α, i , L β, j ] = (β(i + 1) − α(j + 1))L α+β, i+j + αδα+β, 0δ i+j, −2 C 1 + (i + 1)δα+β, 0δ i+j, −2 C 2, where C 1, C 2 are central elements. In this paper, it is proved that a quasi-finite irreducible B-module is either a highest or a lowest weight module. We also give a classification of all highest/lowest weight B-modules.

Derivations and Automorphisms of a Lie Algebra of Block Type

Let [Formula: see text] be a Lie algebra of Block type with basis {Lα,i| α ∈ ℤ, i ∈ ℤ+} and relations [Lα,i,Lβ,j]= ((α-1)(j+1)-(β-1)(i+1))Lα+β, i+j. In the present paper, the derivation algebra and automorphism group of [Formula: see text] are explicitly described. In particular, it is shown that the outer derivation space is 1-dimensional and the inner automorphism group of [Formula: see text] is trivial.

Structure of a Class of Lie Algebras of Block Type

Let ℬ(q) be a class of Lie algebras of Block type with basis {L α, i | α, i ∈ ℤ, i ≥ 0} and relations [L α, i , L β, j ] = (β(i + q) − α(j + q))L α+β, i+j , where q is a positive integer. In this article, it is shown that ℬ(q) are different from each other for distinct positive integers q's. The automorphism groups, the derivation algebras, and the central extensions of all ℬ(q) are also uniformly and explicitly described, which generalize some previous results.

Block type symmetry of bigraded Toda hierarchy

In this paper, we define Orlov-Schulman's operators ML, MR, and then use them to construct the additional symmetries of the bigraded Toda hierarchy. We further show that these additional symmetries form an interesting infinite-dimensional Lie algebra known as a Block type Lie algebra, whose structure theory and representation theory have recently received much attention in the literature. By acting on two different spaces under the weak W-constraints, we find in particular two representations of this Block type Lie algebra.

Whittaker modules for a Lie algebra of Block type

In this paper, we study Whittaker modules for a Lie algebra of Block type. We define Whittaker modules and under some conditions, obtain a bijective correspondence between the set of isomorphism classes of Whittaker modules over this algebra and the set of ideals of a polynomial ring, parallel to a result from the classical setting and the case of the Virasoro algebra.

Quantization of lie algebras of block type

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.

Representations of four-derivation Lie algebras of Block type

Block introduced certain analogues of the Zassenhaus algebras over a field of characteristic 0. The nongraded infinite-dimensional simple Lie algebras of Block type constructed by Xu can be viewed as generalizations of the Block algebras. In this paper, we construct a family of irreducible modules in terms of multiplication and differentiation operators on “polynomials” for four-devivation nongraded Lie algebras of Block type based on the finite-dimensional irreducible weight modules with multiplicity one of general linear Lie algebras. We also find a new series of submodules from which some irreducible quotient modules are obtained.

Lie Bialgebra Structures on Lie Algebras of Block Type

This paper considers Lie bialgebra structures on Lie algebras of Block type and proves that all such Lie bialgebras are triangular coboundary.

Lie bialgebras of a family of Lie algebras of Block type

Lie bialgebra structures on a family of Lie algebras of Block type are shown to be triangular coboundary.

Highest weight representations of a family of Lie algebras of Block type

For an additive subgroup G of a field \( \mathbb{F} \) of characteristic zero, a Lie algebra Open image in new window (G) of Block type is defined with basis {Lα,i | α ∈ G, i ∈ ℤ+} and relations [Lα,i, Lβ,j] = (β − α)Lα+β,i+j (αj − βi)Lα+β,i+j−1. It is proved that an irreducible highest weight Open image in new window (ℤ)-module is quasifinite if and only if it is a proper quotient of a Verma module. Furthermore, for a total order ≻ on G and any Λ ∈ Open image in new window (G)0* (the dual space of Open image in new window (G)0 = span{L0,i | i ∈ ℤ+}), a Verma Open image in new window (G)-module M(Λ, ≻) is defined, and the irreducibility of M(Λ, ≻) is completely determined.

Generalized Verma modules over some Block algebras

In this paper, a class of generalized Verma modules M(V) over some Block type Lie algebra ℬ(G) are constructed, which are induced from nontrivial simple modules V over a subalgebra of ℬ(G). The irreducibility of M(V) is determined.

Quasifinite modules of a Lie algebra related to Block type

Let B be the universal central extension of a graded Lie algebra of Block type. In this paper, it is proved that any quasifinite irreducible B -module is either highest weight, lowest weight or uniformly bounded. Furthermore, the quasifinite irreducible highest weight B -modules are classified, and the intermediate series B -modules are classified and constructed.

Highest weight representations of a Lie algebra of Block type

For a field F of characteristic zero and an additive subgroup G of F, a Lie algebra B(G) of the Block type is defined with the basis {Lα,i, c|α∈G, -1≤i∈Z} and the relations [Lα,i,Lβ,j] = ((i + 1)β- (j + 1)α)Lα+β,i+j +αδα,-βδi+j,-2c,[c, Lα,i] = 0. Given a total order (?) on G compatible with its group structure, and anyα∈B(G)0*, a Verma B(G)-module M(A, (?)) is defined, and the irreducibility of M(A,(?)) is completely determined. Furthermore, it is proved that an irreducible highest weight B(Z )-module is...