Lie Algebras Of Type Research Articles

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.

Filiform Lie algebras of order 3

The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France 98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(2,\mathbb {C})$\end{document}sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases.

INFINITESIMAL CHEREDNIK ALGEBRAS AS W-ALGEBRAS

In this article we establish an isomorphism between universal infinitesimal Cherednik algebras and W-algebras for Lie algebras of the same type and 1-block nilpotent elements. As a consequence we obtain some fundamental results about infinitesimal Cherednik algebras.

CRYSTAL B(λ) IN B(∞) FOR G2TYPE LIE ALGEBRA

A previous work gave a combinatorial description of the crystal B(<TEX>${\infty}$</TEX>), in terms of certain simple Young tableaux referred to as the marginally large tableaux, for finite dimensional simple Lie algebras. Using this result, we present an explicit description of the crystal B(<TEX>${\lambda}$</TEX>), in terms of the marginally large tableaux, for the <TEX>$G_2$</TEX> Lie algebra type. We also provide a new description of B(<TEX>${\lambda}$</TEX>), in terms of Nakajima monomials, that is in natural correspondence with our tableau description.

Free Field Realizations of Induced Modules for Affine Lie Algebras

We construct a free field realization of generalized loop modules induced from admissible diagonal ℤ-graded irreducible modules of nonzero level for a Heisenberg subalgebra of untwisted affine Lie algebra of type A. Such generalized loop modules are irreducible as it was shown in Bekkert et al. [2]. We also extend this construction to modules induced from an arbitrary parabolic subalgebra of whose Levi factor is infinite-dimensional and contains .

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.

Nonlinear Integrable Couplings of Levi Hierarchy and WKI Hierarchy

With the help of the known Lie algebra, a type of new 8-dimensional matrix Lie algebra is constructed in the paper. By using the 8-dimensional matrix Lie algebra, the nonlinear integrable couplings of the Levi hierarchy and the Wadati-Konno-Ichikawa (WKI) hierarchy are worked out, which are different from the linear integrable couplings. Based on the variational identity, the Hamiltonian structures of the above hierarchies are derived.

Crystal [formula omitted] as a subset of crystal [formula omitted] expressed as tableaux for [formula omitted] type

The crystal B(∞)⊗Tλ is known to contain a copy of the irreducible highest weight crystal B(λ) as a sub-crystal. By explicitly identifying the elements belonging to the mentioned sub-crystal, we present two realizations of B(λ) that are based on two realizations of B(∞), for the special linear Lie algebra type. The first description of B(λ) is based on our new realization of B(∞) as the set of all marginally large reverse tableaux and the second is based on a previous realization of B(∞) as the set of all marginally large tableaux. We further present two new Nakajima monomial realizations of B(λ) that correspond naturally to our reverse tableau and tableau realizations.

The Lie algebra of type [formula omitted] is rational over its quotient by the adjoint action

Let G be a split simple group of type G2 over a field k, and let g be its Lie algebra. Answering a question of J.-L. Colliot-Thélène, B. Kunyavskiĭ, V.L. Popov, and Z. Reichstein, we show that the function field k(g) is generated by algebraically independent elements over the field of adjoint invariants k(g)G.

ON CARTAN INVARIANTS FOR GRADED LIE ALGEBRAS OF CARTAN TYPE

Let L = X(m; n), X ∈ {W, S, H, K}, be a graded simple Lie algebra of Cartan type over an algebraically closed field of characteristic p &gt; 3. Then L is a so-called generalized restricted Lie algebra. Let [Formula: see text] be the primitive p-envelope of L, and G = X(m; 1), a subalgebra of [Formula: see text]. In this paper, a close connection between Cartan invariants for [Formula: see text] and U(G, χ) is established, where χ ∈ L* is extended to be a linear function on [Formula: see text] trivially, and 1 ≤ ht (χ) &lt; p-2+δXW. This reduces the study of projective representations of the generalized restricted Lie algebra L to the one of the corresponding restricted Lie algebra G. As a special case, we recover some results in [Shu and Jiang, On Cartan invariants and blocks of Zassenhaus algebras, Comm. Algebra33(10) (2005) 3619–3630].

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.

Nilpotent coadjoint orbits in small characteristic

We show that the numbers of nilpotent coadjoint orbits in the dual of exceptional Lie algebra G2 in characteristic 3 and in the dual of exceptional Lie algebra F4 in characteristic 2 are finite. We determine the closure relation among nilpotent coadjoint orbits in the dual of Lie algebras of type B,C,F4 in characteristic 2 and in the dual of Lie algebra of type G2 in characteristic 3. In each case we give an explicit description of the nilpotent pieces in the dual defined in [2], which are in general unions of nilpotent coadjoint orbits, coincide with the earlier case-by-case definition in [11,26] in the case of classical groups and have nice properties independent of the characteristic of the base field. This completes the classification of nilpotent coadjoint orbits in the dual of Lie algebras of reductive algebraic groups and the determination of closure relation among such orbits in all characteristic.

FUSION RULES AND COMPLETE REDUCIBILITY OF CERTAIN MODULES FOR AFFINE LIE ALGEBRAS

We develop a new method for obtaining branching rules for affine Kac–Moody Lie algebras at negative integer levels. This method uses fusion rules for vertex operator algebras of affine type. We prove that an infinite family of ordinary modules for affine vertex algebra of type A investigated in our previous paper J. Algebra319 (2008) 2434–2450, is closed under fusion. Then, we apply these fusion rules on explicit bosonic realization of level -1 modules for the affine Lie algebra of type [Formula: see text], obtain a new proof of complete reducibility for these representations, and the corresponding decomposition for ℓ ≥ 3. We also obtain the complete reducibility of the associated level -1 modules for affine Lie algebra of type [Formula: see text]. Next, we notice that the category of [Formula: see text] modules at level -2ℓ + 3 has the isomorphic fusion algebra. This enables us to decompose certain [Formula: see text] and [Formula: see text]-modules at negative levels.

Combinatorial bases of principal subspaces for the affine Lie algebra of type [formula omitted

We consider principal subspaces WL(kΛ0) and WN(kΛ0) of standard module L(kΛ0) and generalized Verma module N(kΛ0) at level k≥1 for the affine Lie algebra of type B2(1). By using the theory of vertex operator algebras, we find combinatorial bases of principal subspaces in terms of quasi-particles. From quasi-particle bases, we obtain character formulas for WL(kΛ0) and WN(kΛ0).

On the quantum L-operator for the two-dimensional lattice Toda model

We consider the two-dimensional quantum lattice Toda model for affine and simple Lie algebras of type A. For its known L-operator, the second-order correction in lattice parameter e is found. It is proved that the equation determining the third-order correction in e has no solutions. Bibliography: 9 titles.

The composition Hall algebra of a weighted projective line

Abstract In this article, we deal with properties of the reduced Drinfeld double of the composition subalgebra of the Hall algebra of the category of coherent sheaves on a weighted projective line. This study is motivated by applications in the theory of quantized enveloping algebras of some Lie algebras. We obtain a new realization of the quantized enveloping algebras of affine Lie algebras of simply-laced types as well as some new embeddings between them. Moreover, our approach allows to derive new results on the structure of the quantized enveloping algebras of the toroidal algebras of types D4(1, 1), E6(1, 1), E7(1, 1) and E8(1, 1). In particular, our method leads to a construction of a modular action and allows to define a PBW-type basis for that classes of algebras.

Some remarks on triangulated categories and Kac–Moody algebras

In [L. Peng, J. Xiao, Triangulated categories and Kac–Moody algebras, Invent. Math. 140 (3) (2000) 563–603], the authors proved a theorem for constructing Hall type Lie algebras associated to 2-periodic triangulated categories. In particular, this theorem gives a global realization of symmetrizable Kac–Moody algebras. The purpose of this paper is to provide a refinement (meanwhile, a new proof) of the theorem by applying the approach of derived Hall algebras arising in homologically finite triangulated categories appeared in [B. Toën, Derived Hall algebras, Duke Math. J. 135 (3) (2006) 587–615] and [J. Xiao, F. Xu, Hall algebras associated to triangulated categories, Duke Math. J. 143 (2) (2008) 357–373].

The variety of nilpotent elements and invariant polynomial functions on the special algebra Sn

Abstract In the study of the variety of nilpotent elements in a Lie algebra, Premet conjectured that this variety is irreducible for any finite dimensional restricted Lie algebra. In this paper, with the assumption that the ground field is algebraically closed of characteristic p &gt; 3, we confirm this conjecture for the Lie algebras of Cartan type S˜ n and Sn . Moreover, we show that the variety of nilpotent elements in Sn is a complete intersection. Motivated by the proof of the irreducibility, we describe explicitly the ring of invariant polynomial functions on Sn .

Completable two step nilpotent Lie algebras of type (2, p

A Lie algebra is called complete if its centre is zero and all its derivations are inner. A nilpotent Lie algebra is called completable if it is the maximal nilpotent ideal of a complete solvable Lie algebra. A two-step nilpotent Lie algebra is said to be of type , if dim and dim . In this paper, based on the classification result of M. Gauger, we classify two step completable complex nilpotent Lie algebras of type.

Double extensions of Lie algebras of Kac-Moody type and applications to some hamiltonian systems

We describe some Lie algebras of the Kac-Moody type, construct their double extensions, central and by derivations; we also construct the corresponding Lie groups in some cases. We study the case of the Lie algebra of unimodular vector fields in more detail and use the linear Poisson structure on their regular duals to construct generalizations of some infinite-dimensional Hamiltonian systems, such as magnetohydrodynamics.