New HKT manifolds arising from quaternionic representations

We give a procedure for constructing an 8n-dimensional HKT Lie algebra starting from a 4n-dimensional one by using a quaternionic representation of the latter. The strong (respectively, weak, hyper-K?hler, balanced) condition is preserved by our construction. As an application of our results we obtain a new compact HKT manifold with holonomy in ${SL(n,\mathbb{H})}$ which is not a nilmanifold. We find in addition new compact strong HKT manifolds. We also show that every K?hler Lie algebra equipped with a flat, torsion-free complex connection gives rise to an HKT Lie algebra. We apply this method to two distinguished 4-dimensional K?hler Lie algebras, thereby obtaining two conformally balanced HKT metrics in dimension 8. Both techniques prove to be an effective tool for giving the explicit expression of the corresponding HKT metrics.     

Pluriharmonic maps into Kähler symmetric spaces and Sym’s formula

A construction due to Sym and Bobenko recovers constant mean curvature surfaces in euclidean 3-space from their harmonic Gauss maps. We generalize this construction to higher dimensions and codimensions replacing the surface by a complex manifold and the sphere (the target space of the Gauss map) by a Kähler symmetric space of compact type with its standard embedding into the Lie algebra ${\mathfrak{g}}A construction due to Sym and Bobenko recovers constant mean curvature surfaces in euclidean 3-space from their harmonic Gauss maps. We generalize this construction to higher dimensions and codimensions replacing the surface by a complex manifold and the sphere (the target space of the Gauss map) by a K?hler symmetric space of compact type with its standard embedding into the Lie algebra \mathfrakg{\mathfrak{g}} of its transvection group. Thus we obtain a new class of immersed K?hler submanifolds of \mathfrakg{\mathfrak{g}} and we derive their properties.     

Analytic actions on compact surfaces and fixed points

 Consider an effective real analytic action of a connected Lie group G on a compact connected surface of Euler characteristic χ≠0. We show that if the action has no fixed point then χ≥1 and the Lie algebra 𝒢 of G is isomorphic either to a subalgebra of the affine algebra of ℝ², which is the extension of the ideal of constant vector fields by an irreducible linear subalgebra, or to sl(2,ℝ), o(3), sl(2,ℂ) and sl(3,ℝ). Received: 7 August 2001 Published online: 24 January 2003     

Ideals of identities of representations of nilpotent lie algebras

Let L be a Lie algebra, nilpotent of class 2, over an infinite field K, and suppose that the centre C of L is one dimensional; such Lie algebras are called Heisenberg algebras. Let ρ:L→hom _KV be a finite dimensional representation of the Heisenberg algebra L such that ρ(C) contains non-singular linear transformations of V, and denote l(ρ) the ideal of identities for the representation ρ. We prove that the ideals of identities of representations containing I(ρ) and generated by multilinear polynomials satisfy the ACC. Let sl ₂(L) be the Lie algebra of the traceless 2×2 matrices over K, and suppose the characteristic of K equals 2. As a corollary we obtain that the ideals of identities of representations of Lie algebras containing that of the regular representation of sl ₂(K) and generated by multilinear polynomials, are finitely based. In addition we show that one cannot simply dispense with the condition of multilinearity. Namely, we show that the ACC is violated for the ideals of representations of Lie algebras (over an infinite field of characteristic 2) that contain the identities of the regular representation of sl ₂(K).     

Quantum octonions

This paper constructs a quantum deformation of the complex Cayley dgebra. The method uses the representation theory of U _q(sl(2)), the quantized enveloping algebra of the simple complex Lie algebra s/(2). The paper begins by constructing a quantum deforma-tion of the complex quaternion algebra, since this simpler case illustrates all of the necessary steps. As intermediate results, deformations are constructed of sl(2) and the 7-dimensional simple Malcev algebra.     

Two expanding integrable systems and quasi-Hamiltonian function associated with an equation hierarchy

A Lie algebra sl(2) which is isomorphic to the known Lie algebra A₁ is introduced for which an isospectral Lax pair is presented, whose compatibility condition leads to a soliton-equation hierarchy. By using the trace identity, its Hamiltonian structure is obtained. Especially, as its reduction cases, a Sine equation and a complex modified KdV(cmKdV) equation are obtained,respectively. Then we enlarge the sl(2) into a bigger Lie algebra sl(4) so that a type of expanding integrable model of the hierarchy is worked out. However, the soliton-equation hierarchy is not integrable couplings. In order to generate the integrable couplings, an isospectral Lax pair is introduced. Under the frame of the zero curvature equation, we generate an integrable coupling whose quasi-Hamiltonian function is derived by employing the variational identity. Finally, two types of computing formulas of the constant γ are obtained, respectively.     

Quasi-particle fermionic formulas for (k, 3)-admissible configurations

We construct new monomial quasi-particle bases of Feigin-Stoyanovsky type subspaces for the affine Lie algebra sl(3;ℂ)∧ from which the known fermionic-type formulas for (k, 3)-admissible configurations follow naturally. In the proof we use vertex operator algebra relations for standard modules and coefficients of intertwining operators.     

Aspherical Kähler manifolds with solvable fundamental group

We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition, we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical Kähler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of $\mathbb{C}^{n}We survey recent developments which led to the proof of the Benson-Gordon conjecture on K?hler quotients of solvable Lie groups. In addition, we prove that the Albanese morphism of a K?hler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical K?hler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of \mathbbCⁿ\mathbb{C}^{n} by a discrete group of complex isometries.     

Algebraic properties of first integrals for systems of second-order ordinary differential equations of maximal symmetry

Symmetries of the first integrals for scalar linear or linearizable secondorder ordinary di?erential equations (ODEs) have already been derived and shown to exhibit interesting properties. One of these is that the symmetry algebra sl(3, IR) is generated by the three triplets of symmetries of the functionally independent first integrals and its quotient. In this paper, we first investigate the Lie-like operators of the basic first integrals for the linearizable maximally symmetric system of two second-order ODEs represented by the free particle system, obtainable from a complex scalar free particle equation, by splitting the corresponding complex basic first integrals and its quotient as well as their associated symmetries. It is proved that the 14 Lie-like operators corresponding to the complex split of the symmetries of the functionally independent first integrals I₁, I₂ and their quotient I₂/I₁ are precisely the Lie-like operators corresponding to the complex split of the symmetries of the scalar free particle equation in the complex domain. Then, it is shown that there are distinguished four symmetries of each of the four basic integrals and their quotients of the two-dimensional free particle system which constitute four-dimensional Lie algebras which are isomorphic to each other and generate the full symmetry algebra sl(4, IR) of the free particle system. It is further shown that the (n + 2)-dimensional algebras of the n + 2 first integrals of the system of n free particle equations are isomorphic to each other and generate the full symmetry algebra sl(n + 2, IR) of the free particle system.     

The tetrahedron algebra and its finite-dimensional irreducible modules

Recently Terwilliger and the present author found a presentation for the three-point sl₂ loop algebra via generators and relations. To obtain this presentation we defined a Lie algebra ? by generators and relations and displayed an isomorphism from ? to the three-point sl₂ loop algebra. In this paper we classify the finite-dimensional irreducible ?-modules.     

The embedding ofU q(sl (2)) and sine algebras in generalized Clifford algebras

In this note, we establish the connection between certain quantum algebras and generalized Clifford algebras (GCA). Precisely, we embed the quantum tori Lie algebra andU _q(sl (2)) in GCA.     

Quantum differential operators on the quantum plane

The universal enveloping algebra of a Lie algebra acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or quantum group) is a deformation of a universal enveloping algebra and acts not through the differential operators of its representation ring but through the quantised differential operators of its representation ring. We present this situation for the quantum group of sl₂.     

Verma type modules for toroidal lie algebras

We study the structure of imaginary Verma modules induced from the"natural"Borel subalgebra of a toroidal Lie algebra. In particular, we establish a criterion of irreducibility for imaginary Verma modules and describe their submodules and irreducible quotients. We also describe the structure of Verma type modules in the case of sl(2)-toroidal Lie algebra over two variables.     

Nonembeddable Factors of the Enveloping Algebras of Semisimple Lie Algebras

It is shown that the enveloping algebra of every (finite dimensional,complex) semisimple Lie algebra has a factor ring which cannotbe embedded in any Artinian ring. The proof helps to clarifythe connection between primary decomposition and embeddability,which was obscured in the original proof [3] that U(sl₂(C))admits a nonembeddable factor.     

Balanced Deep Matrix Algebras

This paper continues the study of associative and Lie deep matrix algebras, DM(X,\mathbbK){\mathcal{DM}}(X,{\mathbb{K}}) and \mathfrakgld(X,\mathbbK){\mathfrak{gld}}(X,{\mathbb{K}}), and their subalgebras. After a brief overview of the general construction, balanced deep matrix subalgebras, BDM(X,\mathbbK){\mathcal{BDM}}(X,{\mathbb{K}}) and \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}), are defined and studied for an infinite set X. The global structures of these two algebras are studied, devising a depth grading on both as well as determining their ideal lattices. In particular, \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) is shown to be semisimple. The Lie algebra \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) possesses a deep Cartan decomposition and is locally finite with every finite subalgebra naturally enveloped by a semi-direct product of \mathfraksl_n{\mathfrak{{sl}_n}}’s. We classify all associative bilinear forms on \mathfraksl₂\mathfrakd{\mathfrak{sl}_2\mathfrak{d}} (a natural depth analogue of \mathfraksl₂{\mathfrak{{sl}_2}}) and \mathfrakbld{\mathfrak{bld}}.     

Projective functors and the restriction of a Verma module to a subalgebra of Levi type

We observe that the restriction of a Verma module over a semi-simple Lie algebra to a subalgebra of Levi type may be viewed as a projective functor. By simple arguments we prove that this restriction can be decomposed into a direct sum of standard indecomposables in the category O. For the restriction problem from sl(n+1) to gl(n) we describe the complete answer. We study the properties of the modules with Verma flag also and prove that any module with Verma flag is a submodule of some projective.     

Generalized complex and Dirac structures on homogeneous spaces

The aim of this paper is to study generalized complex geometry (Hitchin, 2002) [6] and Dirac geometry (Courant, 1990) [3], (Courant and Weinstein, 1988) [4] on homogeneous spaces. We offer a characterization of equivariant Dirac structures on homogeneous spaces, which is then used to construct new examples of generalized complex structures. We consider Riemannian symmetric spaces, quotients of compact groups by closed connected subgroups of maximal rank, and nilpotent orbits in sln(R). For each of these cases, we completely classify equivariant Dirac structures. Additionally, we consider equivariant Dirac structures on semisimple orbits in a semisimple Lie algebra. Here equivariant Dirac structures can be described in terms of root systems or by certain data involving parabolic subalgebras.     

Malcev superalgebras with trival nucleus

The nucleus of a Malcev superalgebra M measures how far it is from being a Lie superalgebraM being a Lie superalgebra if and only if its nucleus is the whole M. This paper is devoted to study Malcev superalgebras in the opposite direction, that is, with trivial nucleus. The odd part of any finite-dimensional Malcev superalgebra with trivial nucleus is shown to be contained in the solvable radical. For algebraically closed fields, any such superalgebra splits as the sum of its solvable radical and a semisimple Malcev algebra contained in the even part, which is a direct sum of copies of sl(2, F) and the seven-dimensional simple non-Lie Malcev algebra, obtained from the Cayley-Dickson algebra.