Related papers: Hypercomplex structures on special linear groups

We classify indefinite simply connected hyper-Kaehler symmetric spaces. Any such space without flat factor has commutative holonomy group and signature (4m,4m). We establish a natural 1-1 correspondence between simply connected…

Let G be compact Lie group. It is shown that the cotangent bundle of the complexification of G admits a hyperkahler structure which is invariant under left and right translations by elements of G. The proof is to realize the cotangent…

We discuss conditions for the integrability of an almost complex structure defined on the total space of an induced Hopf S^3-bundle over a Sasakian manifold . As an application, we obtain an uncountable family of inequivalent complex…

We construct a Hennings type logarithmic invariant for restricted quantum $\mathfrak{sl}(2)$ at a $2\mathsf{p}$-th root of unity. This quantum group $U$ is not braided, but factorizable. The invariant is defined for a pair: a 3-manifold $M$…

This paper gives a new proof of a result of Geoff Mess that the linear holonomy group of a complete flat Lorentz 3-manifold cannot be cocompact in SO(2,1). The proof uses a signed marked Lorentzian length-spectrum invariant developed by…

The problem of finding optimal curves (the longest arcs) for sub-Lorentzian structures is an optimal control problem with an unbounded control set and a concave cost functional. The question of existence of an optimal solution is nontrivial…

Let $(N, J)$ be a simply connected $2n$-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on $N$ compatible with $J$ to be minimal, if it minimizes the norm of the…

Given a hyperbolic knot $K$ and any $n\geq 2$ the abelian representations and the holonomy representation each give rise to an $(n-1)$-dimensional component in the $\operatorname{SL}(n,\Bbb{C})$-character variety. A component of the…

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\"ahler, balanced) condition is…

The invariants in D=4, N=4 supergravity are discussed up to the three-loop order (where one expects a general R^4 structure). Because there is an anomaly in the rigid SL(2,R) symmetry of this theory, the analysis of possible restrictions on…

A new type of high-dimensional Lie superalgebras is constructed, including Lie superalgebras spo(4,2) and osp(4,2). Based on it, two different coupled nonisospectral super AKNS hierarchies and their bi-Hamiltonian structures are obtained.…

In the present paper we review the progress of the project of classification and construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we called earlier…

It is known that a connected and simply-connected Lie group admits only one left-invariant Riemannian metric up to scaling and isometry if and only if it is isomorphic to the Euclidean space, the Lie group of the real hyperbolic space, or…

In this paper we briefly survey the classical problem of understanding which Lie algebras admit a complex structure, put in the broader perspective of almost complex structures with special properties. We focus on the different behavior of…

It is well known that the category of super Lie groups (SLG) is equivalent to the category of super Harish-Chandra pairs (SHCP). Using this equivalence, we define the category of unitary representations (UR's) of a super Lie group. We give…

This is a collection of notes on the properties of left-invariant metrics on the eight-dimensional compact Lie group SU(3). Among other topics we investigate the existence of invariant pseudo-Riemannian Einstein metrics on this manifold. We…

In [3], after defining notions of LS category in the simplicial context, the authors show that the geometric simplicial LS category is non-decreasing under strong collapses. However, they do not give examples where it increases strictly,…

A theory of finite type invariants for arbitrary compact oriented 3-manifolds is proposed, and illustrated through many examples arising from both classical and quantum topology. The theory is seen to be highly non-trivial even for…

We investigate left-invariant Hitchin and hypo flows on $5$-, $6$- and $7$-dimensional Lie groups. They provide Riemannian cohomogeneity-one manifolds of one dimension higher with holonomy contained in $SU(3)$, $G_2$ and $Spin(7)$,…

A bi-invariant differential 2-form on a Lie group G is a highly constrained object, being determined by purely linear data: an Ad-invariant alternating bilinear form on the Lie algebra of G. On a compact connected Lie group these have an…