Classification of Lie algebras with naturally graded quasi-filiform nilradicals

The whole class of complex Lie algebras g$\mathfrak{g}$ having a naturally graded nilradical with characteristic sequence c(g)=(dimg−2,1,1)$c\left(\mathfrak{g}\right)=(dim\mathfrak{g}-2,1,1)$ is classified. It is shown that up to one exception, such Lie algebras are solvable.     

Infinitesimal supersymmetries over Lie groups and their classification for gl(1,1) and sl(1,1)

Infinitesimal supersymmetries over classical Lie groups that are not necessarily induced by a Lie supergroup are described. They yield a notion of supersymmetry that is less rigid than the assumption of a Lie supergroup action but still implies an underlying action of a Lie group. In contrast to Lie supergroups, the arising representation-theoretical Lie supergroups (RTLSG) occur as families associated to Harish–Chandra superpairs. However morphisms of RTLSGs directly correspond to morphisms of Harish–Chandra superpairs. Particular RTLSGs can be derived from the explicit constructions of Lie supergroups given by Kostant and Koszul. The Lie superalgebras $\mathfrak{g}\mathfrak{l}(1,1)$ or $\mathfrak{s}\mathfrak{l}(1,1)$ appearing also in higher dimensional classical Lie superalgebras, provide interesting first examples of RTLSGs. A classification of RTLSGs associated to real and complex $\mathfrak{g}\mathfrak{l}(1,1)$- and $\mathfrak{s}\mathfrak{l}(1,1)$-Harish–Chandra superpairs is given by parameter spaces and complete sets of invariants. The underlying Lie group is assumed to be connected but possibly not simply connected.     

Weak mirror symmetry of complex symplectic Lie algebras

A complex symplectic structure on a Lie algebra h$\mathfrak{h}$ is an integrable complex structure J$J$ with a closed non-degenerate (2,0)$(2,0)$-form. It is determined by J$J$ and the real part Ω$\Omega$ of the (2,0)$(2,0)$-form. Suppose that h$\mathfrak{h}$ is a semi-direct product g?V$\mathfrak{g}?V$, and both g$\mathfrak{g}$ and V$V$ are Lagrangian with respect to Ω$\Omega$ and totally real with respect to J$J$. This note shows that g?V$\mathfrak{g}?V$ is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of Ω$\Omega$ and J$J$ are isomorphic.     

Addendum to: “Ricci-flat holonomy: A classification” [J. Geom. Phys. 57 (6) 1457–1475]: The case of spin(10)

This note fills a hole in the author’s previous paper “Ricci-flat holonomy: A classification”, by dealing with irreducible holonomy algebras that are subalgebras or real forms of C⊕spin(10,C)$\mathbb{C}\oplus \mathfrak{s}\mathfrak{p}\mathfrak{i}\mathfrak{n}(10,\mathbb{C})$. These all turn out to be of Ricci type.     

Projective geometry from Poisson algebras

By analogy with the Poisson algebra of quadratic forms on the symplectic plane and with the concept of duality in the projective plane introduced by Arnold (2005) [1], where the concurrence of the triangle altitudes is deduced from the Jacobi identity, we consider the Poisson algebras of the first degree harmonics on the sphere, on the pseudo-sphere and on the hyperboloid, to obtain analogous duality concepts and similar results for spherical, pseudo-spherical and hyperbolic geometry. Such algebras, including the algebra of quadratic forms, are isomorphic either to the Lie algebra of the vectors in R³${\mathbb{R}}^3$, with the vector product, or to algebra s_l2(R)$s{l}_2\left(\mathbb{R}\right)$. The Tomihisa identity, introduced in (Tomihisa, 2009) [3] for the algebra of quadratic forms, holds for all these Poisson algebras and has a geometrical interpretation. The relationships between the different definitions of duality in projective geometry inherited by these structures are shown here.     

Omni-Lie 2-algebras and their Dirac structures

We introduce the notion of omni-Lie 2-algebra, which is a categorification of Weinstein’s omni-Lie algebras. We prove that there is a one-to-one correspondence between strict Lie 2-algebra structures on 2-sub-vector spaces of a 2-vector space V$\mathbb{V}$ and Dirac structures on the omni-Lie 2-algebra gl(V)⊕V$\mathfrak{g}\mathfrak{l}\left(\mathbb{V}\right)\oplus \mathbb{V}$. In particular, strict Lie 2-algebra structures on V$\mathbb{V}$ itself one-to-one correspond to Dirac structures of the form of graphs. Finally, we introduce the notion of twisted omni-Lie 2-algebra to describe (non-strict) Lie 2-algebra structures. Dirac structures of a twisted omni-Lie 2-algebra correspond to certain (non-strict) Lie 2-algebra structures, which include string Lie 2-algebra structures.     

Non Abelian structures and the geometric phase of entangled qudits

In this work, we address some important topological and algebraic aspects of two-qudit states evolving under local unitary operations. The projective invariant subspaces and evolutions are connected with the common elements characterizing the su(d)$\mathfrak{s}\mathfrak{u}\left(d\right)$ Lie algebra and their representations. In particular, the roots and weights turn out to be natural quantities to parametrize cyclic evolutions and fractional phases. This framework is then used to recast the coset contribution to the geometric phase in a form that generalizes the usual monopole-like formula for a single qubit.     

A note on the hidden conformal structure of non-extremal black holes

We study, following Bertini et al. [1], the hidden conformal symmetry of the massless Klein–Gordon equation in the background of the general, charged, spherically symmetric, static black-hole solution of a class of d  -dimensional Lagrangians which includes the relevant parts of the bosonic Lagrangian of any ungauged supergravity. We find that a hidden SL(2,R)$\mathit{SL}(2,\mathbb{R})$ symmetry appears at the near event- and Cauchy-horizon limits. We extend the two sl(2)$\mathfrak{sl}(2)$ algebras to two full Witt algebras (Virasoro algebras with vanishing central charges). We comment on the implications of the possible existence of an associated quantum conformal field theory.     

Generalizations of Maxwell (super)algebras by the expansion method

The Lie algebras expansion method is used to show that the four-dimensional spacetime Maxwell (super)algebras and some of their generalizations can be derived in a simple way as particular expansions of o(3,2)$o(3,2)$ and osp(N|4)$\mathit{osp}(N|4)$.     

On the Riemannian geometry of Seiberg–Witten moduli spaces

We construct a natural L²$L^2$-metric on the perturbed Seiberg–Witten moduli spaces _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$ of a compact 4-manifold M$M$, and we study the resulting Riemannian geometry of _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$. We derive a formula which expresses the sectional curvature of _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$ in terms of the Green operators of the deformation complex of the Seiberg–Witten equations. In case M$M$ is simply connected, we construct a Riemannian metric on the Seiberg–Witten principal U(1)$U\left(1\right)$ bundle P→_Mμ+$\mathfrak{P}\to {\mathfrak{M}}_{{\mu }^{+}}$ such that the bundle projection becomes a Riemannian submersion. On a Kähler surface M$M$, the L²$L^2$-metric on _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$ coincides with the natural Kähler metric on moduli spaces of vortices.