Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExact G$_2$-structures on compact quotients of Lie groups
265
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We show that the compact quotient $Gammabackslashmathrm{G}$ of a seven-dimensional simply connected Lie group $mathrm{G}$ by a co-compact discrete subgroup $Gammasubsetmathrm{G}$ does not admit any exact $mathrm{G}_2$-structure which is induced by a left-invariant one on $mathrm{G}$.
rate research
Read More
We prove continuity results for abstract epimorphisms of locally compact groups onto finitely generated groups.
We characterize the structure of a seven-dimensional Lie algebra with non-trivial center endowed with a closed G$_2$-structure. Using this result, we classify all unimodular Lie algebras with non-trivial center admitting closed G$_2$-structures, up to isomorphism, and we show that six of them arise as the contactization of a symplectic Lie algebra. Finally, we prove that every semi-algebraic soliton on the contactization of a symplectic Lie algebra must be expanding, and we determine all unimodular Lie algebras with center of dimension at least two that admit semi-algebraic solitons, up to isomorphism.
Let $G$ be a connected, simply-connected, compact simple Lie group. In this paper, we show that the isometry group of $G$ with a left-invariant pseudo-Riemannan metric is compact. Furthermore, the identity component of the isometry group is compact if $G$ is not simply-connected.
We consider seven-dimensional unimodular Lie algebras $mathfrak{g}$ admitting exact $G_2$-structures, focusing our attention on those with vanishing third Betti number $b_3(mathfrak{g})$. We discuss some examples, both in the case when $b_2(mathfrak{g}) eq0$, and in the case when the Lie algebra $mathfrak{g}$ is (2,3)-trivial, i.e., when both $b_2(mathfrak{g})$ and $b_3(mathfrak{g})$ vanish. These examples are solvable, as $b_3(mathfrak{g})=0$, but they are not strongly unimodular, a necessary condition for the existence of lattices on the simply connected Lie group corresponding to $mathfrak{g}$. More generally, we prove that any seven-dimensional (2,3)-trivial strongly unimodular Lie algebra does not admit any exact $G_2$-structure. From this, it follows that there are no compact examples of the form $(Gammabackslash G,varphi)$, where $G$ is a seven-dimensional simply connected Lie group with (2,3)-trivial Lie algebra, $Gammasubset G$ is a co-compact discrete subgroup, and $varphi$ is an exact $G_2$-structure on $Gammabackslash G$ induced by a left-invariant one on $G$.
In this paper, we study the analytic continuation to complex time of the Hamiltonian flow of certain $Gtimes T$-invariant functions on the cotangent bundle of a compact connected Lie group $G$ with maximal torus $T$. Namely, we will take the Hamiltonian flows of one $Gtimes G$-invariant function, $h$, and one $Gtimes T$-invariant function, $f$. Acting with these complex time Hamiltonian flows on $Gtimes G$-invariant Kahler structures gives new $Gtimes T$-invariant, but not $Gtimes G$-invariant, Kahler structures on $T^*G$. We study the Hilbert spaces ${mathcal H}_{tau,sigma}$ corresponding to the quantization of $T^*G$ with respect to these non-invariant Kahler structures. On the other hand, by taking the vertical Schrodinger polarization as a starting point, the above $Gtimes T$-invariant Hamiltonian flows also generate families of mixed polarizations $mathcal{P}_{0,sigma}, sigma in {mathbb C}, {rm Im}(sigma) >0$. Each of these mixed polarizations is globally given by a direct sum of an integrable real distribution and of a complex distribution that defines a Kahler structure on the leaves of a foliation of $T^*G$. The geometric quantization of $T^*G$ with respect to these mixed polarizations gives rise to unitary partial coherent state transforms, corresponding to KSH maps as defined in [KMN1,KMN2].
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
