Lorentzian Algebraic Ricci Solitons on the Heisenberg Group

RESUMO

There exist three nonequivalent left invariant Lorentzian metrics on the Heisenberg group $H_{2n+1}$, or equivalently, three nonequivalent Lorentzian inner products on the Heisenberg Lie algebra $\mathfrak{h}_{2n+1}$, denoted by $\mu$, $\nu$, and $\phi$. We show that, in a specific case, $\mu$ is an algebraic Ricci soliton that is shrinking. Moreover, $\nu$ is an algebraic Ricci soliton only on the three-dimensional Heisenberg Lie algebra $\mathfrak{h}_3$ and it is shrinking. Finally, we show that $\phi$ is a steady algebraic Ricci soliton on $\mathfrak{h}_{2n + 1}$ for $n > 1$. However, for $n = 1$, $\phi$ is flat.