The Borcherds invariant-subalgebra conjecture for orbifolded M-theory in D=1

Let ${\mathfrak{e}}_{10}$ be the hyperbolic Kac–Moody algebra, and let $\mathcal{U}_{2}^{{\mathbbm{Z}}_{n}}$ be the automorphism defining the orbifold action. Let ${\mathfrak{g}}_{\mathrm{inv}}$ denote the real invariant subalgebra. A Borcherds invariant-subalgebra conjecture asserts that the invariant subalgebra of ${\mathfrak{e}}_{10}$ under $\mathcal{U}_{2}^{{\mathbbm{Z}}_{n}}$ is the direct sum of a $\mathfrak{u}(1)$ factor and a Borcherds algebra with degenerate Cartan matrix, characterized by one isotropic imaginary simple root $\beta_I$ of multiplicity one and nine real simple roots, modulo its centre and derivation. This gives a closed-form Borcherds description of the $D=1$ invariant algebra, despite the possibility that invariant subalgebras under finite-order automorphisms need not be Kac–Moody algebras. The supplied text gives no resolution status.