Vladimir Sobol

The Symmetry Centre of an Integral Lorentzian Lattice: Cell, Breaking, and Two Roots (Geometry of the vacuum, floor 0)

We describe the symmetry centre of an integral Lorentzian lattice of rank 4 and show that three previously unrelated features of it reduce to two roots, not to one. The fundamental domain of the group generated by the mirrors of I_(3,1) is a finite-volume hyperbolic simplex with...

The Symmetry Centre of an Integral Lorentzian Lattice: Cell, Breaking, and Two Roots (Geometry of the vacuum, floor 0)

We describe the symmetry centre of an integral Lorentzian lattice of rank 4 and show that three previously unrelated features of it reduce to two roots, not to one. The fundamental domain of the group generated by the mirrors of I_(3,1) is a finite-volume hyperbolic simplex with...

Crystallography from a lift condition (Geometry of the vacuum, floor 0)

A single deep wedge in the orthogonal Lie algebra so(5,3) induces on its transverse quotient a metric of signature exactly (3,1), and the equivariance of the degree-0 floor built over it forces that same metric as the unique so(W)-equivariant bracket of one family. On this struct...

Non-Crystallographic Orders of Integral Lorentzian Lattices and Where They Are Realized (Geometry of the vacuum, floor 0)

Finite-order integral isometries of a lattice of signature (3,1) have order in {1,2,3,4,6}, so orders 5, 8 and 12 are excluded there. This paper asks where those orders are realized instead, and answers on two levels. First, one rank up: for each n in {5,8,12} the cyclotomic carr...

Crystallography from a lift condition (Geometry of the vacuum, floor 0)

A single deep wedge in the orthogonal Lie algebra so(5,3) induces on its transverse quotient a metric of signature exactly (3,1), and the equivariance of the degree-0 floor built over it forces that same metric as the unique so(W)-equivariant bracket of one family. On this struct...