A construction of Exceptional Weyl Group W(F4) and W(E8) using Quaternion, and the lattice in 16-dimensional Euclidean space
Misaki Ohta, University of the Ryukyus.
It is mentioned that there is a subalgebra isomorphic to the alternating group 2 · A4 as a subalgebra of the Quaternion over integers and half-integers called Hurwitz quaternionic integers H in the book by J.H.Conway and Neil J. A. Sloane. In this paper, I have followed this book and extended Quaternion over integers and half-integers to have duality, and proved that a subalgebra in it isomorphic to Exceptional Weyl group W(F4). I have also found a method of constructing the 16-dimensional lattice Λ16 which seems to be isomorphic to the lattice called the Barnes-Wall lattice ΛBarnes-Wall , which is currently considered to be very dense (although this remains to be discussed) using the Dual Quaternion. Lastly, I briefly mention how to construct an exceptional Weyl group W(E8) using an Octonion and Dual Quaternion.
PAPER(ARXIV)
My researchgate account
On the automorphism of Barns Wall Lattice ΛBW16 and rank 4 tensor of quaternions.
Misaki Ohta, University of the Ryukyus.
In a previous paper, I found that the Weyl group W (F4) and Barns-Wall Lattice BW16 can
be constructed using the rank 2 tensor of the quaternion. In the present paper, I describe how
I were able to construct an algebra, which is the subalgebra of the direct product of Hurwitz
4 21 5 2 Quaternionic integers H , isomorphic to the automorphism Aut(BW16) order 2 · 3 · 5 · 7
of Barns Wall Lattice BW16 by functionally extending the rank of the tensor product of quaternions to 4.
be constructed using the rank 2 tensor of the quaternion. In the present paper, I describe how
I were able to construct an algebra, which is the subalgebra of the direct product of Hurwitz
4 21 5 2 Quaternionic integers H , isomorphic to the automorphism Aut(BW16) order 2 · 3 · 5 · 7
of Barns Wall Lattice BW16 by functionally extending the rank of the tensor product of quaternions to 4.
On exchange relations on pentagons and the Mathieu Group 2 · M 12 and Ternary Golay Code.
Misaki Ohta, University of the Ryukyus.
Just as the geometry of the root system E8 is well described by the exchange relation of 7 points on the figure by the octonion operation, in this paper I show that the exchange relation of 11 points on the pentagon corresponds to the mathieu group M12.