相关论文: Quaternionic Root Systems and Subgroups of the $Au…
The automorphism groups of the 27 lines on the smooth cubic surface or the 28 bitangents to the general quartic plane curve are well-known to be closely related to the Weyl groups of $E\_6$ and $E\_7$. We show how classical…
We study the problem of quadruple extensions of simple Lie algebras. We find that, adding a new simple root $\alpha_{+4}$, it is not possible to have an extended Kac-Moody algebra described by a Dynkin-Kac diagram with simple links and no…
$E_8$ is prominent in mathematics and theoretical physics, and is generally viewed as an exceptional symmetry in an eight-dimensional space very different from the space we inhabit; for instance the Lie group $E_8$ features heavily in…
Brundan and Kleshchev showed that some parts of the representation theory of the affine Hecke-Clifford superalgebras and its finite-dimensional "cyclotomic" quotients are controlled by the Lie theory of type $A^{(2)}_{2l}$ when the quantum…
Each of the local isometry groups arising in 3d gravity can be viewed as the group of unit (split) quaternions over a ring which depends on the cosmological constant. In this paper we explain and prove this statement, and use it as a…
The extended affine Weyl group of a root system is the semidirect product of the corresponding Weyl group by its coweight lattice. The stabilizer subgroup of the extended affine Weyl group with respect to the corresponding fundamental…
In this work we are interested in the general problem of the determination of the normed division algebras. Our fundamental results are obtained in the particular subclass of those 8-dimensional quadratic flexible real division algebras. We…
A classification of hadrons and their interactions at low energies according to SU(4) allows to identify combinations of the fifteen mesons $\pi$, $\omega$ and $\rho$ within the spin-isospin decomposition of the regular representation…
We develop further quaternionic analysis introducing left and right doubly regular functions. We derive Cauchy-Fueter type formulas for these doubly regular functions that can be regarded as another counterpart of Cauchy's integral formula…
Four types of discrete transforms of Weyl orbit functions on the finite point sets are developed. The point sets are formed by intersections of the dual-root lattices with the fundamental domains of the affine Weyl groups. The finite sets…
This paper is a continuation of [5]. Using the root categories, we define the compact real forms of the complex semisimple Lie algebras, and maximal compact subgroups of the Chevalley groups over $\mathbb{C}$. In [7], Lusztig used the…
This paper extends the seven-dimensional Fano plane to a 15-dimensional Fano volume, which is related to sedenions. The Fano plane visualises the octonions and their structure as seven quaternions and is derived from a calibration in…
After reviewing the algebraic structures that underlie the geometries of N=2 Maxwell-Einstein supergravity theories (MESGT) in five and four dimensions with symmetric scalar manifolds, we give a unified realization of their three…
We present the subalgebra structure of sl(3,O), a particular real form of E6 chosen for its relevance to particle physics through the connection between its associated Lie group SL(3,O) and generalized Lorentz groups. Given the…
We associate the lepton-quark families with the vertices of the 4D polytopes 5-cell and the rectified 5-cell derived from the SU(5) Coxeter-Dynkin diagram. The off-diagonal gauge bosons are associated with the root poytope (1000)A4 whose…
That finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter-Dynkin diagrams and indecomposable irreducible root systems, is arguably one of…
In our joint papers [FL1-FL2] we revive quaternionic analysis and show deep relations between quaternionic analysis, representation theory and four-dimensional physics. As a guiding principle we use representation theory of various real…
As is well-known, the real quaternion division algebra $ {\cal H}$ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra ${\cal O}$ can not be algebraically isomorphic to any matrix algebras…
The Dunkl--Dirac operator is a deformation of the Dirac operator by means of Dunkl derivatives. We investigate the symmetry algebra generated by the elements supercommuting with the Dunkl--Dirac operator and its dual symbol. This symmetry…
We analyze the geometrical background under which many Lie groups relevant to particle physics are endowed with a (possibly multiple) hexagonal structure. There are several groups appearing, either as special holonomy groups on the…