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All possible permutations in the discrete $S_4$ group are classified by three rotation angles associated with the orthogonal group $O(3)$. We construct a spinor representation ${\bf 2}_D$ of $O(3)$, which is transformed by three 4$\times$4…
We show how the rigid conformal supersymmetries associated with a certain class of pseudo-Riemannian spin manifolds define a Lie superalgebra. The even part of this superalgebra contains conformal isometries and constant R-symmetries. The…
Starting from the four normed division algebras - the real numbers, complex numbers, quaternions and octonions - a systematic procedure gives a 3-cocycle on the Poincare Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure…
We study real lines on certain Moishezon threefolds which are potentially twistor spaces of 3CP^2. Here, line means a smooth rational curve whose normal bundle is O(1)^2 and the reality implies the invariance under an anti-holomorphic…
We clarify the structure of subgroups generated by conjugate graph automorphisms of order $3$ of $O_8^+(2)$ and $O_8^+(3)$. As a result, we obtain a correction to a paper by S. Guest which, in turn, plays an important role in proving the…
We use algebraic geometry to study the anomaly-free representations of an arbitrary gauge Lie algebra for 4-dimensional spacetime fermions. For irreducible representations, the problem reduces to studying the Lie algebras $\mathfrak{su}_n$…
We study $N=1$ supersymmetric $SO(10)$ gauge theory with a field in the spinorial representaition and $N_f$ ($\leq8$) fields in the defining representation. It is shown that this theory for $N_f=7,8$ has a dual description, which is $N=1$…
The purpose of this article is to give a geometric interpretation to the so-called "twelve surfaces of Darboux", or "Darboux wreath", which appear by applying repeatedly certain simple transformations to a given infinitesimal isometric…
The article is devoted to the question whether the orbit space of a compact linear group is a topological manifold and a homological manifold. In the paper, the case of a simple three-dimensional group is considered. An upper bound is…
We investigate dimensional constraints arising from representation theory when abstract graph edges possess internal degrees of freedom but lack geometric properties. We prove that such internal degrees of freedom can only encode…
Representations of $\text{SO}(4,2)$ are constructed using $4\times4$ and $2\times2$ matrices with elements in $\mathbb{H}'\otimes\mathbb{C}$, and the known isomorphism between the conformal group and $\text{SO}(4,2)$ is written explicitly…
We classify all three-dimensional connected topological loops such that the group topologically generated by their left translations is the four-dimensional connected Lie group $G$ which has trivial center and precisely two one-dimensional…
We study briefly some properties of real Clifford algebras and identify them as matrix algebras. We then show that the representation space on which Clifford algebras act are spinors and we study in details matrix representations. The…
It is well known that anomaly cancellations for D_16 Lie algebra are at the root of the first string revolution. For E_8 Lie algebra, cancellation of anomalies is the principal fact leading to the existence of heterotic string. They are in…
We explore how introducing a non-trivial Mordell-Weil group changes the structure of the Coulomb phases of a five-dimensional gauge theory from an M-theory compactified on an elliptically fibered Calabi-Yau threefolds with a I$_2$+I$_4$…
To solve the path integral for quantum gravity, one needs to regularise the space-times that are summed over. This regularisation usually is a discretisation, which makes it necessary to give up some paradigms or symmetries of continuum…
We construct the E theory analogue of the particles that transform under the Poincare group, that is, the irreducible representations of the semi-direct product of the Cartan involution subalgebra of E11 with its vector representation. We…
The truncation formulae of $D=11$ supergravity on $S^7$ to $D=4$ ${\cal N} =8$ SO(8)-gauged supergravity are completed to include the full non-linear dependence of the $D=11$ three-form potential $\hat A_{(3)}$ on the $D=4$ fields, and…
All results concern characteristic 2. Two procedures that to every simple Lie algebra assign simple Lie superalgebras, most of the latter new, are offered. We prove that every simple finite-dimensional Lie superalgebra is obtained as the…
This paper presents a systematic study of the structure of non-solvable cyclic metric Lie algebras. A cyclic metric is a symmetric bilinear form satisfying a cyclic cocycle condition, which arises naturally in the contexts of…