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A complex $\omega$-Lie algebra is a vector space $L$ over the complex field, equipped with a skew symmetric bracket $[-,-]$ and a bilinear form $\omega$ such that $$[[x,y],z]+[[y,z],x]+ [[z,x],y]=\omega(x,y)z+\omega(y,z)x+\omega(z,x)y$$ for…
Rank-2 and rank-3 tensors are almost all identifiable with only few exceptions. We classify them all together with the dimensions and the structures of all the sets evincing the rank.
Classifying isomorphism classes of group gradings on algebras presents a compelling challenge, particularly within the realms of non-simple and infinite-dimensional algebras, which have been relatively unexplored. This study focuses on a…
A Cayley (resp. bi-Cayley) graph on a dihedral group is called a {\em dihedrant} (resp. {\em bi-dihedrant}). In 2000, a classification of trivalent arc-transitive dihedrants was given by Maru\v si\v c and Pisanski, and several years later,…
This paper is a contribution to the development of the non associative algebras theory. More precisely, this work deals with the classification of the complex 4-dimensional Leibniz algebras. Note that the classification of 4-dimensional…
In this paper we classify all simply connected five dimensional nilpotent Lie groups which admit $(\alpha,\beta)$-metrics of Berwald and Douglas type defined by a left invariant Riemannian metric and a left invariant vector field. During…
In this paper we will give a short and direct proof that Wolfgang Kuehnel's 9-vertex triangulation of the complex projective plane really is the complex projective plane. The idea of our proof is to recall the trisection of the complex…
Real forms of a complex reductive group are classified in terms of Galois cohomology $H^1(\Gamma,G_{ad})$ where $G_{ad}$ is the adjoint group. Alternatively, the theory of the Cartan involution gives a description in terms of cohomology…
We construct two categorifications of the Lusztig--Vogan module associated to a real reductive algebraic group. The first categorification is given by semisimple complexes in an equivariant derived category, and the second is constructed as…
Defining the real Lie superalgebra as real $Z_2$--graded vector space we classify real Manin supertriples and Drinfel'd superdoubles of superdimensions (2,2), (4,2) and (2,4). They can be used for construction of sigma-models on supergroups…
The study of irreducible subvarieties has recently seen a surge of interest due to connections with convex geometry. In this paper, we study cohomology classes of Grassmannians that are realizable by irreducible subvarieties. We completely…
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…
We classify all Gieseker semi-stable sheaves on the complex projective plane that have dimension 1 and multiplicity 6. We decompose their moduli spaces into strata which occur naturally as quotients modulo actions of certain algebraic…
We classify the non-degenerate two-step nilpotent Lie algebras of dimension 8 over the field of real numbers, using known results over complex numbers. We write explicit structure constants for these real Lie algebras.
The V-systems are special finite sets of covectors which appeared in the theory of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. Several families of V-systems are known but their classification is an open problem. We…
In this article we give a classification of the sub-groups in PSL(2,Z) and of the conjugacy classes of these sub-groups by the mean of an combinatorial invariant: some trivalent diagrams (dotted or not). We give explicit formulae enabling…
Given a braided pivotal category $\mathcal C$ and a pivotal module tensor category $\mathcal M$, we define a functor $\mathrm{Tr}_{\mathcal C}:\mathcal M \to \mathcal C$, called the associated categorified trace. By a result of…
We study the subrank of real order-three tensors and give an upper bound to the subrank of a real tensor given its complex subrank. Using similar arguments to those used by Bernardi-Blekherman-Ottaviani, we show that all subranks between…
This paper presents a full catalogue, up to conjugacy and subgroups of finite index, of all matrix groups $H < {\rm GL}(3,\mathbb{R})$ that give rise to a continuous wavelet transform with associated irreducible quasi-regular…
The explicit matrix realizations of the reversion anti-automorphism and the spin group depend on the set of matrices chosen to represent a basis of 1 -vectors for a given Clifford algebra. On the other hand, there are iterative procedures…