Francisco Javier Navarro Izquierdo

• Phd in Mathematics
• Professor (Associate) at Universidad de Cádiz

A non-commutative Poisson algebra is a Lie algebra endowed with a, not necessarily commutative, associative product in such a way that the Lie and associative products are compatible via the Leibniz identity. If we consider a non-commutative Poisson algebra \(\mathcal {P}\) of arbitrary dimension, over an arbitrary base field \({\mathbb F}\), a bas...

A non-commutative Poisson algebra is a Lie algebra endowed with a, not necessarily commutative, associative product in such a way that the Lie and associative products are compatible via the Leibniz identity. If we consider a non-commutative Poisson algebra \({\mathfrak P}\) of arbitrary dimension, over an arbitrary base field \({\mathbb F}\), a ba...

Let \(\mathfrak {R}\) be a ring graded by an arbitrary set A. We show that \(\mathfrak {R}\) decomposes as the sum of the well-described graded ideals plus (maybe) a certain subgroup. We also provide a context where the graded simplicity of \(\mathfrak {R}\) is characterized and where a second Wedderburn-type theorem in the category of arbitrarily...

We associate an adequate graph to any pair (V,W) where V is a graded module over a graded linear space W, in such a way that allows us to study the inner algebraic structure of (V,W). In particular, the homogeneous indecomposability, the homogeneous semisimplicity and the homogeneous simplicity of (V,W) are characterized in terms of this graph. Som...

We consider maps f:A×R→A between arbitrary non-empty sets A and R and show that if f fixes some element in A, then this map induces an adequate decomposition of A as the disjoint-pointed union of well-described f-invariant subsets (submodules). If A is furthermore a division set-module, it is shown that the above decomposition is by means of the fa...

Let A be a non-empty set. An augmented ternary map over A is any map f : A × A × A ? A ? with ?/ A. We show that any augmented ternary map f over A induces a decomposition on A as the orthogonal disjoint union of well-described ideals. If (A, f) is furthermore a division f-triple, it is shown that the above decomposition is through the family of it...

Let V be a linear space of arbitrary dimension and over an arbitrary base field F, endowed with a bilinear map f:V×V→V. A basis B={vi}i∈I of V is an f-basis if for any i,j∈I we have that f(vi,vj)∈Fvk for some k∈I. We associate to any triplet (V,f,B) an adequate graph (V,E). By arguing on this graph we show that V decomposes as a direct sum of stron...

Let (A,〈⋅,…,⋅〉) be an n-algebra of arbitrary dimension and over an arbitrary base field 𝔽. A basis ℬ = {ei}i∈I of A is said to be multiplicative if for any i1,…,in ∈ I, we have either 〈ei1,…,ein〉 = 0 or 0≠〈ei1,…,ein〉∈ 𝔽ej for some (unique) j ∈ I. If n = 2, we are dealing with algebras admitting a multiplicative basis while if n = 3 we are speaking...

Let (T, ⟨ ·, ·, · ⟩) be a triple system of arbitrary dimension, over an arbitrary base field 𝔽 and in which any identity on the triple product is not supposed. A basis ℬ = {ei}i∈I of T is called multiplicative if for any i, j, k ∈ I we have that ⟨ ei, ej, ek ⟩ ∈ 𝔽er for some r ∈ I. We show that if T admits a multiplicative basis then it decomposes...

Fix an infinite set I and consider the associative matrix algebra where is a base field with . For any couple of bijective maps , such that and , we introduce a linear subspace of . We endow it with a structure of (non-associative) algebra for a certain bilinear product, and obtain a wide class of non-associative algebras containing, in particular,...

Let V and W be two vector spaces over a base field (Formula presented.). It is said that V is a module over W if it is endowed with a bilinear map (Formula presented.), (Formula presented.). A basis (Formula presented.) of V is called multiplicative with respect to the basis (Formula presented.) of W if for any (Formula presented.) we have either (...

Let be an algebra of arbitrary dimension, over an arbitrary base field and in which any identity on the product is not supposed. A basis of is called multiplicative if for any we have that for some . We show that if admits a multiplicative basis then it decomposes as the direct sum of well-described ideals admitting each one a multiplicative basis....