A Lie algebra gQ over Q is said to be R-universal if every homomorphism from gQ to gl(n,R) is conjugate to a homomorphism into gl(n,Q) (for every n). By using Galois cohomology, we provide a short proof of the known fact that every real semisimple Lie algebra has an R-universal Q-form. We also provide a classification of the R-universal Lie algebras that are semisimple.

We define the Homomorphism Extension (HomExt) problem: given a group G, a subgroup M ≤ G and a homomorphism φ : M → H , decide whether or not there exists a homomorphism φ̃ : G → H extending φ, i.e., φ̃|M = φ. This problem arose in the context of list-decoding homomorphism codes but is also of independent interest, both as a problem in computational group theory and as a new and natural problem i...

A brief comment about items (3) and (4). If H is a subgroup of G the inclusion map is an injective homomorphism from H to G . On the other hand, the image of an injective homomorphism α : H −→ G is a subgroup that is isomorphic to H . So the study of injective homomorphisms is (up to isomorphism) the study of subgroups. After studying subgroups, we defined normal subgroup and showed several equ...

Let Σg → E → Σh be a surface bundle over a surface with monodromy representation ρ : π1Σh → Mod(Σg) contained in the Torelli group Ig. In this paper we express the cup product structure in H∗(E,Z) in terms of the Johnson homomorphism τ : Ig → ∧(H1(Σg ,Z)). This is applied to the question of obtaining an upper bound on the maximal n such that p1 : E → Σh1 , . . . , pn : E → Σhn are fibering maps...

In the group of (continuous) Jordan automorphisms, with the uniform topology, on a semisimple Banach algebra, we show that the connected component of the identity consists of automorphisms. P. Civin and B. Yood have shown that a Jordan homomorphism (that is, a homomorphism that preserves the product xoy = | (xy+yx)) from a Banach algebra onto a semisimple Banach algebra is continuous provided t...

For any semifinite von Neumann algebra ${\mathcal M}$ and $1\leq p<\infty$, we introduce a natutal $S^1$-valued noncommutative $L^p$-space $L^p({\mathcal M};S^1)$. We say that bounded map $T\colon L^p({\mathcal M})\to N})$ is $S^1$-bounded (resp. $S^1$-contractive) if $T\otimes I_{S^1}$ extends to contractive) $T\overline{\otimes} from $ M};S^1)$ into N};S^1)$. show completely positive $S^1$-bo...

A signed bipartite (simple) graph (G,σ) is said to be C−4-critical if it admits no homomorphism C−4 (a negative 4-cycle) but each of its proper subgraphs does. To motivate the study graphs, we show that notion 4-coloring graphs and captured, through simple operations, by C−4. In particular, 4-color theorem equivalent to: Given a planar G, obtained from G replacing edge with path length 2 maps W...

Let r; n be xed natural numbers. We prove that for n-manifolds the set of all linear natural operators T ! T T (r) is a nitely dimensional vector space over R. We construct explicitly the bases of the vector spaces. As a corollary we nd all linear natural operators T ! T r. All manifolds and maps are assumed to be innnitely diierentiable. 0. Let r; n be xed natural numbers. Given a manifold M w...

Classically, an abelian group $G$ is said to be slender if every homomorphism from the countable product $\mathbb Z^{\mathbb N}\mathbb N$ factors through projection some finite Z^n$. Various authors have proposed ge