Because of the associativity of fusion products ⊠, the above hypothesis is equivalent to the condition that for each irreducible module W , there is an irreducible module W̃ such that HomV (W̃ ⊠ W,V ) 6= 0. In the previous paper [6], we have introduced a concept of ”Semi-Rigidity” and then proved that if W is semi-rigid, then W is flat for the fusion products ⊠, that is, 0 → W ⊠ A → W ⊠ B → W ⊠ C...

Let X and A be sets and α : X → A a map between them. We call a map μ : X ×X ×X → A an approximate Mal’tsev operation with approximation α, if it satisfies μ(x, y, y) = α(x) = μ(y, y, x) for all x, y ∈ X. Note that if A = X and the approximation α is an identity map, then μ becomes an ordinary Mal’tsev operation. We prove the following two characterization theorems: a category X is a Mal’tsev c...

It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle Ed1,...,dn on P N defined as the kernel of a general epimorphism φ : O(−d1)⊕ · · · ⊕ O(−dn) O is (semi)stable. In this note, we restrict our attention to the case of syzygy bundles Ed,n on P associated to n generic forms f1, . . . , fn ∈ K[X0, X1, . . . , XN ] of the same degree d. Our first goal is to pro...

An algebra extension A |B is right depth two if its tensor-square A⊗B A is in the Dress category Add BAA. We consider necessary conditions for right, similarly left, D2 extensions in terms of partial A-invariance of twosided ideals in A contracted to the centralizer. Finite dimensional algebras extending central simple algebras are shown to be depth two. Following P. Xu, left and right bialgebr...

We continue our examination of absolute CR -epic spaces, or spaces with the property that any embedding induces an epimorphism, in the category of commutative rings, between their rings of continuous functions. We examine more closely the deleted plank construction, which generalizes the Dieudonné construction, and yields absolute CR -epic spaces which are not Lindelöf. For the Lindelöf case, a...

A q partial group is defined to be a partial group, that is, a strong semilattice of groups S= [E(S);Se,φe, f ] such that S has an identity 1 and φ1,e is an epimorphism for all e ∈ E(S). Every partial group S with identity contains a unique maximal q partial group Q(S) such that (Q(S))1 = S1. This Q operation is proved to commute with Cartesian products and preserve normality. With Q extended t...

Abstract. A formal notion of a Typ T of a self-dual linear code over a finite left Rmodule V is introduced which allows to give explicit generators of a finite complex matrix group, the associated Clifford-Weil group C(T ) ≤ GL|V |(C), such that the complete weight enumerators of self-dual isotropic codes of Type T span the ring of invariants of C(T ). This generalizes Gleason’s 1970 theorem to...

We propose a new approach to superrigidity phenomena and implement it for lattice representations and measurable cocycles with Homeo+(S ) as the target group. We are motivated by Ghys’ theorem stating that any representation ̺ : Γ → Homeo+(S ) of an irreducible lattice Γ in a semi-simple real Lie group G of higher rank, either has a finite orbit or, up to a semi-conjugacy, extends to G which act...

Let C \mathcal C be a concrete c...

Let $n,m\in \mathbb{N}$, and let $B_{n,m}(\mathbb{R}P^2)$ be the set of $(n + m)$-braids projective plane whose associated permutation lies in subgroup $S_n\times S_m$ symmetric group $S_{n+m}$. We study splitting problem following generalisation Fadell-Neuwirth short exact sequence: $$1\rightarrow B_m(\mathbb{R}P^2 \setminus \{x_1,\dots,x_n\})\rightarrow B_{n,m}(\mathbb{R}P^2)\xrightarrow{\bar...