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Spinors in braided geometry

Mićo Đurđević, Zbigniew Oziewicz (1996)

Banach Center Publications

Let V be a ℂ-space, σ ∈ E n d ( V ⊗ 2 ) be a pre-braid operator and let F ∈ l i n ( V ⊗ 2 , ℂ ) . This paper offers a sufficient condition on (σ,F) that there exists a Clifford algebra Cl(V,σ,F) as the Chevalley F-dependent deformation of an exterior algebra C l ( V , σ , 0 ) ≡ V ∧ ( σ ) . If σ ≠ σ - 1 and F is non-degenerate then F is not a σ-morphism in σ-braided monoidal category. A spinor representation as a left Cl(V,σ,F)-module is identified with an exterior algebra over F-isotropic ℂ-subspace of V. We give a sufficient condition on braid σ that the spinor representation...

Split extensions and semidirect products of unitary magmas

Marino Gran, George Janelidze, Manuela Sobral (2019)

Commentationes Mathematicae Universitatis Carolinae

We develop a theory of split extensions of unitary magmas, which includes defining such extensions and describing them via suitably defined semidirect product, yielding an equivalence between the categories of split extensions and of (suitably defined) actions of unitary magmas on unitary magmas. The class of split extensions is pullback stable but not closed under composition. We introduce two subclasses of it that have both of these properties.

Split structures.

Rosebrugh, Robert, Wood, R.J. (2004)

Theory and Applications of Categories [electronic only]

Splitting maps and norm bounds for the cyclic cohomology of biflat Banach algebras

Yemon Choi (2010)

Banach Center Publications

We revisit the old result that biflat Banach algebras have the same cyclic cohomology as C, and obtain a quantitative variant (which is needed in separate, joint work of the author on the simplicial and cyclic cohomology of band semigroup algebras). Our approach does not rely on the Connes-Tsygan exact sequence, but is motivated strongly by its construction as found in [2] and [5].

Squared Hopf algebras and reconstruction theorems

Volodymyr Lyubashenko (1997)

Banach Center Publications

Given an abelian 𝑉-linear rigid monoidal category 𝑉, where 𝑉 is a perfect field, we define squared coalgebras as objects of cocompleted 𝑉 ⨂ 𝑉 (Deligne's tensor product of categories) equipped with the appropriate notion of comultiplication. Based on this, (squared) bialgebras and Hopf algebras are defined without use of braiding. If 𝑉 is the category of 𝑉-vector spaces, squared (co)algebras coincide with conventional ones. If 𝑉 is braided, a braided Hopf algebra can be obtained from a squared...

Stable short exact sequences and the maximal exact structure of an additive category

Wolfgang Rump (2015)

Fundamenta Mathematicae

It was recently proved that every additive category has a unique maximal exact structure, while it remained open whether the distinguished short exact sequences of this canonical exact structure coincide with the stable short exact sequences. The question is answered by a counterexample which shows that none of the steps to construct the maximal exact structure can be dropped.

Stable tubes in extriangulated categories

Li Wang, Jiaqun Wei, Haicheng Zhang (2022)

Czechoslovak Mathematical Journal

Let 𝒳 be a semibrick in an extriangulated category. If 𝒳 is a τ -semibrick, then the Auslander-Reiten quiver Γ ( ℱ ( 𝒳 ) ) of the filtration subcategory ℱ ( 𝒳 ) generated by 𝒳 is ℤ 𝔸 ∞ . If 𝒳 = { X i } i = 1 t is a τ -cycle semibrick, then Γ ( ℱ ( 𝒳 ) ) is ℤ 𝔸 ∞ / τ 𝔸 t .

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