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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...

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...

Sur les quasi-limites

Syméon Bozapalides (1976)

Cahiers de Topologie et Géométrie Différentielle Catégoriques

The affineness criterion for quantum Hom-Yetter-Drinfel'd modules

Shuangjian Guo, Shengxiang Wang (2016)

Colloquium Mathematicae

Quantum integrals associated to quantum Hom-Yetter-Drinfel’d modules are defined, and the affineness criterion for quantum Hom-Yetter-Drinfel’d modules is proved in the following form. Let (H,α) be a monoidal Hom-Hopf algebra, (A,β) an (H,α)-Hom-bicomodule algebra and B = A c o H . Under the assumption that there exists a total quantum integral γ: H → Hom(H,A) and the canonical map β : A ⊗ B A → A ⊗ H , a ⊗ B b ↦ S - 1 ( b [ 1 ] ) α ( b [ 0 ] [ - 1 ] ) ⊗ β - 1 ( a ) β ( b [ 0 ] [ 0 ] ) , is surjective, we prove that the induction functor A ⊗ B - : ̃ ( k ) B → A H is an equivalence of categories.

The compositional construction of Markov processes II

L. de Francesco Albasini, N. Sabadini, R. F. C. Walters (2011)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We add sequential operations to the categorical algebra of weighted and Markov automata introduced in [L. de Francesco Albasini, N. Sabadini and R.F.C. Walters, 
arXiv:0909.4136]. The extra expressiveness of the algebra permits the description of hierarchical systems, and ones with evolving geometry. We make a comparison with the probabilistic automata of Lynch et al. [SIAM J. Comput. 37 (2007) 977–1013].

The compositional construction of Markov processes II

L. de Francesco Albasini, N. Sabadini, R. F.C. Walters (2011)

RAIRO - Theoretical Informatics and Applications

We add sequential operations to the categorical algebra of weighted and Markov automata introduced in [L. de Francesco Albasini, N. Sabadini and R.F.C. Walters, 
arXiv:0909.4136]. The extra expressiveness of the algebra permits the description of hierarchical systems, and ones with evolving geometry. We make a comparison with the probabilistic automata of Lynch et al. [SIAM J. Comput.37 (2007) 977–1013].

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