The Planar Algebra of a Coaction
نویسنده
چکیده
We study actions of “compact quantum groups” on “finite quantum spaces”. According to Woronowicz and to general C-algebra philosophy these correspond to certain coactions v : A → A ⊗ H . Here A is a finite dimensional C-algebra, and H is a certain special type of Hopf ∗-algebra. If v preserves a positive linear form φ : A → C, a version of Jones’ “basic construction” applies. This produces a certain C-algebra structure on A, plus a coaction vn : A ⊗n → A ⊗H , for every n. The elements x satisfying vn(x) = x⊗1 are called fixed points of vn. They form a C -algebra Qn(v). We prove that under suitable assumptions on v the graded union of the algebras Qn(v) is a spherical C -planar algebra.
منابع مشابه
Adjunctions between Hom and Tensor as endofunctors of (bi-) module category of comodule algebras over a quasi-Hopf algebra.
For a Hopf algebra H over a commutative ring k and a left H-module V, the tensor endofunctors V k - and - kV are left adjoint to some kinds of Hom-endofunctors of _HM. The units and counits of these adjunctions are formally trivial as in the classical case.The category of (bi-) modules over a quasi-Hopf algebra is monoidal and some generalized versions of Hom-tensor relations have been st...
متن کاملCoverings of skew products and crossed products by coactions
Consider a projective limit G of finite groups Gn. Fix a compatible family δn of coactions of the Gn on a C ∗-algebra A. From this data we obtain a coaction δ of G on A. We show that the coaction crossed product of A by δ is isomorphic to a direct limit of the coaction crossed products of A by the δn. If A = C∗(Λ) for some k-graph Λ, and if the coactions δn correspond to skewproducts of Λ, then...
متن کاملar X iv : h ep - t h / 94 03 15 4 v 1 2 5 M ar 1 99 4 Interrelations between Quantum Groups and Reflection Equation ( Braided ) Algebras
We show that the differential complex Ω B over the braided matrix algebra BM q (N) represents a covariant comodule with respect to the coaction of the Hopf algebra Ω A which is a differential extension of GL q (N). On the other hand, the algebra Ω A is a covariant braided comodule with respect to the coaction of the braided Hopf algebra Ω B. Geometrical aspects of these results are discussed.
متن کاملA new algebra which transmutes to the braided algebra
We find a new braided Hopf structure for the algebra satisfied by the entries of the braided matrix BSLq(2). A new nonbraided algebra whose coalgebra structure is the same as the braided one is found to be a two parameter deformed algebra. It is found that this algebra is not a comodule algebra under adjoint coaction. However, it is shown that for a certain value of one of the deformation param...
متن کاملWeakly multiplicative coactions of quantized function algebras
A condition is identified which guarantees that the coinvariants of a coaction of a Hopf algebra on an algebra form a subalgebra, even though the coaction may fail to be an algebra homomorphism. A Hilbert Theorem (finite generation of the subalgebra of coinvariants) is obtained for such coactions of a cosemisimple Hopf algebra. This is applied for two coactions α, β : A → A⊗O, where A is the co...
متن کامل