Idempotent States and the Inner Linearity Property

• Autorzy: Banica T. , Franz U. , Skalski A.

Identyfikatory

DOI

10.4064/ba60-2-3

• Języki publikacji: EN

Abstrakty

EN

We find an analytic formulation of the notion of Hopf image, in terms of the associated idempotent state. More precisely, if π:A→M_n(C) is a finite-dimensional representation of a Hopf C*-algebra, we prove that the idempotent state associated to its Hopf image A′ must be the convolution Cesàro limit of the linear functional φ=tr∘π. We then discuss some consequences of this result, notably to inner linearity questions.

Słowa kluczowe

EN

idempotent state; Hopf image; inner linearity

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2012
• Tom: Vol. 60, no 2
• Strony: 123--132
• Opis fizyczny: Bibliogr. 20 poz.

Twórcy

autor

Banica T.

• Department of Mathematics Cergy-Pontoise University 95000 Cergy-Pontoise, France

autor

Franz U.

• Department of Mathematics University of Franche-Comté 16 route de Gray 25030 Besançon Cedex, France

autor

Skalski A.

• Institute of Mathematics Polish Academy of Sciences Sniadeckich 8 P.O. Box 21 00-956 Warszawa, Poland

Bibliografia

• [1] N. Andruskiewitsch and J. Bichon, Examples of inner linear Hopf algebras, Rev. Un. Mat. Argentina 51 (2010), 7–18.
• [2] T. Banica, Quantum permutations, Hadamard matrices, and the search for matrix models, arXiv:1109.4888.
• [3] T. Banica and J. Bichon, Hopf images and inner faithful representations, Glasgow Math. J. 52 (2010), 677–703.
• [4] T. Banica, J. Bichon and J.-M. Schlenker, Representations of quantum permutation algebras, J. Funct. Anal. 257 (2009), 2864–2910.
• [5] T. Banica and B. Collins, Integration over the Pauli quantum group, J. Geom. Phys. 58 (2008), 942–961.
• [6] B. Collins and P. Sniady, Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Comm. Math. Phys. 264 (2006), 773–795.
• [7] S. Dascalescu, C. Nastasescu and S. Raianu, Hopf Algebras. An Introduction, Dekker, 2001.
• [8] M. S. Dijkhuizen and T. H. Koornwinder, CQG algebras: A direct algebraic approach to compact quantum groups, Lett. Math. Phys. 32 (1994), 315–330.
• [9] U. Franz and A. Skalski, On idempotent states on quantum groups, J. Algebra 322 (2009), 1774–1802.
• [10] —, —, A new characterisation of idempotent states on finite and compact quantum groups, C. R. Math. Acad. Sci. Paris 347 (2009), 991–996.
• [11] U. Franz, A. Skalski and R. Tomatsu, Idempotent states on compact quantum groups and their classification on Uq(2), SUq(2), and SOq(3), J. Noncommut. Geom., to appear.
• [12] V. F. R. Jones, Planar algebras I, arXiv:math/9909027.
• [13] Y. Kawada and K. Itô, On the probability distribution on a compact group I, Proc. Phys.-Math. Soc. Japan 22 (1940), 977–998.
• [14] A. Pal, A counterexample on idempotent states on a compact quantum group, Lett. Math. Phys. 37 (1996), 75–77.
• [15] P. Salmi and A. Skalski, Idempotent states on locally compact quantum groups, Quart. J. Math., to appear; arXiv:1102.2051.
• [16] S. Vaes, Strictly outer actions of groups and quantum groups, J. Reine Angew. Math. 578 (2005), 147–184.
• [17] A. Van Daele, The Haar measure on a compact quantum group, Proc. Amer. Math. Soc. 123 (1995), 3125–3128.
• [18] S. Z. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195–211.
• [19] S. L. Woronowicz, Compact matrix pseudogroups, ibid. 111 (1987), 613–665.
• [20] —, Compact quantum groups, in: Symétries quantiques, North-Holland, 1998, 845–884.