Generalized (co)integrals on coideal subalgebras

Given a a Hopf algebra $H$, its left coideal subalgebra $A$ and a non-zero multiplicative functional $\mu$ on $A$, we define the space of left $\mu$-integrals $L^A_\mu\subset A$. We observe that $\dim L^A_\mu=1$ if $A$ is a Frobenius algebra and we conclude this equality for finite dimensional left coideal subalgebras of a weakly finite Hopf algebra. In general we prove that if $\dim L^A_\mu>0$ then $\dim A <\infty$. Given a group-like element $g\in H$ we define the space $L^g_{ A}\subset A'$ of $g$-cointegrals on $A$ and linking this concept with the theory of $\mu$-integrals we observe that: - every semisimple left coideal subalgebra $A\subset H$ which is preserved by the antipode squared admits a faithful $1$-cointegral; - every unimodular finite dimensional left coideal subalgebra $A\subset H$ admitting a faithful $1$-cointegral is preserved by the antipode square; - every non-degenerate right group-like projection in a cosemisimple Hopf algebra is a two sided group-like projection. Finally we list all $\varepsilon$-integrals for left coideals subalgebras in Taft algebras and we list all $g$-cointegrals on them.Comment: The results are essentially the same but the presentation of them and their proofs was improved. In Section 5 the list of $\varepsilon$-integrals was completed with the case missing in the previous versio

Group-like projections for locally compact quantum groups

Let $\mathbb{G}$ be a locally compact quantum group. We give a 1-1 correspondence between group-like projections in $L^\infty(\mathbb{G})$ preserved by the scaling group and idempotent states on the dual quantum group $\widehat{\mathbb{G}}$. As a byproduct we give a simple proof that normal integrable coideals in $L^\infty(\mathbb{G})$ which are preserved by the scaling group are in 1-1 correspondence with compact quantum subgroups of $\mathbb{G}$.Comment: Typos corrected, reference added. Accepted for a publication in JOT. arXiv admin note: text overlap with arXiv:1606.0057

On the Hopf (co)center of a Hopf algebra

The notion of Hopf center and Hopf cocenter of a Hopf algebra is investigated by the extension theory of Hopf algebras. We prove that each of them yields an exact sequence of Hopf algebras. Moreover the exact sequences are shown to satisfy the faithful (co)flatness condition. Hopf center and cocenter are computed for $\mathsf{U}_q(\mathfrak{g})$ and the Hopf algebra $\textrm{Pol}(\mathbb{G}_q)$, where $\mathbb{G}_q$ is the Drinfeld-Jimbo quantization of a compact semisimple simply connected Lie group $\mathbb{G}$ and $\mathfrak{g}$ is a simple complex Lie algebra.Comment: 24 pages + references; more minor corrections / additions. Accepted to J. Algebr

Quantum families of quantum group homomorphisms

The notion of a quantum family of maps has been introduced in the framework of C*-algebras. As in the classical case, one may consider a quantum family of maps preserving additional structures (e.g. quantum family of maps preserving a state). In this paper we define a quantum family of homomorphisms of locally compact quantum groups. Roughly speaking, we show that such a family is classical. The purely algebraic counterpart of the discussed notion, i.e. a quantum family of homomorphisms of Hopf algebras, is introduced and the algebraic counterpart of the aforementioned result is proved. Moreover, we show that a quantum family of homomorphisms of Hopf algebras is consistent with the counits and coinverses of the given Hopf algebras. We compare our concept with weak coactions introduced by Andruskiewitsch and we apply it to the analysis of adjoint coaction.Comment: Minor corrections. This version is accepted for a publication in Communications in Algebr

Embeddable Quantum Homogeneous Spaces

We discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups. On the von Neumann algebra level we find an interesting duality for such objects. A definition of a quantum homogeneous space is proposed along the lines of the pioneering work of Vaes on induction and imprimitivity for locally compact quantum groups. The concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces. Among various examples we single out the quantum analog of the quotient of the Cartesian product of a quantum group with itself by the diagonal subgroup, analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles.Comment: 20 pages, 1 figure, exposition improved in Section 7, several references adde

Shifts of group-like projections and contractive idempotent functionals for locally compact quantum groups

A one to one correspondence between shifts of group-like projections on a locally compact quantum group ${\mathbb{G}}$ which are preserved by the scaling group and contractive idempotent functionals on the dual $\hat{\mathbb{G}}$ is established. This is a generalization of the Illie-Spronk's correspondence between contractive idempotents in the Fourier-Stieltjes algebra of a locally compact group $G$ and cosets of open subgroups of $G$. We also establish a one to one correspondence between non-degenerate, integrable, ${\mathbb{G}}$-invariant ternary rings of operators $X\subset L^\infty({\mathbb{G}})$, preserved by the scaling group and contractive idempotent functionals on ${\mathbb{G}}$. Using our results we characterize coideals in $L^\infty(\hat{\mathbb{G}})$ admitting an atom preserved by the scaling group in terms of idempotent states on ${\mathbb{G}}$. We also establish a one to one correspondence between integrable coideals in $L^\infty({\mathbb{G}})$ and group-like projections in $L^\infty(\hat{\mathbb{G}})$ satisfying an extra mild condition. Exploiting this correspondence we give examples of group like projections which are not preserved by the scaling group.Comment: Section 5 added, establishing a one to one correspondence between non-degenerate, integrable, ${\mathbb{G}}$-invariant ternary rings of operators $X\subset L^\infty({\mathbb{G}})$, preserved by the scaling group and contractive idempotent functionals on ${\mathbb{G}}$. A 1-1 correspondence between integrable coideals and group-like projections were extended beyond the compact cas

Open quantum subgroups of locally compact quantum groups

The notion of an open quantum subgroup of a locally compact quantum group is introduced and given several equivalent characterizations in terms of group-like projections, inclusions of quantum group C*-algebras and properties of respective quantum homogenous spaces. Open quantum subgroups are shown to be closed in the sense of Vaes and normal open quantum subgroups are proved to be in 1-1 correspondence with normal compact quantum subgroups of the dual quantum group.Comment: 30 pages, v2 simplifies the proof of Theorem 5.8 (and drops the properness assumption) as well as corrects several small points. The final version will appear in Advances in Mathematic

Kawada-It\^o-Kelley Theorem for Quantum Semigroups

Idempotent states on locally compact quantum semigroups with weak cancellation properties are shown to be Haar states on a certain sub-object described by an operator system with comultiplication. We also give a characterization of the situation when this sub-object is actually a compact quantum subgroup. In particular we reproduce classical results on idempotent probability measures on locally compact semigroups with cancellation.Comment: 12 page

The canonical central exact sequence for locally compact quantum groups

For a locally compact quantum group $\mathbb{G}$ we define its center, $\mathscr{Z}(\mathbb{G})$, and its quantum group of inner automorphisms, $\mathrm{Inn}(\mathbb{G})$. We show that one obtains a natural isomorphism between $\mathrm{Inn}(\mathbb{G})$ and $\mathbb{G}/\!\mathscr{Z}(\mathbb{G})$, we characterize normal quantum subgroups of a compact quantum group as those left invariant by the action of the quantum group of inner automorphisms and discuss several examples.Comment: v4: 14 pages, modified title. The final version of the paper will appear in Mathematische Nachrichte

Integrable actions and quantum subgroups

We study homomorphisms of locally compact quantum groups from the point of view of integrability of the associated action. For a given homomorphism of quantum groups $\Pi\colon\mathbb{H}\to\mathbb{G}$ we introduce quantum groups $\mathbb{H}/\!\ker{\Pi}$ and $\overline{\mathrm{im}\,\Pi}$ corresponding to the classical quotient by kernel and closure of image. We show that if the action of $\mathbb{H}$ on $\mathbb{G}$ associated to $\Pi$ is integrable then $\mathbb{H}/\!\ker\Pi\cong\overline{\mathrm{im}\,\Pi}$ and characterize such $\Pi$. As a particular case we consider an injective continuous homomorphism $\Pi\colon{H}\to{G}$ between locally compact groups $H$ and $G$. Then $\Pi$ yields an integrable action of $H$ on $L^\infty\;\!\!(G)$ if and only if its image is closed and $\Pi$ is a homeomorphism of $H$ onto $\mathrm{im}\,\Pi$. We also give characterizations of open quantum subgroups and of compact quantum subgroups in terms of integrability and show that a closed quantum subgroup always gives rise to an integrable action. Moreover we prove that quantum subgroups closed in the sense of Woronowicz whose associated homomorphism of quantum groups yields an integrable action are closed in the sense of Vaes.Comment: 18 page