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Compact and discrete subgroups of algebraic quantum groups I
Operator Algebras · Submitted 15 Feb 2007
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We find several associated objects that generalize the classical objects associated to a compact open subgroup of a locally compact group.
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The abstract, sentence by sentence
Let $G$ be a locally compact group. Consider the C$^*$-algebra $C_0(G)$ of continuous complex functions on $G$, tending to 0 at infinity. The product in $G$ gives rise to a coproduct $\Delta_G$ on the C$^*$-algebra $C_0(G)$. A locally compact {\it quantum} group is a pair $(A,\Delta)$ of a C$^*$-algebra $A$ with a coproduct $\Delta$ on $A$, satisfying certain conditions. The definition guarantees that the pair $(C_0(G),\Delta_G)$ is a locally compact quantum group and that conversely, every locally compact quantum group $(A,\Delta)$ is of this form when the underlying C$^*$-algebra $A$ is abelian. Assume now that $G$ is a locally compact group with a compact open subgroup $K$. The algebra of complex functions on $G$ of {\it polynomial type} is a dense multiplier Hopf $^*$-algebra with positive integrals (i.e. an algebraic quantum group}. The characteristic function of $K$ is a group-like projection in this algebraic quantum group. In this paper, we study group-like projections in an arbitrary algebraic quantum group. We find several associated objects that generalize the classical objects associated to a compact open subgroup of a locally compact group.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $G$ be a locally compact group. | background 1.00 | 0.04 | 0.01 | asserted 0.77 |
| s1 | Consider the C$^*$-algebra $C_0(G)$ of continuous complex functions on $G$, tending to 0 at infinity. | background 1.00 | 0.04 | 0.02 | asserted 0.86 |
| s2 | The product in $G$ gives rise to a coproduct $\Delta_G$ on the C$^*$-algebra $C_0(G)$. | background 1.00 | 0.10 | 0.02 | asserted 1.00 |
| s3 | A locally compact {\it quantum} group is a pair $(A,\Delta)$ of a C$^*$-algebra $A$ with a coproduct $\Delta$ on $A$, satisfying certain conditions. | background 1.00 | 0.05 | 0.02 | asserted 1.00 |
| s4 | The definition guarantees that the pair $(C_0(G),\Delta_G)$ is a locally compact quantum group and that conversely, every locally compact quantum group $(A,\Delta)$ is of this form when the underlying C$^*$-algebra $A$ is abelian. | background 1.00 | 0.09 | 0.04 | asserted 1.00 |
| s5 | Assume now that $G$ is a locally compact group with a compact open subgroup $K$. | background 0.97 | 0.09 | 0.02 | asserted 0.70 |
| s6 | The algebra of complex functions on $G$ of {\it polynomial type} is a dense multiplier Hopf $^*$-algebra with positive integrals (i.e. an algebraic quantum group}. | background 0.79 | 0.40 | 0.06 | asserted 1.00 |
| s7 | The characteristic function of $K$ is a group-like projection in this algebraic quantum group. | background 0.57 | 0.45 | 0.04 | asserted 1.00 |
| s8 | In this paper, we study group-like projections in an arbitrary algebraic quantum group. | contribution 0.99 | 0.62 | 0.07 | asserted 0.97 |
| s9 | We find several associated objects that generalize the classical objects associated to a compact open subgroup of a locally compact group. | contribution 0.61 | 0.94 | 0.34 | asserted 1.00 |
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The paper
Original abstract
Let $G$ be a locally compact group. Consider the C$^*$-algebra $C_0(G)$ of continuous complex functions on $G$, tending to 0 at infinity. The product in $G$ gives rise to a coproduct $\Delta_G$ on the C$^*$-algebra $C_0(G)$. A locally compact {\it quantum} group is a pair $(A,\Delta)$ of a C$^*$-algebra $A$ with a coproduct $\Delta$ on $A$, satisfying certain conditions. The definition guarantees that the pair $(C_0(G),\Delta_G)$ is a locally compact quantum group and that conversely, every locally compact quantum group $(A,\Delta)$ is of this form when the underlying C$^*$-algebra $A$ is abelian. Assume now that $G$ is a locally compact group with a compact open subgroup $K$. The algebra of complex functions on $G$ of {\it polynomial type} is a dense multiplier Hopf $^*$-algebra with positive integrals (i.e. an algebraic quantum group}. The characteristic function of $K$ is a group-like projection in this algebraic quantum group. In this paper, we study group-like projections in an arbitrary algebraic quantum group. We find several associated objects that generalize the classical objects associated to a compact open subgroup of a locally compact group.
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References
How the literature uses it
6 arXiv papers cite this in 21 sentences
Papers citing it · 12 sentences without a confident label
Builds on it 1 paper
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The following fact extends equivalence (i)$⟹$(ii) in Proposition 1.10 and a part of Theorem 2.2 of [ref] , with the same proofs remaining valid.
On idempotent states on quantum groups · Group-like projections in the multiplier algebra and the construction of corresponding quantum subhypergroups · 1.00
Compares with it 1 paper
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A warning is in place here - contrary to the situation in [ref] we cannot expect here in general the invariance of $p$ under the modular group, so also if $h$ is not right-invariant we cannot expect $h_0$ to be right-invariant.
On idempotent states on quantum groups · Group-like projections in the multiplier algebra and the construction of corresponding quantum subhypergroups · 0.82
Cites it as support 1 paper
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This follows from Corollary [ref] and [ref] .
A New Characterisation of Idempotent States on Finite and Compact Quantum Groups · Characterisations of idempotents states on finite quantum groups · 0.97
Cites it as background 4 papers
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A non-zero element $p∈𝖠$ in a compact quantum group $(𝖠,Δ)$ is called a group-like projection [ref] , if it is a projection, i.e. $p^2=p=p^*$, and satisfies $Δ(p)(1⊗p)=p⊗p$.
A New Characterisation of Idempotent States on Finite and Compact Quantum Groups · Preliminaries · 0.95
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Landstad in [ref] and further investigated in [ref] and [ref] .
On idempotent states on quantum groups · Group-like projections in the multiplier algebra and the construction of corresponding quantum subhypergroups · 0.88
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We start with a general result, see [ref] for similar results in the operator algebraic framework and [ref] for the framework of algebraic quantum groups.
Quantum flag manifolds, quantum symmetric spaces and their associated universal K-matrices · Characters of $_q(Z_ν)$ · 0.83
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