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Quantum \( SL _2\), infinite curvature and Pitman’s 2M-X theorem

Abstract

The classical theorem by Pitman states that a Brownian motion minus twice its running infimum enjoys the Markov property. On the one hand, Biane understood that Pitman’s theorem is intimately related to the representation theory of the quantum group \({{\mathcal {U}}}_q\left( {\mathfrak sl}_2 \right) \), in the so-called crystal regime \(q \rightarrow 0\). On the other hand, Bougerol and Jeulin showed the appearance of exactly the same Pitman transform in the infinite curvature limit \(r \rightarrow \infty \) of a Brownian motion on the hyperbolic space \({{\mathbb {H}}}^3 = SL _2({{\mathbb {C}}})/ SU _2\). This paper aims at understanding this phenomenon by giving a unifying point of view. In order to do so, we exhibit a presentation \({{\mathcal {U}}}_q^\hbar \left( {\mathfrak sl}_2 \right) \) of the Jimbo–Drinfeld quantum group which isolates the role of curvature r and that of the Planck constant \(\hbar \). The simple relationship between parameters is \(q=e^{-r}\). The semi-classical limits \(\hbar \rightarrow 0\) are the Poisson–Lie groups dual to \( SL _2({{\mathbb {C}}})\) with varying curvatures \(r \in {{\mathbb {R}}}_+\). We also construct classical and quantum random walks, drawing a full picture which includes Biane’s quantum walks and the construction of Bougerol–Jeulin. Taking the curvature parameter r to infinity leads indeed to the crystal regime at the level of representation theory (\(\hbar >0\)) and to the Bougerol–Jeulin construction in the classical world (\(\hbar =0\)). All these results are neatly in accordance with the philosophy of Kirillov’s orbit method.

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Acknowledgements

During a conference in Reims, R.C. had the opportunity to meet Kirillov and present some of the ideas behind this paper, while still in their infancy. Kirillov said with a smile that these ideas were not completely absurd and that now, “you have to work”. Needless to say, one could not hope for better words of encouragement.

F.C. and R.C. acknowledge the support of the Grant PEPS JC 2017 “Quantum walks on quantum groups” and the Grant ANR-18-CE40-0006 MESA funded by the French National Research Agency (ANR). Finally, the authors express their gratitude to the anonymous referees for their useful comments.

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Correspondence to Reda Chhaibi.

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To our teachers, Philippe Biane and Philippe Bougerol

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A. Non-commutative topological considerations

A. Non-commutative topological considerations

Throughout the paper, we avoided the matters of completions of algebras. Since this is definitely not the main focus of the paper, we chose to postpone these topological considerations to this appendix.

Already, let us explain why, in most of our proofs, completing into a Von Neuman algebra would have been an overkill. In the entire paper, one can perform functional calculus at the level of the matrix algebras obtained after representation. For example, the first instance where we invoked elements belonging to a completion was Theorem 1.6 where the Casimir element \(C_{{\mathfrak {g}}}\) is defined as a square-root. The simplest way of defining the object is the following. One has to remember that, in the computation of non-commutative moments, \(C_{{\mathfrak {g}}}^2\) is represented as a Hermitian matrix before taking the trace. At that level, functional calculus is available for Hermitian matrices and the square-root is perfectly well-defined.

Nevertheless, since the machinery exists (see [32, 33] and references therein), it is possible to have a more intrinsic point of view and complete any of the algebras \({{\mathcal {A}}}\) considered in the paper into a \(C^*\)-algebra or a Von Neumann algebra. The general technique relies on the state \(\tau \). In order to form a Banach algebra, we define the norm d given by:

$$\begin{aligned} \forall X \in {{\,\mathrm{End}\,}}(V), \ d(X) = \lim _{p \rightarrow \infty } \tau \left( (X X^*)^p \right) ^{\frac{1}{p}}, \end{aligned}$$

then we complete \({{\mathcal {A}}}\) thanks to d. In order to form a Von Neumann algebra, there is the Gelfand–Naimark–Segal construction (GNS for short). This is done in several steps which we detail for the quantum group \({{\mathcal {U}}}_q^\hbar ({\mathfrak sl}_2) = {{\mathcal {A}}}\). The other algebras considered in the paper are tensor products or degenerations.

Step 1: Linear structure Consider a representation

$$\begin{aligned} \rho : {{\mathcal {A}}}= {{\mathcal {U}}}_q^\hbar ({\mathfrak sl}_2) \longrightarrow \bigoplus _{n=0}^\infty {{\,\mathrm{End}\,}}\left( V^q(n\hbar ) \right) , \end{aligned}$$

which can be taken to be the faithful Peter–Weyl isomorphism. For shorter notations, let \(\rho _{n\hbar } = \rho _{| {{\,\mathrm{End}\,}}(V^q(n\hbar ))}\) be the restriction to the nth component. Then define a trace via

$$\begin{aligned} \tau (a) = \sum _{n=0}^\infty d_n \frac{{{\,\mathrm{Tr}\,}}( \rho _{n\hbar }(a) )}{\dim V^q(n\hbar )}, \end{aligned}$$

where

$$\begin{aligned} \sum _{n=0}^\infty d_n = 1. \end{aligned}$$

From this normalized trace \(\tau \), one forms the scalar product \(\langle a, b \rangle := \tau ( a^{\dagger } b )\) and considers the completion into a Hilbert space H. Thus we embed the algebra into a linear space, but the multiplicative structure is missing.

Step 2: Multiplicative structure The algebra \({{\mathcal {A}}}\) acts on H via multiplication. Moreover, seeing \(A \in {{\mathcal {A}}}\) as an operator on H, we write for \(b \in H\):

$$\begin{aligned} A(b) := A \times b, \end{aligned}$$

and from the Cauchy–Schwarz inequality:

$$\begin{aligned} \Vert A (b) \Vert _{H} = \sqrt{ \tau \left[ \rho (A^\dagger A) b^\dagger b \right] } \le \sqrt{ \tau ( \rho (A^\dagger A) ) } \Vert b \Vert _{H}. \end{aligned}$$

Hence the A acts necessarily as a bounded operator. As such the algebra \({{\mathcal {A}}}\) is identified to a subalgebra of B(H), the algebra of bounded operators on H.

Step 3: Completions Upon completion for the operator norm, one obtains a \(C^*\) algebra. Upon completion with respect to the weak-* topology, one obtains a Von Neumann algebra. Going even further, one obtains the unbounded operators on H affiliated to the algebra \({{\mathcal {A}}}\).

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Chapon, F., Chhaibi, R. Quantum \( SL _2\), infinite curvature and Pitman’s 2M-X theorem. Probab. Theory Relat. Fields 179, 835–888 (2021). https://doi.org/10.1007/s00440-020-01002-8

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  • DOI: https://doi.org/10.1007/s00440-020-01002-8

Keywords

  • Orbit method
  • Jimbo–Drinfeld’s quantum groups
  • Non-commutative (\(=\) quantum) probability
  • Quantum random walks
  • Brownian motion on \({{\mathbb {H}}}^3 = SL _2({{\mathbb {C}}})/ SU _2\)
  • Infinite curvature

Mathematics Subject Classification

  • Primary 46L53
  • Secondary 58B32
  • 60B99