This document provides an overview of algebraic aspects of quantum Lévy processes. It begins with background on algebraic terminology and stochastic processes like Lévy processes. It then defines quantum Lévy processes and describes some of their basic properties, including the correspondence between quantum Lévy processes and Schürmann triples. The document also discusses the Lévy-Khinchin decomposition property for quantum Lévy processes and provides examples and counterexamples. It concludes by mentioning some open questions and known results regarding classification of quantum Lévy processes over different algebraic structures.

1. Algebraic Aspects of Quantum Lévy
Process
Rei Mizuta
The Uniersity of Tokyo, Japan
August 29, 2018

2. Notation and Preliminaries
Quantum Lévy Process
Appendix
Contents
1 Notation and Preliminaries
Algebraic Terminology
Stochastic Background
2 Quantum Lévy Process
Definition and Basic Property
Lévy-Khinchin decomposition property in QLP
3 Appendix
Proofs
2 / 23

3. Notation and Preliminaries
Quantum Lévy Process
Appendix
Algebraic Terminology
Stochastic Background
Notation
For pre-Hilbert space D, L(D) denotes
fF : D ! D j F is linear adjointableg
Definition
A: a unital C-*-algebra is called *-bialgebra if there is
unital *-homs. ´ : A ! A ˙ A and › : A ! C such
that
(1 ˙ ´)´ = (´ ˙ 1)´ and
(1 ˙ ›)´ = (› ˙ 1)´ = id.
Example
G:Group ,› : C[G] ! C :coefficient of trivial
representation.
Then, (C[G]; ›) is *-bialgebra.
3 / 23

4. Notation and Preliminaries
Quantum Lévy Process
Appendix
Algebraic Terminology
Stochastic Background
Hochschild Cohomology
Let A be unital algebra over a comm. ring R and M
be A-bimodule.
Definition (Hochschild Cohomology)
H˜(A; M) denotes the cohomology of following
cochain complex.
1 Cn(A; M) := fffi: A˙n ! M; ffi R-linearg
2 for ffi 2 Cn`1(A; M)
dffi(a1 ˙ ´ ´ ´ ˙ an) := a1ffi(a2 ˙ ´ ´ ´ ˙ an)
+
n`1X
i=1
(`1)i
ffi
“
a1 ˙ ´ ´ ´ ˙ (aiai+1) ˙ ´ ´ ´ ˙ an
”
+ (`1)n
ffi(a1 ˙ ´ ´ ´ ˙ an`1)an
4 / 23

5. Notation and Preliminaries
Quantum Lévy Process
Appendix
Algebraic Terminology
Stochastic Background
Classifier Space
Let G be a group.
Theorem ([5])
There is a path-connected topological space X which
satisfies following conditions.
ı1(X) ‰= G.
ın(X) ‰= 0 for all n – 2.
Remark ([5])
Moreover, X := BG can be constructed as a
composition of following two functors.
B: : Grp ! Set´op
j:j : Set´op
! Top
5 / 23

6. Notation and Preliminaries
Quantum Lévy Process
Appendix
Algebraic Terminology
Stochastic Background
Some Facts
Theorem ([6])
H˜(G) ‰= H˜(Z[G]; Z) ‰= H˜
sing(BG).
where H˜(G) denotes group cohomology[6]
Example
G := Z2 then BG = T2
G := Fn then BG = _nS1
Theorem
H˜(C[G]; ›C›) ‰= H˜(Z[G]; Z) ˙Z C
6 / 23

7. Notation and Preliminaries
Quantum Lévy Process
Appendix
Algebraic Terminology
Stochastic Background
Lévy Process
Definition (Lévy Process[2])
R-valued Stochastic Process Xt is said Lévy Process if
X0 = 0 a.e. and it admits following three conditions.
1 (Independence of increments) Random variables
Xt1 ` Xs1 ; : : : ; Xtn ` Xsn are independent for all
n 2 N and all 0 » s1 » t1 » s2 » ´ ´ ´ » tn.
2 (Stationarity of increments) The distribution
Xt ` Xs depends only on the difference t ` s.
3 (Continuity) limh!0 P (jXt+h ` Xtj > ›) = 0 for
all t – 0; › > 0
Example
Brownian motion.
Poisson process. 7 / 23

8. Notation and Preliminaries
Quantum Lévy Process
Appendix
Algebraic Terminology
Stochastic Background
Lévy-Khinchin decomposition
Let Xt be R-valued Lévy Process.
Theorem (Lévy-Khinchin decomposition[2])
ffiXt (u) := E[eiuXt ] =
exp(t(aiu ` 1
2
ff2u2 +
R
Rnf0g(eiux ` 1 ` iux1jxj<1)‌))
where
a 2 R; ff – 0.
‌ is Borel measure on R n f0g s.t.
R
Rnf0g min(x2; 1)‌ < 1.
Conversely, for each ffi(u) as above form, there is a
Lévy Process whose characteristic function is ffi(u).
8 / 23

9. Notation and Preliminaries
Quantum Lévy Process
Appendix
Definition and Basic Property
Lévy-Khinchin decomposition property in QLP
Definition of QLP
Let A be *-bialgebra,
and (B; ffi) be the tuple of unital C-*-algebra and a
state.
Definition (Quantum Lévy Process[3])
The family of unital *-hom. (js;t)0»s»t : A ! B is
called Quantum Lévy Process if it satisfies following
four conditions.
1 (Increment property)
jrs ? jst = jrt for all 0 » r » s » t
jtt = 1 ´ › for all 0 » t:
where jrs ? jst(a) := (jrs ˙ jst)(´(a)) for a 2 A
9 / 23

10. Notation and Preliminaries
Quantum Lévy Process
Appendix
Definition and Basic Property
Lévy-Khinchin decomposition property in QLP
Definition of QLP (contd.)
Definition (Quantum Lévy Process[3] (contd.),)
2 (Independence of increments) The family
(jst)0»s»t is independent, i.e. js1t1 ; : : : ; jsntn are
independent for all n 2 N and all
0 » s1 » t1 » s2 » ´ ´ ´ » tn.
3 (Stationarity of increments) The distribution
’s;t = ffi(js;t) of js;t depends only on the
difference t ` s.
4 (Weak continuity) ’s;t converge to ’s;s in
distribution for t & s.
10 / 23

11. Notation and Preliminaries
Quantum Lévy Process
Appendix
Definition and Basic Property
Lévy-Khinchin decomposition property in QLP
Schürmann triple
Definition (Schürmann triple[1])
A tuple (ȷ; ”; ) is called Schürmann triple if there is
a C-pre-Hilbert space D and
ȷ : A ! L(D) is unital *-hom.
” : A ! D is linear map satisfying
”(ab) = ȷ(a)”(b) + ”(a)›(b).
: A ! C is linear map satisfying
(a˜b) = h”(a); ”(b)i + ›(a˜) (b) + (a˜)›(b)
Remark
Last condition is equivalent that
a ˙ b 7! h”(a˜); ”(b)i is 0 in H2(A; ›C›).
”(1) = 0; (1) = 0 and 8a 2 ker›; (a˜a) – 0 follow
by definition. 11 / 23

12. Notation and Preliminaries
Quantum Lévy Process
Appendix
Definition and Basic Property
Lévy-Khinchin decomposition property in QLP
Basic Properties
Definition
two QLPs are equivalent if their distributions are
same, where QLP’s dist. is defined by ’t := ffi(j0;t)
Theorem ([1])
Let (js;t)0»s»t be QLP, then There is a Schürmann
triple (ȷ; ”; ) s.t. exp(t ) = ’t.
we call generator of QLP.
Conversely, Let (ȷ; ”; ) be a Schürmann triple, then
there is a QLP s.t. exp(t ) = ’t.
Remark ([1])
In the construction of latter direction, (B; ffi) is always
obtained as symmetric Fock space.
12 / 23

13. Notation and Preliminaries
Quantum Lévy Process
Appendix
Definition and Basic Property
Lévy-Khinchin decomposition property in QLP
Classical Example
Let (Xt)0»t be real valued Lévy Process with finite
moments.
Example ([3])
A := C[X] with ´(X) := X ˙ 1 + 1 ˙ X; ›(X) := 0.
Then, js;t : A ! p–1Lp(˙; C); f 7! f(Xt ` Xs) is
QLP.
Proposition ([3])
Let be generator in this setting. TFAE.
K3 := fpqr j p; q; r 2 ker›g ȷ ker .
(ȷ; ”; ) : Sch. tri. ) ȷ = 1 ´ ›.
Xt = aBt + bt for some a; b 2 R.
In general, we call a generator Gaussian if the first or
second of above conditions follows. 13 / 23

14. Notation and Preliminaries
Quantum Lévy Process
Appendix
Definition and Basic Property
Lévy-Khinchin decomposition property in QLP
Lévy-Khinchin decomposition property
Definition (Lévy-Khinchin decomposition property[7])
Schürmann triple (ȷ; ”; ) satisfies
Lévy-Khinchin decomposition property if There is two
Schürmann triples (ȷG; ”G; G); (ȷP ; ”P ; P ) whose
represantation space DG and DP is orthogonal
subspace of —D such that
ȷ = ȷG ˘ ȷP ; ” = ”G + ”P and = G + P .
DG = f(ȷ(a) ` ›(a))v j a 2 A; v 2 Dg?
Remark
Existence of (ȷG; ”G); (ȷP ; ”P ) is trivial.
14 / 23

15. Notation and Preliminaries
Quantum Lévy Process
Appendix
Definition and Basic Property
Lévy-Khinchin decomposition property in QLP
Some Notations
Definition
Let (A; ›) be a tuple of C-*-algebra and unital *-hom
from A to C, it is said to satisfy
(H2Z) if H2(A; ›C›) = 0.
(AC) if all ȷ; ” admits gen. fct. . i.e. (ȷ; ”; )
becomes Schürmann triple.
(LK) if all Schürmann triples over (A; ›) admit
Lévy-Khinchin decomposition property.
Theorem ([7])
(H2Z) ) (AC) ) (LK)
15 / 23

16. Notation and Preliminaries
Quantum Lévy Process
Appendix
Definition and Basic Property
Lévy-Khinchin decomposition property in QLP
Counterexamples (in Group Algebras)
Theorem (counterexamples[7])
G := ı1(clo. ori. surf. of genus 2). (:(LK))
G := ha; b; r j aba`1b`1r2 = (ra)2 = (rb)2 = ei.
((LK) ^ :(AC))
A := Chx; y; j x˜ = x; x2y = `y; y˜y = 0i
›(x) = ›(y) = 0.((AC) ^ :(H2Z))
Remark ([7])
A group algebra (and trivial representation) which
satisfies (AC) ^ :(H2Z) is unknown.
16 / 23

17. Notation and Preliminaries
Quantum Lévy Process
Appendix
Definition and Basic Property
Lévy-Khinchin decomposition property in QLP
Known Results (in Quantum Groups)
Theorem (LK property of CMQGs[10][8][9])
Q. Symmetry Algebra As(n; d) admits (H2Z).[10]
Free Orthogonal Q. Group O
+
n does not admit
(LK).[8]
SUq(2) for q 2 (0; 1) admits (LK).[8]
SUq(d) does not admit (AC) for
q 2 (0; 1); d – 3.[8]
Remark ([10][8])
Whether SUq(d) has (LK) for q 2 (0; 1); d – 3 is
unknown.
17 / 23

18. Notation and Preliminaries
Quantum Lévy Process
Appendix
Definition and Basic Property
Lévy-Khinchin decomposition property in QLP
Future Work and Known Results
Trying to find group algebra counterexamples which
satisfy :(H2Z) ^ (AC).
Proposition
finite groups always satisfy (H2Z).
Trying to classify QLP over group algebra.
Remark (classification of QLP[4][9])
All QLPs of Quantum Permutation Group[9] S
+
n and
SUq(2)[4] are classified.
18 / 23

19. Notation and Preliminaries
Quantum Lévy Process
Appendix
Proofs
Proof of Prop 1/2
Proof
According to [7], we obtain following exact sequence
0 ! H1
! K0
1 ! (K1 ˙C[G] K1)0
! H2
! 0
where K1 := ker›.
Obviously,
dimK1 = jGj ` 1
dimK1 ˙C[G] K1 = jGj ` 1
dimH1 = 0.
Then (H2Z) follows by dimension theorem.
19 / 23

20. References I
[1] Schürmann, M. White noise on bialgebras. Lecture
notes in mathematics, Springer-Verlag, 1993.
[2] A, David. Lévy Processes―From Probability to
Finance and Quantum Groups, Notices of the
American Mathematical Society. Providence, RI:
American Mathematical Society. 51 (11): 1336 ‒
1347, 2004.
[3] U. Franz, The Theory of Quantum Lévy
Processes, arXiv:math/0407488, 2004.
20 / 23

21. References II
[4] M. Schürmann and M. Skeide, Infinitesimal
generators on the quantum group SUq(2). Infin.
Dimens. Anal. Quantum Probab. Relat. Top. 1
(1998), no. 4, 573 ‒ 598.
[5] J-L.Loday, Free loop space and homology
arXiv:1110.0405, 2011.
[6] K. S. Brown, Group cohomology, Springer Science
& Business Media, 1982
21 / 23

22. References III
[7] U. Franz, M. Gerhold and A. Thom, On the
Lévy-Khinchin decomposition of generating
functionals, arXiv:1510.03292, 2015.
[8] B. Das, U. Franz, A. Kula, A. Skalski,
Lévy-Khintchine decompositions for generating
functionals on algebras associated to universal
compact quantum groups, arXiv:1711.02755, 2018.
[9] U. Franz, A. Kula, A. Skalski, Lévy Processes on
Quantum Permutation Groups, arXiv:1510.08321,
2016.
22 / 23

23. References IV
[10] J. Bichon, U. Franz, M. Gerhold, Homological
properties of quantum permutation algebras,
arXiv:1704.00589, 2017.
23 / 23