Benvegnù, Alberto and Spera, Mauro:
Low-Dimensional Pure Braid Group Representations Via Nilpotent Flat Connections
Bollettino dell'Unione Matematica Italiana Serie 9 6 (2013), fasc. n.3, p. 643-672, (English)
pdf (367 Kb), djvu (239 Kb). | MR 3202844
Sunto
In this note we discuss low-dimensional matrix representations of pure braid group (on three and four strands) obtained via holonomy of suitable nilpotent flat connections. Flatness is directly enforced by means of the Arnol'd relations. These explicit representations are used to investigate Brunnian and “nested” Brunnian phenomena.
Referenze Bibliografiche
[2]
K. AOMOTO,
Fonctions hyperlogarithmiques et groupes de monodromie unipotents,
J. Fac. Sci. Tokio,
25 (
1978), 149-156. |
MR 509582 |
Zbl 0416.32020[3]
P.K. ARAVIND,
Borromean entanglement of the GHZ state, in
Potentiality, Entanglement and Passion-at-a-Distance eds
R.S. Cohen,
M. Horne,
J. Stachel (
Kluwer Academic Publishers, Boston,
1997), pp. 53-59. |
fulltext (doi) |
MR 1739812 |
Zbl 1006.81504[4]
V. I. ARNOL'D,
The cohomology ring of colored braids,
Mat. Zametki,
5 No 2 (
1969), 227-231. (Russian) English transl. in
Trans. Moscow Math. Soc.,
21 (
1970), 30-52. |
MR 242196[5]
E. ARTIN,
Theorie der Zöpfe,
Abh. Math. Sem. Hamburg Univ.,
4 (
1925), 42-72;
Theory of braids,
Ann. Math., 48 (
1947), 101-126. |
fulltext (doi) |
MR 3069440[7]
A. BENVEGNÙ and
M. SPERA,
On Uncertainty, Braiding and Entanglement in Geometric Quantum Mechanics,
Rev. Math. Phys.,
18 (
2006), 1075-1102. |
fulltext (doi) |
MR 2287641[8]
M. BERGER,
Third order link invariants,
J. Phys. A: Math. Gen.,
23 (
1990), 2787- 2793. |
MR 1062985 |
Zbl 0711.57008[9]
M. BERGER,
Third order braid invariants,
J. Phys. A: Math. Gen.,
24 (
1991), 4027- 4036. |
MR 1126646 |
Zbl 0747.57002[13]
J.S. BIRMAN and
T.E. BRENDLE,
Braids: A Survey, Ch.2 in
Handbook of Knot Theory eds.
W. Menasco and
M. Thistlethwaite (
Elsevier B.V.
2005), pp. 19-103. |
fulltext (doi) |
MR 2179260 |
Zbl 1094.57006[15]
K.-T. CHEN,
Collected Papers of K.-T. Chen (eds
P. Tondeur and
R. Hain),
Contemporary Mathematicians (
Birkäuser, Boston, MA
2001). |
MR 1847673 |
Zbl 0977.01042[16]
N.W. EVANS and
M.A. BERGER,
A hierarchy of linking integrals, in
Topological Aspects of the Dynamics of Fluids and Plasmas, eds.
H.K. Moffatt et al. (
Kluwer, Dordrecht, The Netherlands,
1992), pp. 237-248. |
MR 1232234 |
Zbl 0799.57004[17]
R. HAIN,
The Geometry of the Mixed Hodge Structure on the Fundamental Group,
Proc. Symp. Pure Math.,
46 (
1987), 247-282. |
MR 927984[19]
L. KAUFFMAN and
S. LOMONACO,
Quantum entanglement and topological entanglement,
New J. Phys.,
4 (
2002) 73.1-73.18. |
fulltext (doi) |
MR 1963025[24]
M. KONTSEVICH,
Vassiliev's knot invariants,
Adv. Sov. Math. 16, Part 2 (
1993), 137-150, (AMS Providence, RI). |
MR 1237836 |
Zbl 0839.57006[25]
R. E. LAWRENCE,
Homological representations of the Hecke algebra,
Commun. Math. Phys.,
135 (
1990), 141-191. |
MR 1086755 |
Zbl 0716.20022[26]
T.Q.T. LE and
J. MURAKAMI,
Kontsevich's integral for the Homfly polynomial and relations between values of multiple zeta functions,
Topology Appl.,
62 (
1995), 193- 206. |
fulltext (doi) |
MR 1320252 |
Zbl 0839.57007[28] P. PAPI and C. PROCESI, Invarianti di nodi, Quaderno U.M.I. 45 (Pitagora, Bologna, 1998) (in Italian).
[32]
G. WECHSUNG,
Functional Equations of Hyperlogarithms, in
Structural properties of Polylogarithms (ed.
L. Lewin), Ch. 8, (
AMS Providence, RI,
1991), pp. 171-184. |
fulltext (doi) |
MR 1148379