INDEX 245
algebra, 8
bivector field, 9
bracket, 8
degenerate, 8
nondegenerate, 8
universal, 149
manifold, 8
map, 8
structure, 8
compatible, 67
generically symplectic, 107
homogeneous, 73
submanifold, 8
regular, 107
Poisson-Lie group, 9
Polarization, 4
Polynomiality, 127
Pre-symplectic structure, 111
compatible, 111
Pseudoline arrangement, 30
Ptolemy relation, 116
Puncture, 115
Quasi-Cartan
companion, 50
matrix, 49
positive, 50
Quasihomogeneous polynomial, 48
R-matrix
classical, 11
trigonometric, 190
Rank
of a Poisson structure, 9
of a cluster algebra, 39
Reduced
decomposition, 5
word, 5
Representation by a network, 186
Resolution, 20
degenerate, 21
Root
almost positive, 59
height, 4
negative, 4
positive, 4
simple positive, 4
system, 3
Schubert cell, 24
real refined, 109
Seed (of geometric type), 37
Seeds
adjacent, 39
distance between, 39
equivalent, 40
Serre relations, 4
Sink, 142
Skew-symmetrizer, 37
Sklyanin bracket, 11
Source, 142
Space
of edge weights, 142, 176
of face and trail weights, 180
of face weights, 166, 181
of Grassmannian loops, 191
Spin, 125
Spun
edge, 125
flip, 125
geodesic arc, 125
intersection number, 127
triangulation, 125
Whitehead move, 125
Stable variable, 37
Standard Poisson-Lie structure, 11
Star, 118
Stasheff
pentagon, 16
polytope, 17
Subdiagram, 54
Subtraction-free rational expression, 144
Subtriangulation, 18
Symplectic
leaf, 9
generic, 107
manifold, 8
τ-cluster, 69
τ-coordinates, 69
Teichm¨uller
cluster, 118
space, 115
decorated, 115
Thurston shear coordinates, 119
Toda
flow, 199
hierarchy (finite nonperiodic), 199
lattice, 200
relativistic, 231
Toric
action
extension of, 103
global, 103
local, 102
regular locus, 104
actions compatible, 103
chart, 27
flow, 103
Trail, 181
weight, 181
Transvection, 29
symplectic, 72
Transversal flags, 2
Tree edge, 54
Triangulation
ideal, 115
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