22:
1971:
But the trace algebra is a strictly small subalgebra (there are fewer invariants). This provides an involutive action on the torus that needs to be accounted for to yield the Culler–Shalen character variety. The involution on this torus yields a 2-sphere. The point is that up to
1476:
1712:
has shown that in this case the ring of invariants is in fact generated by only traces. Since trace functions are invariant by all inner automorphisms, the Culler–Shalen construction essentially assumes that we are acting by
348:
627:
of prime characteristic. In this generality, character varieties are only algebraic sets and are not actual varieties. To avoid technical issues, one often considers the associated reduced space by dividing by the
872:
1643:
1807:
2026:, and geometric structures on topological spaces, given generally by the observation that, at least locally, equivalent objects in these categories are parameterized by conjugacy classes of
1759:
1706:
1157:
924:
636:). However, this does not necessarily yield an irreducible space either. Moreover, if we replace the complex group by a real group we may not even get an algebraic set. In particular, a
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2004:
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for the bundles or a fixed topological space for the geometric structures, the holonomy homomorphism is a group homomorphism from
1764:
2006:-conjugation all points are distinct, but the trace identifies elements with differing anti-diagonal elements (the involution).
86:
58:
43:
1481:
This character variety appears in the theory of the sixth
Painleve equation, and has a natural Poisson structure such that
2126:(or quantization) of the character variety. It is closely related to topological quantum field theory in dimension 2+1.
65:
1716:
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1114:
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442:
2135:
412:
72:
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32:
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54:
2154:
Horowitz, R.D. (1972). "Characters of Free Groups
Represented in the Two-Dimensional Special Linear Group".
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1865:
1071:
2336:
564:
1975:
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1471:{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}+x^{2}+y^{2}+z^{2}-(ab+cd)x-(ad+bc)y-(ac+bd)z+abcd+xyz-4=0.}
1028:
507:
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closures intersect. This is the weakest equivalence relation on the set of conjugation orbits,
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2015:
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498:
486:
234:
206:
163:
131:
127:
2218:"A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation"
2181:
Magnus, W. (1980). "Rings of Fricke
Characters and Automorphism Groups of Free Groups".
1221:
is free of rank three. Then the character variety is isomorphic to the hypersurface in
2089:
2033:
1905:
1162:
929:
722:
699:
679:
529:
502:
356:
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183:
140:
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are
Casimir functions, so the symplectic leaves are affine cubic surfaces of the form
2438:
2202:
2123:
2019:
989:
123:
2425:
2023:
1657:
1025:; its coordinate ring is isomorphic to the complex polynomial ring in 3 variables,
555:
1652:
This construction of the character variety is not necessarily the same as that of
2030:
homomorphisms of flat connections. In other words, with respect to a base space
2119:
1653:
1108:
gives a closed real three-dimensional ball (semi-algebraic, but not algebraic).
993:
119:
21:
343:{\displaystyle {\mathfrak {R}}(\pi ,G)=\operatorname {Hom} (\pi ,G)/\!\sim \,.}
1859:
633:
135:
2167:
2014:
There is an interplay between these moduli spaces and the moduli spaces of
2027:
2234:
2217:
1111:
Another example, also studied by Vogt and Fricke–Klein is the case with
2194:
2400:
867:{\displaystyle {\mathcal {M}}_{B}(X,G)={\mathfrak {R}}(\pi _{1}(X),G)}
2114:
The coordinate ring of the character variety has been related to
664:
is free we always get an honest variety; it is singular however.
1638:{\displaystyle xyz+x^{2}+y^{2}+z^{2}+c_{1}x+c_{2}y+c_{3}z=c_{4}}
415:, and two homomorphisms are defined to be equivalent (denoted
15:
1802:{\displaystyle {\mathfrak {R}}=\operatorname {Hom} (\pi ,H)}
797:
623:
Here more generally one can consider algebraically closed
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1660:(generated by evaluations of traces), although when
46:. Unsourced material may be challenged and removed.
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2339:; Sikora, Adam S. (2000). "On skein algebras and
2311:-characters and the Kauffman bracket skein module
332:
946:is the Riemann sphere punctured three times, so
743:, is the character variety of the surface group
1754:{\displaystyle G=\mathrm {SL} (n,\mathbb {C} )}
1701:{\displaystyle G=\mathrm {SL} (n,\mathbb {C} )}
1179:is the Riemann sphere punctured four times, so
1152:{\displaystyle G=\mathrm {SL} (2,\mathbb {C} )}
919:{\displaystyle G=\mathrm {SL} (2,\mathbb {C} )}
2156:Communications on Pure and Applied Mathematics
478:{\displaystyle \operatorname {Hom} (\pi ,G)/G}
8:
2377:{\displaystyle {\rm {SL}}_{2}(\mathbb {C} )}
2304:{\displaystyle {\rm {SL}}_{2}(\mathbb {C} )}
1902:, the conjugation action is trivial and the
672:An interesting class of examples arise from
404:{\displaystyle \operatorname {Hom} (\pi ,G)}
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988:is free of rank two, then Henri G. Vogt,
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106:Learn how and when to remove this message
2146:
996:proved that the character variety is
7:
44:adding citations to reliable sources
1770:
828:
283:
2352:
2349:
2279:
2276:
2222:Proc. Japan Acad. Ser. A Math. Sci
1983:
1980:
1961:{\displaystyle S^{1}\times S^{1}.}
1895:{\displaystyle G=\mathrm {SO} (2)}
1879:
1876:
1730:
1727:
1677:
1674:
1128:
1125:
1101:{\displaystyle G=\mathrm {SU} (2)}
1085:
1082:
895:
892:
14:
2315:Commentarii Mathematici Helvetici
613:{\displaystyle \mathbb {C} ^{G}.}
2122:. The skein module is roughly a
1999:{\displaystyle \mathrm {SO} (2)}
1922:-character variety is the torus
1243:{\displaystyle \mathbb {C} ^{7}}
1214:{\displaystyle \pi =\pi _{1}(X)}
1018:{\displaystyle \mathbb {C} ^{3}}
981:{\displaystyle \pi =\pi _{1}(X)}
778:{\displaystyle \pi =\pi _{1}(X)}
20:
644:. On the other hand, whenever
31:needs additional citations for
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696:is a Riemann surface then the
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1:
2410:10.1016/S0040-9383(98)00062-7
1061:{\displaystyle \mathbb {C} }
519:{\displaystyle \mathbb {C} }
2450:Group actions (mathematics)
2110:Connection to skein modules
2079:{\displaystyle \pi _{1}(M)}
2466:
2136:Geometric invariant theory
546:-character variety is the
2320:(1997), no. 4, 521–542.
638:maximal compact subgroup
548:spectrum of prime ideals
159:finitely generated group
2384:-character varieties".
2086:to the structure group
1828:{\displaystyle G\neq H}
1512:{\displaystyle a,b,c,d}
1250:given by the equation
497:Formally, and when the
435:) if and only if their
2378:
2305:
2168:10.1002/cpa.3160250602
2100:
2080:
2044:
2010:Connection to geometry
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1708:they do agree, since
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541:
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430:
428:{\displaystyle \sim }
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368:
345:
266:
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218:
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152:
2343:
2270:
2216:Iwasaki, K. (2002).
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2034:
1976:
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1906:
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1851:{\displaystyle \pi }
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718:character variety of
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657:{\displaystyle \pi }
648:
565:
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508:
501:is defined over the
443:
419:
377:
357:
278:
255:
244:{\displaystyle \pi }
235:
216:{\displaystyle \pi }
207:
202:character variety of
184:
173:{\displaystyle \pi }
164:
141:
40:improve this article
2337:Przytycki, JĂłzef H.
2235:10.3792/pjaa.78.131
1838:For instance, when
229:group homomorphisms
225:equivalence classes
55:"Character variety"
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2301:
2195:10.1007/BF01214715
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1996:
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1098:
1068:. Restricting to
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1015:
978:
936:
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864:
775:
741:Betti moduli space
729:
706:
686:
654:
642:semi-algebraic set
640:generally gives a
632:of 0 (eliminating
610:
554:(i.e., the affine
552:ring of invariants
536:
516:
475:
425:
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363:
340:
261:
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190:
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147:
2099:{\displaystyle G}
2043:{\displaystyle M}
2016:principal bundles
1915:{\displaystyle G}
1172:{\displaystyle X}
939:{\displaystyle X}
732:{\displaystyle X}
709:{\displaystyle G}
689:{\displaystyle X}
539:{\displaystyle G}
366:{\displaystyle G}
264:{\displaystyle G}
193:{\displaystyle G}
150:{\displaystyle G}
116:
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923:
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878:For example, if
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713:
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707:
695:
693:
692:
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674:Riemann surfaces
663:
661:
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616:
611:
606:
605:
572:
545:
543:
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537:
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523:
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485:, that yields a
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476:
471:
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410:
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372:
370:
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353:More precisely,
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24:
16:
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2210:
2180:
2179:
2175:
2153:
2152:
2148:
2144:
2132:
2112:
2106:of the bundle.
2088:
2087:
2057:
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2051:
2032:
2031:
2012:
1974:
1973:
1945:
1932:
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1903:
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1710:Claudio Procesi
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503:complex numbers
499:reductive group
495:
487:Hausdorff space
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12:
11:
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2394:(1): 115–148.
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2300:
2296:
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2281:
2278:
2264:Doug Bullock,
2257:
2208:
2173:
2162:(6): 635–649.
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2020:vector bundles
2011:
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1911:
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1888:
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1862:of rank 2 and
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2445:Moduli theory
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2116:skein modules
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2024:Higgs bundles
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1010:
995:
991:
990:Robert Fricke
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124:moduli theory
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110:
107:
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88:
85:
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78:
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71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
2391:
2385:
2331:
2317:
2265:
2260:
2228:(7): 131–5.
2225:
2221:
2211:
2186:
2182:
2176:
2159:
2155:
2149:
2113:
2013:
1970:
1837:
1658:Peter Shalen
1651:
1480:
1110:
877:
740:
717:
671:
622:
556:GIT quotient
496:
352:
201:
117:
102:
96:January 2017
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
2124:deformation
2120:knot theory
1654:Marc Culler
994:Felix Klein
493:Formulation
413:conjugation
126:, given an
120:mathematics
2439:Categories
2252:1058.34125
2189:: 91–103.
2142:References
1860:free group
634:nilpotents
66:newspapers
2266:Rings of
2203:120977131
2059:π
1943:×
1846:π
1820:≠
1788:π
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1403:−
1376:−
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1187:π
961:π
954:π
838:π
758:π
751:π
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586:π
580:
456:π
450:
423:∼
390:π
384:
334:∼
316:π
310:
292:π
239:π
211:π
168:π
136:Lie group
132:reductive
128:algebraic
2426:14740329
2387:Topology
2130:See also
2028:holonomy
1809:even if
1648:Variants
668:Examples
373:acts on
2418:1710996
2325:1600138
2244:1930217
2183:Math. Z
630:radical
550:of the
118:In the
80:scholar
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2422:S2CID
2396:arXiv
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231:from
87:JSTOR
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1656:and
1159:and
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2406:doi
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