Vector bundles on degenerations of elliptic curves and Yang-Baxter equations [Elektronische Ressource] / Thilo Grunwald-Henrich. Mathematisch-Naturwissenschaftliche Fakultät

Thilo Grunwald-Henrich

Sujets

• Mathematics

Informations

• Publié par: rheinische_friedrich-wilhelms-universitat_bonn
• Publié le: 01 janvier 2011
• Nombre de lectures: 12
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Vector bundles on degenerations of elliptic curves and
Yang-Baxter equations
Dissertation
zur
Erlangung des Doktorgrades (Dr. rer. nat.)
der
Mathematisch-Naturwissenschaftlichen Fakultät
der
Rheinischen Friedrich-Wilhelms-Universität Bonn
vorgelegt von
Thilo Grunwald-Henrich
aus
Bonn
Bonn 2011Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultät der
Rheinischen Friedrich-Wilhelms-Universität Bonn
1. Gutachter: PD Dr. Igor Burban
2. Gutachter: Prof. Dr. Catharina Stroppel
Tag der Promotion: 25.10.2011
Erscheinungsjahr: 2011Summary
In this thesis, we study connections between vector bundles on degenerations of elliptic
curvesandtheclassical,quantumandassociativeYang-Baxterequation. Letg =sl (C)n
and let U denote the universal enveloping algebra of g. The classical Yang-Baxter
equation (CYBE) is given as follows:! " ! " ! "
12 23 12 13 13 23r (y ,y),r (y ,y) + r (y ,y),r (y ,y) + r (y ,y),r (y ,y) =0,1 2 2 3 1 2 1 3 1 3 2 3
2 ijwhere r :C ! g"g is a meromorphic function and r (y,y):g"g!U"U"U isi j
the embedding given by (i,j). This equation plays an important role in mathematical
physics, representation theory and integrable systems.
In 1982, Belavin and Drinfeld gave a classification of solutions of the CYBE. In par-
ticular, they proved that any solution of the CYBE is either elliptic, trigonometric or
rational. Moreover, they described all elliptic and trigonometric solutions. Their work
has been extended by Stolin, who gave a certain classification of rational solutions.
2 3 2 3 2Result A. Let E = V(wv # 4u #g uw #g w )$P be a Weierstraß cubic curve,2 3
0<d<n a pairof coprimeintegers andA = Ad(P), whereP is a simple vectorbundle
of rank n and degree d on E. Consider the map
!1# #$ %! res ev !y y= =1 0 2# #g#!A #! H A(y ) #!A #! g,1y y1 2
where res istheresidue map, ev istheevaluation map andthefirstandthelastmapsy y1 2
are induced by a certain trivialization of A. Then the tensor r (y ,y) % g"g,1 2(E,n,d)
obtained from the map above using the Killing form, is a solution of the CYBE.
This result extends an earlier construction given in works of Polishchuk and Burban-
Kreußler. The core of our method is the computation of certain triple Massey products$ %
bin the derived category D Coh(E) .
Result B. Let E be a cuspidal cubic curve. Then the solution r from above is(E,n,d)
rational. We explicitly describe the Stolin triple (L,B,k) (whereL is a Lie subalgebra
of g, B is a 2-cocycle ofL and k% ) such that r =r .(E,n,d) (L,B,k)
ResultC.Wehavefoundnewellipticsolutionsoftheassociative Yang-Baxter equation
of the form & &$ %
r(v,y)= & !(v,y) e "e ,kl i,j+k j,i+l
0"k"n#1 1"i"n#l
0"l"n#1 1"j"n#k
where !(v,y) is the Kronecker function and& are certain di!erential operators. Thiskl
leads to new identities for the higher derivatives of the Kronecker function.
Result D. We elaborate a relation between solutions of the associative, classical and
quantum Yang-Baxter equations, generalizing results of Polishchuk.4
Contents
1. Introduction 8
1.1. Organization of the material 14
1.2. Acknowledgement 15
Part 1. Yang-Baxter equations: Interplay 16
2. The classical Yang-Baxter equation 17
3. The associative Yang-Baxter equation 19
4. Relationship with the quantum Yang-Baxter equation 21
5. Poles of solutions of the AYBE 22
6. Uniqueness of lifts from CYBE to AYBE 24
7. Quantization of solutions of CYBE coming from solutions of AYBE 27
Part 2. Triple Massey products and the Yang-Baxter equations 34
8. Triple Massey products and the AYBE 36
8.1. Algebraic Triple Massey products 36
8.2. Geometric Massey Products 39
9. Triple Massey products and the CYBE 40
9.1. Preliminaries from linear algebra 41
9.2. Triple Massey products revisited 42
9.3. On the sheaf of traceless endomorphism of a simple vector bundle 44
9.4. Residues and traces 46
9.5. Algebraic versus geometric Massey products 49
9.6. Genus one fibrations and the CYBE 55
Part 3. Vector bundles on degenerations of elliptic curves 60
10. Vector bundles on a one-dimensional complex torus 61
11. The category of triples and Matrix problems 63
11.1. The category of Triples 63
11.2. Reduction to Matrix problems 66
11.3. Matrix problem for the cuspidal cubic curve 66
11.4. Primary reduction 68
12. Di!erential Biquivers 70
12.1. Di!erential biquivers 70
12.2. Small reduction 72
12.3. Di!erential biquiver for the cuspidal cubic curve 73
13. Vector bundles on the cuspidal cubic curve 74
13.1. Classification 74
13.2. Algorithm for construction of simple vector bundles 755
13.3. Hom and Ext vanishing 76
Part 4. From vector bundles on Weierstraß cubic curves to solutions
of the Yang-Baxter equations 77
14. From vector bundles on the elliptic curve to solutions of the AYBE 78
14.1. Construction of the elliptic solutions r of the AYBE 78B
14.2. Identification of the geometric method and Algorithm 14.1 79
14.3. Remarks on the solutions r 82B
15. From vector bundles on the cuspidal cubic curve to solutions of the AYBE 86
15.1. Construction of the rational solutions r of the AYBE 86(n,d)
15.2. Identification of the geometric method and Algorithm 15.1 88
15.3. Obtaining rational solutions of the CYBE from solutions of the AYBE 90
16. From vector bundles on the cuspidal cubic curve to solutions of the CYBE 91
16.1. Construction of the rational solutions c of the CYBE 91(n,d)
16.2. The solution c for a particular choice of basis 92(n,d)
16.3. Identification of the geometric method and Algorithm 16.1 93
Part 5. Computations of elliptic solutions of the AYBE 97
17. Solution obtained from a diagonal matrix 98
18.n attached to a Jordan block 99
19. Combinatorial proofs 108
19.1. Proof of Proposition 18.3 108
19.2. Proof of Lemma 18.20 118
Part 6. Theory of rational solutions 121
20. Classification of rational solutions 122
21. The rational solution s 126(n,n#d)
21.1. Algorithm: Construction of s 126(n,n#d)
21.2. The solution s for a particular choice of basis 127(n,n#d)
21.3. Verification of the construction of s 130(n,n#d)
22. Connections between the solutions s and c 137(n,n#d) (n,d)
22.1. The map " 137J
22.2. Gauge equivalence of s and c 139(n,n#d) (n,d)
22.3. Structure results 141
23. Explicit computation of s 147(n,1)
Part 7. PC Implementations 154
23.1. The program for r and c 155(n,d) (n,d)
23.2. The program for s 159(n,n#d)6
References 1637
To my wife and family8
1. Introduction
In this thesis, I incorporate and explain in greater detail the results presented in [12],
[13] and [26]. Although these papers are quite di!erent from each other with respect
to the methods and tools we use - analytical, combinatorial, algebro-geometric and
Lie-theoretic - there still is a common denominator for the studies we pursue. Namely,
we aim for a better understanding of the Yang-Baxter equations and their solutions by
application of the theory of coherent sheaves on degenerations of elliptic curves.
The Yang-Baxter equations – or to be more precise, the classical Yang-Baxter equation
(CYBE), the quantum Yang-Baxter equation (QYBE) and the associative Yang-Baxter
equation(AYBE)–areimportantobjectsappearinginmathematicalphysics,especially
in integrable systems and statistical mechanics. Moreover they are studied in the con-
text of representation theory. There are di!erent versions for each of these equations,
di!ering from each other with respect to the number of spectral variables involved. The
version of the CYBE that we shall be mostly interested in is of the following form. Let
2g = sl (C) and r:(C ,0)! g"g be a meromorphic function. Then r is a solution ofn
the CYBE if it satisfies the equality! " ! " ! "
12 23 12 13 13 23r (y ,y),r (y ,y) + r (y ,y),r (y ,y) + r (y ,y),r (y ,y) =0.1 2 2 3 1 2 1 3 1 3 2 3$ %$3ij $2Here r = # 'r denotes r followed by the obvious inclusions g !U g withU(g)ij
denoting the universal enveloping algebra of g, e.g. # (a"b)=a"1"b. We shall13
focus our studies on solutions r which satisfy two additional assumptions. Firstly, we
will assume that r is unitary:
12 21r (y ,y)=#r (y ,y).1 2 2 1
Secondly, r will always be non-degenerate, that is its image under the isomorphism$ %
g"g#! End(g),a "b(! c(! tr(ac)·b
is an invertible operator for some (and hence, for a generic) value of the spectral pa-
rameters (y ,y). On the set of solutions of the CYBE there exists a natural action of1 2
the group of holomorphic function germs $:(C,0)#! Aut(g) given by the rule$ %
r(y ,y)(!r˜(y ,y)= $(y )"$(y ) r(y ,y).1 2 1 2 1 2 1 2
It is easy to see that r˜(y ,y) is again a solution of the CYBE. Moreover, r˜(y ,y)1 2 1 2
is unitary respectively non-degenerate provided r(y ,y) is unitary respectively non-1 2
degenerate. The solutionsr(y ,y) andr˜(y ,y) related by the equality above are called1 2 1 2
gauge equivalent.
It was shown by Belavin and Drinfeld [3] that any non-degenerate solution of the
CYBE is either elliptic (two-periodic), trigonometric (one-periodic) or rational. More-
over, they classified all elliptic and trigonometric solutions completely [3, Proposition9
5.1 and Theorem 6.1]. Especial