Garside groups and Yang-Baxter equation

0.37 MB·English

by  Fabienne Chouraqui

Garside groups and Yang-Baxter equation
1 Fabienne Chouraqui
1
0
2
Abstract
n
a
We establish a one-to-one correspondence between a class of Gar-
J
side groups admitting a certain presentation and the structure groups
3
of non-degenerate, involutive and braided set-theoretical solutions of
2
thequantumYang-Baxterequation. Wealsocharacterizeindecompos-

] able solutions in terms of ∆-pure Garside groups.
R
G
1 Introduction
.
h
t
a
ThequantumYang-Baxterequationisanequationinthefieldofmathemat-
m
ical physics and it lies in the foundation of the theory of quantum groups.
[
Let R : V ⊗V → V ⊗V be a linear operator, where V is a vector space. The

3 quantum Yang-Baxter equation is the equality R12R13R23 = R23R13R12 of
v
linear transformations on V ⊗V ⊗V, where Rij means R acting on the ith
7
2 and jth components.
8
Aset-theoreticalsolutionofthisequationisasolutionforwhichV isavector
4
. spacespannedbyasetX andRisthelinearoperatorinducedbyamapping
2
X×X → X×X. ThestudyofthesewassuggestedbyDrinfeld[10]. Etingof,
1
9 Soloviev and Schedler study set-theoretical solutions (X,S) of the quantum
0 Yang-Baxter equation that are non-degenerate, involutive and braided [11].
:
v To each such solution, they associate a group called the structure group and
i
X they show that this group satisfies some properties. They also give a clas-
r sification of such solutions (X,S) up to isomorphism, when the cardinality
a
of X is up to eight. As an example, there are 23 solutions for X with four
elements, 595 solutions for X with six elements and 34528 solutions for X
witheightelements. Inthispaper, weestablishaone-to-onecorrespondence
between non-degenerate, involutive and braided set-theoretical solutions of
the quantum Yang-Baxter equation (up to isomorphism) and Garside pre-
sentations which satisfy some additional conditions up to t-isomorphism (a
notion that will be defined below). The main result is as follows.
1
Theorem 1. (i) Let X be a finite set, and (X,S) be a non-degenerate,
involutive and braided set-theoretical solution of the quantum Yang-Baxter
equation. Then the structure group of (X,S) is a Garside group.
(ii) Conversely, assume that Mon(cid:104)X | R(cid:105) is a Garside monoid such that:
- the cardinality of R is n(n−1)/2, where n is the cardinality of X and each
side of a relation in R has length 2 and
- if the word x x appears in R, then it appears only once.
i j
Then there exists a function S : X×X → X×X such that (X,S) is a non-
degenerate, involutive and braided set-theoretical solution and Gp(cid:104)X | R(cid:105) is
its structure group.
The main idea of the proof is to express the right and left complement
on the generators in terms of the functions that define (X,S). Moreover,
we prove that the structure group of a set-theoretical solution satisfies some
specific constraints. Picantin defines the notion of ∆-pure Garside group
in [16] and he shows that the center of a ∆-pure Garside group is a cyclic
subgroup that is generated by some exponent of its Garside element.
Theorem 2. Let X be a finite set, and (X,S) be a non-degenerate, involu-
tive and braided set-theoretical solution of the quantum Yang-Baxter equa-
tion. Let G be the structure group of (X,S) and M the monoid with the
same presentation. Then
(i) The right least common multiple of the elements in X is a Garside ele-
ment in M.
(ii) The (co)homological dimension of G is equal to the cardinality of X.
(iii) The group G is ∆-pure Garside if and only if (X,S) is indecomposable.
Point (i) above means that G is Garside in the restricted sense of [9].
Let us observe that, independently, Gateva-Ivanova and Van den Bergh de-
fine in [14] monoids and groups of left and right I-type and they show that
they yield solutions to the quantum Yang-Baxter equation. They show also
that a monoid of left I-type is cancellative and has a group of fractions that
is torsion-free and Abelian-by-finite. Jespers and Okninski extend their re-
sultsin[15],andestablishacorrespondencebetweengroupsofI-typeandthe
structure group of a non-degenerate, involutive and braided set-theoretical
solution. Using our result, this makes a correspondence between groups of
I-type and the class of Garside groups studied in this paper. They also re-
mark that the defining presentation of a monoid of I-type satisfies the right
cube condition, as defined by Dehornoy in [7, Prop.4.4]. So, the necessity
of being Garside can be derived from the combination of the results from
[15, 14]. Our methods in this paper are different as we use the tools of
2
reversing and complement developed in the theory of Garside monoids and
groupsandourtechniquesofproofareuniformthroughoutthepaper. Itcan
be observed that our results imply some earlier results by Gateva-Ivanova.
Indeed, she shows in [13] that the monoid corresponding to a special case of
non-degenerate, involutive and braided set-theoretical solution (square-free)
has a structure of distributive lattice with respect to left and right divisi-
bility and that the left least common multiple of the generators is equal to
their right least common multiple and she calls this element the principal
monomial.
Thepaperisorganizedasfollows. Insection2,wegivesomepreliminarieson
Garside monoids. In section 3, we give the definition of the structure group
of a non-degenerate, involutive and braided set-theoretical solution and we
showthatitisGarside,usingthecriteriadevelopedbyDehornoyin[4]. This
implies that this group is torsion-free from [6] and biautomatic from [4]. In
section 4, we show that the right least common multiple of the generators is
a Garside element and that the (co)homological dimension of the structure
group of a non-degenerate, involutive and braided set-theoretical solution
is equal to the cardinality of X. In section 5, we give the definition of a
∆-pure Garside group and we show that the structure group of (X,S) is
∆-pure Garside if and only if (X,S) is indecomposable. In section 6, we es-
tablish a converse to the results of section 3, namely that a Garside monoid
satisfying some additional conditions defines a non-degenerate, involutive
and braided set-theoretical solution of the quantum Yang-Baxter equation.
Finally, in section 7, we address the case of non-involutive solutions. There,
we consider the special case of permutation solutions that are not involutive
and we show that their structure group is Garside. We could not extend
this result to general solutions, although we conjecture this should be true.
At the end of the section, we give the form of a Garside element in the case
of permutation solutions.
Acknowledgment. This work is a part of the author’s PhD research, done
at the Technion under the supervision of Professor Arye Juhasz. I am very
grateful to Professor Arye Juhasz, for his patience, his encouragement and
his many helpful remarks. I am also grateful to Professor Patrick Dehornoy
for his comments on this paper.
3
2 Garside monoids and groups
All the definitions and results in this section are from [4] and [5]. In this
paper, if the element x of M is in the equivalence class of the word w, we
say that w represents x.
2.1 Garside monoids
Let M be a monoid and let x,y,z be elements in M. The element x is a
left divisor of z if there is an element t such that z = xt in M and z is
a right least common multiple (right lcm) of x and y if x and y are left
divisors of z and additionally if there is an element w such that x and y are
left divisors of w, then z is left divisor of w. We denote it by z = x∨y.
The complement at right of y on x is defined to be an element c ∈ M such
that x∨y = xc, whenever x∨y exists. We denote it by c = x\y and by
definition, x∨y = x(x\y). Dehornoy shows that if M is left cancellative
and1istheuniqueinvertibleelementinM, thentherightlcmandtheright
complement of two elements are unique, whenever they exist [4]. We refer
the reader to [5, 4] for the definitions of the left lcm and the left and right
gcd of two elements. An element x in M is an atom if x (cid:54)= 1 and x = yz
implies y = 1 or z = 1. The norm (cid:107) x (cid:107) of x is defined to be the supremum
ofthelengthsofthedecompositionsofxasaproductofatoms. Themonoid
M is atomic if M is generated by its atoms and for every x in M the norm
of x is finite. It holds that if all the relations in M are length preserving,
then M is atomic, since each element x of M has a finite norm as all the
words which represent x have the same length.
A monoid M is Gaussian if M is atomic, left and right cancellative,
and if any two elements in M have a left and right gcd and lcm. If ∆ is
an element in M, then ∆ is a Garside element if the left divisors of ∆ are
the same as the right divisors, there is a finite number of them and they
generate M. A monoid M is Garside if M is Gaussian and it contains a
Garside element. A group G is a Gaussian group (respectively a Garside
group) if there exists a Gaussian monoid M (respectively a Garside monoid)
such that G is the fraction group of M. A Gaussian monoid satisfies both
left and right Ore’s conditions, so it embeds in its group of fractions (see
[3]). As an example, braid groups and Artin groups of finite type [12], torus
knot groups [18] are Garside groups.
Definition 2.1. [4, Defn.1.6] Let M be a monoid. M satisfies:
- (C ) if 1 is the unique invertible element in M.
0
- (C ) if M is left cancellative.
1
4
- (C˜ ) if M is right cancellative.
1
- (C ) if any two elements in M with a right common multiple admit a right
2
lcm.
- (C ) if M has a finite generating set P closed under \, i.e if x,y ∈ P then
3
x\y ∈ P.
Theorem 2.2. [4, Prop. 2.1] A monoid M is a Garside monoid if and only
if M satisfies the conditions (C ), (C ), (C˜ ), (C ), and (C ).
0 1 1 2 3
2.2 Recognizing Garside monoids
Let X be an alphabet and denote by (cid:15) the empty word in X . Let f be a
∗
partialfunctionofX×X intoX ,f isacomplement onX iff(x,x) = (cid:15)holds
∗
for every x in X, and f(y,x) exists whenever f(x,y) does. The congruence
onX generatedbythepairs(xf(x,y),yf(y,x))with(x,y)inthedomainof
∗
f is denoted by “≡+”. The monoid associated with f is X / ≡+ or in other
∗
words the monoid Mon(cid:104)X | xf(x,y) = yf(y,x)(cid:105) (with (x,y) in the domain
of f). The complement mapping considered so far is defined on letters only.
Its extension on words is called word reversing (see [5]).
Example 2.3. Let M be the monoid generated by X = {x ,x ,x ,x ,x }
1 2 3 4 5
and defined by the following 10 relations.
x2 = x2 x x = x x x x = x x x x = x x x x = x x
1 2 2 5 5 2 1 2 3 4 1 5 5 1 1 3 4 2
x2 = x2 x x = x x x x = x x x x = x x x x = x x
3 4 2 4 3 1 3 5 5 3 2 1 4 3 4 5 5 4
Then the complement f is defined totally on X×X and the monoid associ-
ated to f, X / ≡+, is M. As an example, f(x ,x ) = x and f(x ,x ) = x
∗ 1 2 1 2 1 2
areobtainedfromtherelationx2 = x2,sinceitholdsthatf(x ,x ) = x \x .
1 2 1 2 1 2
Let f be a complement on X. For u,v,w ∈ X , f is coherent at (u,v,w)
∗
if either ((u\v)\(u\w))\((v\u)\(v\w)) ≡+ (cid:15) holds, or neither of the
words((u\v)\(u\w)),((v\u)\(v\w))exists. Thecomplementf is coherent
if it is coherent at every triple (u,v,w) with u,v,w ∈ X . Dehornoy shows
∗
that if the monoid is atomic then it is enough to show the coherence of f on
its set of atoms. Moreover, he shows that if M is a monoid associated with
a coherent complement, then M satisfies C and C (see [5, p.55]).
1 2
Proposition 2.4. [4, Prop.6.1] Let M be a monoid associated with a com-
plement f and assume that M is atomic. Then f is coherent if and only if
f is coherent on X.
Example 2.5. In example 2.3, we check if ((x \ x ) \ (x \ x )) \ ((x \
1 2 1 3 2
x )\(x \x )) = (cid:15) holds in M. We have x \x = x and x \x = x , so
1 2 3 1 2 1 1 3 2
(x \x )\(x \x ) = x \x = x . Additionally,x \x = x andx \x = x ,
1 2 1 3 1 2 1 2 1 2 2 3 4
5
so (x \x )\(x \x ) = x \x = x . At last, ((x \x )\(x \x ))\((x \
2 1 2 3 2 4 1 1 2 1 3 2
x )\(x \x )) = x \x = (cid:15).
1 2 3 1 1
3 Structure groups are Garside
3.1 The structure group of a set-theoretical solution
All the definitions and results in this subsection are from [11].
Aset-theoreticalsolutionofthequantumYang-Baxterequationisapair
(X,S), where X is a non-empty set and S : X2 → X2 is a bijection. Let S
1
and S denote the components of S, that is S(x,y) = (S (x,y),S (x,y)). A
2 1 2
pair (X,S) is nondegenerate if the maps X → X defined by x (cid:55)→ S (x,y)
2
and x (cid:55)→ S (z,x) are bijections for any fixed y,z ∈ X. A pair (X,S) is
1
braided if S satisfies the braid relation S12S23S12 = S23S12S23, where the
map Sii+1 : Xn → Xn is defined by Sii+1 = id ×S ×id , i < n.
Xi 1 Xn i 1
− −−
A pair (X,S) is involutive if S2 = id , that is S2(x,y) = (x,y) for all
X2
x,y ∈ X.
Let α : X ×X → X ×X be the permutation map, that is α(x,y) = (y,x),
and let R = α ◦ S. The map R is called the R−matrix corresponding to
S. Etingof, Soloviev and Schedler show in [11], that (X,S) is a braided set
if and only if R satisfies the quantum Yang-Baxter equation R12R13R23 =
R23R13R12, where Rij means acting on the ith and jth components and
that (X,S) is a symmetric set if and only if in addition R satisfies the uni-
tary condition R21R = 1. They define the structure group G of (X,S) to
be the group generated by the elements of X and with defining relations
xy = tz when S(x,y) = (t,z). They show that if (X,S) is non-degenerate
and braided then the assignment x → f is a right action of G on X.
x
We use the notation of [11], that is if X is a finite set, then S is defined
by S(x,y) = (g (y),f (x)), x,y in X. Here, if X = {x ,...,x } is a finite
x y 1 n
set and y = x for some 1 ≤ i ≤ n, then we write f ,g instead of f ,g
i i i y y
and S(i,j) = (g (j),f (i)). The following claim from [11] translates the
i j
properties of a solution (X,S) in terms of the functions f ,g and it will be
i i
very useful in this paper.
Claim 3.1. (i) S is non-degenerate ⇔ f , g are bijective, 1 ≤ i ≤ n.
i i
(ii) S is involutive ⇔ g f (i) = i and f g (j) = j, 1 ≤ i,j ≤ n.
gi(j) j fj(i) i
(iii) S is braided ⇔ g g = g g , f f = f f ,
i j gi(j) fj(i) j i fj(i) gi(j)
and f g (j) = g f (j), 1 ≤ i,j,k ≤ n.
g (k) i f (i) k
fj(i) gj(k)
6
Example 3.2. LetX = {x ,x ,x ,x ,x } and S(i,j) = (g (j),f (i)). As-
1 2 3 4 5 i j
sume
f = g = (1,2,3,4)(5) f = g = (1,4,3,2)(5)
1 1 2 2
f = g = (1,2,3,4)(5) f = g = (1,4,3,2)(5)
3 3 4 4
Assume also that the functions f and g are the identity on X. Then a
5 5
case by case analysis shows that (X,S) is a non-degenerate, involutive and
braided solution. Its structure group is generated by X and defined by the
10 relations described in example 2.3.
3.2 Structure groups are Garside
In this subsection, we prove the following result.
Theorem 3.3. The structure group G of a non-degenerate, braided and
involutive set-theoretical solution of the quantum Yang-Baxter equation is a
Garside group.
In order to prove that the group G is a Garside group, we show that the
monoid M with the same presentation isaGarsidemonoid. Forthat, weuse
theGarsiditycriteriongiveninTheorem2.2,thatisweshowthatM satisfies
the conditions (C ) , (C ) , (C ), (C ), and (C˜ ). We refer the reader to [14,
0 1 2 3 1
Lemma 4.1] for the proof that M is right cancellative (M satisfies (C˜ )).
1
We first show that M satisfies the conditions (C ). In order to do that, we
0
describe the defining relations in M and as they are length-preserving, this
implies that M is atomic.
Claim 3.4. Assume (X,S) is non-degenerate. Let x and x be different
i j
elements in X. Then there is exactly one defining relation x a = x b, where
i j
a,b are in X. If in addition, (X,S) is involutive then a and b are different.
For a proof of this result, see [14, Thm. 1.1]. Using the same arguments,
if (X,S) is non-degenerate and involutive there are no relations of the form
ax = ax , where i (cid:54)= j. We have the following direct result from claim 3.4.
i j
Proposition 3.5. Assume (X,S) is non-degenerate and involutive. Then
the complement f is totally defined on X×X, its range is X and the monoid
associated to f is M. Moreover, M is atomic.
Now, we show that M satisfies the conditions (C ), (C ) and (C ). From
1 2 3
Proposition 3.5, we have that there is a one-to-one correspondence between
the complement f and the monoid M with the same presentation as the
structure group, so we say that M is coherent (by abuse of notation). In
order to show that the monoid M satisfies the conditions (C ) and (C ), we
1 2
7
show that M is coherent (see [5, p.55]). Since M is atomic, it is enough
to check its coherence on X (from Proposition 2.4). We show that any
triple of generators (a,b,c) satisfies the following equation: (a\b)\(a\c) =
(b\a)\(b\c), where the equality is in the free monoid X , since the range
∗
of f is X. In the following lemma, we establish a correspondence between
the right complement of generators and the functions g that define (X,S).
i
Lemma 3.6. Assume (X,S) is non-degenerate. Let x ,x be different ele-
i j
ments in X. Then x \x = g 1(j).
i j i−
Proof. If S(i,a) = (j,b), then x \x = a. But by definition of S, we have
i j
that S(i,a) = (g (a),f (i)), so g (a) = j which gives a = g 1(j).
i a i i−
Lemma 3.7. Assume (X,S) is non-degenerate and involutive. Let x ,x
i k
be elements in X. Then g 1(i) = f (i)
k− gi−1(k)
Proof. SinceS isinvolutive,wehavefromclaim3.1thatforeveryx ,x ∈ X,
i j
g f (i) = i. We replace in this formula j by g 1(k) for some 1 ≤ k ≤ n,
gi(j) j i−
then we obtain i = g f (i) = g f (i). So, we have g 1(i) =
gi(gi−1(k)) gi−1(k) k gi−1(k) k−
f (i).
gi−1(k)
Proposition 3.8. Assume (X,S) is non-degenerate, involutive and braided.
Every triple (x ,x ,x ) of generators satisfies the following equation: (x \
i k m i
x ) \ (x \ x ) = (x \ x ) \ (x \ x ). Furthermore, M is coherent and
k i m k i k m
satisfies the conditions (C ) and (C ).
1 2
Proof. If x = x or x = x or x = x , then the equality holds trivially.
i k i m k m
So, assume that (x ,x ,x ) is a triple of different generators. This implies
i k m
that g 1(k) (cid:54)= g 1(m) and g 1(i) (cid:54)= g 1(m), since the functions g are
i− i− k− k− i
bijective. Using the formulas for all different 1 ≤ i,k,m ≤ n from lemma
3.6, we have: (x \x )\(x \x ) = g 1 (x \x ) = g 1 g 1(m) and
i k i m x−i\xk i m g−i−1(k) i−
(x \ x ) \ (x \ x ) = g 1 (x \ x ) = g 1 g 1(m). We prove that
k i k m x−k\xi k m g−k−1(i) k−
g 1 g 1(m) = g 1 g 1(m) by showing that g g = g g for all
g−i−1(k) i− g−k−1(i) k− i gi−1(k) k gk−1(i)
1 ≤ i,k ≤ n. Since S is braided, we have from claim 3.1, that g g =
i gi−1(k)
g g = g g . But, from lemma 3.7, f (i) = g 1(i),
gi(gi−1(k)) fgi−1(k)(i) k fgi−1(k)(i) gi−1(k) k−
so g g = g g . The monoid M is then coherent at X but since M
i gi−1(k) k gk−1(i)
is atomic, M is coherent. So, M satisfies the conditions (C ) and (C ).
1 2
Now, using the fact that M satisfies (C ) and (C ), we show that it
1 2
satisfies also (C ).
3
8
Proposition 3.9. Assume (X,S) is non-degenerate, involutive and braided.
Then there is a finite generating set that is closed under \, that is M satisfies
the condition (C ).
3
Proof. From claim 3.4, for any pair of generators x ,x there are unique
i j
a,b ∈ X such that x a = x b, that is any pair of generators x ,x has a right
i j i j
common multiple. Since from Proposition 3.8, M satisfies the condition
(C ), we have that x and x have a right lcm and the word x a (or x b)
2 i j i j
represents the element x ∨x , since this is a common multiple of x and x
i j i j
of least length. So, it holds that x \x = a and x \x = b, where a,b ∈ X.
i j j i
So, X ∪{(cid:15)} is closed under \.
4 Additional properties of the structure group
4.1 The right lcm of the generators is a Garside element
The braid groups and the Artin groups of finite type are Garside groups
which satisfy the condition that the right lcm of their set of atoms is a
Garside element. Dehornoy and Paris considered this additional condition
as a part of the definition of Garside groups in [9] and in [4] it was removed
from the definition. Indeed, Dehornoy gives the example of a monoid that
is Garside and yet the right lcm of its atoms is not a Garside element [4].
He shows that the right lcm of the simple elements of a Garside monoid is
a Garside element, where an element is simple if it belongs to the closure
of a set of atoms under right complement and right lcm [4]. We prove the
following result:
Theorem 4.1. Let G be the structure group of a non-degenerate, involutive
and braided solution (X,S) and let M be the monoid with the same presen-
tation. Then the right lcm of the atoms (the elements of X) is a Garside
element in M.
In order to prove that, we show that the set of simple elements χ, that
is the closure of X under right complement and right lcm, is equal to the
closure of X under right lcm (denoted by X∨), where the empty word (cid:15) is
added. So, this implies that ∆, the right lcm of the simple elements, is the
rightlcmoftheelementsinX. Weusethewordreversingmethoddeveloped
by Dehornoy and the diagrams for word reversing. We illustrate in example
1 below the definition of the diagram and we refer the reader to [4] and [5]
for more details. Here, reversing the word u 1v using the diagram amounts
−
to computing a right lcm for the elements represented by u and v (see [5,
p.65]).
9
Example 4.2. Let us consider the monoid M defined in example 2.3. We
illustrate the construction of the reversing diagram.
(a) The reversing diagram of the word x 1x is constructed in figure 4.1 in
−3 1
the following way. First, we begin with the left diagram and then using the
defining relation x x = x x in M, we complete it in the right diagram.
1 2 3 4
We have x \x = x and x \x = x .
1 3 2 3 1 4
x1 x1
x3 x3 x2
x
4
Figure 4.1: Reversing diagram of x 1x
−3 1
(b) The reversing diagram of the word x 2x2 is described in figure 4.2:
−4 1
we begin with the left diagram and then we complete it using the defining
relations in the right diagram. So, we have x2x2 = x2x2 in M and since
1 2 4 3
x1 x1 x1 x1
x4 x4 x3 x2
x x
2 4
x4 x4 x1 x2
x x
3 3
Figure 4.2: Reversing diagram of x 2x2
−4 1
x2 = x2 and x2 = x2, it holds that x4 = x4 = x4 = x4 in M. So, a word
1 2 3 4 1 2 3 4
representing the right lcm of x2 and x2 is the word x4 or the word x4. We
1 4 1 4
obtain from the diagram that the word x2 represents the element x2 \ x2
2 1 4
and the word x2 represents the element x2\x2.
3 4 1
In order to prove that χ = X∨∪{(cid:15)}, we show that every complement of
simple elements is the right lcm of some generators. The following technical
lemma is the basis of induction for the proof of Theorem 4.1.
Lemma 4.3. (i) It holds that M \X ⊆ X ∪{(cid:15)}.
(ii) It holds that M \(∨jj==1kxij) ⊆ X∨∪{(cid:15)}, where xij ∈ X, 1 ≤ j ≤ k.
Proof. It holds that S(X × X) ⊆ X × X, so X \ X ⊆ X ∪ {(cid:15)} and this
implies inductively (i) M \X ⊆ X ∪{(cid:15)} (see the reversing diagram).
Let u ∈ M, then using the following rule of calculation on complements
10