Table Of ContentTECHNISCHE UNIVERSITEIT EINDHOVEN
Department of Mathematics and Computer Science
Generalized Brauer Algebras
B.J. Frenk
Supervisor: Prof. dr. Arjeh M. Cohen
Eindhoven, August 2006
Contents
1 Introduction 1
2 Knot Theory 2
2.1 Links, knots and diagrams . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Braids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 The BMW Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Coxeter groups 14
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Admissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Re(cid:13)ection subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 The action on sets of commuting re(cid:13)ections . . . . . . . . . . . . . . 37
4 Generalized Brauer algebras 53
4.1 The classical Brauer algebra . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 De(cid:12)nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 A representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Parametrizationof the idempotents . . . . . . . . . . . . . . . . . . . 65
4.5 Annihilated sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.6 A linear spanning set . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A Notation 77
Chapter 1
Introduction
This thesis is the result of nearly a year of research under the supervision of dr.
ArjehCohen. Itbeganwiththequestionofgivinganeatdescriptionoftheclassical
Braueralgebra in terms of the Coxetergroup of type A , which is nothing but the
n
symmetricgrouponn+1symbols,anditsaccompanyingrootsystem. Thepurpose
was to extend such a description to other simply laced, spherical Coxeter graphs.
One of the reasonswas to shed more light on the newly introduced BMW algebras
of simply lacedtype. See forexample [4] or[22] forthe relationbetweenthe BMW
algebra of type A and the classical Brauer algebra.
TheBMWalgebrasofsimplylacedtypeareageneralizationofanalgebraintro-
ducedbyBirman,WenzlandMurakami. Theirpurposewastogiveadescriptionof
the Kau(cid:11)manlink invariantin terms of a traceon an algebra. As such the original
BMW algebras are strongly connected to knot theory in R3. The connection is
outlined in the (cid:12)rst chapter of this thesis. The BMW algebra of type D seems to
n
have some connection to knot theory on a cylinder.
There is also a connection to the representation theory of Artin groups. In
particular,theLawrence-KrammerlikerepresentationsoftheArtingroupsofsimply
lacedtypefactorthroughttheBMWalgebraofthesametype. Thiswas(cid:12)rstshown
byZinno[29]fortheoriginalBMWalgebraandlaterbyGijsbers[13],fortheother
types.
In the (cid:12)rst chapter the construction of the original BMW algebra in terms of
the Kau(cid:11)man link invariant is outlined. It is fairly self contained, and begins with
basic facts of knot theory. Proofs are usually omitted.
The second chapter deals with Coxeter groups. Some results are proved that
areof importance for the description of the generalizedBraueralgebra in the third
chapter.
The third chapter is where the generalized Brauer algebras are de(cid:12)ned. A rep-
resentation of those algebras is constructed, and by means of the representation
we arrive at an upper bound for the dimension of the generalized Brauer algebras.
TheseupperboundscoincidewiththeupperboundsforthedimensionoftheBMW
algebra, as in [13]. It is conjectured that the bounds are tight.
1
knot
link
isotopic
Chapter 2
Knot Theory
In this section we will not be concerned with exact technical details, but instead
we try to give an overview of the application of the classical Brauer algebra in
mathematical knot theory. The path from knots and links to the Braueralgebrais
schematically depicted in Figure 2.1.
Links Link Diagrams Braids
(cid:0)! !
# #
Kau(cid:11)man Polynomial BMW Algebra
!
#
Brauer Algebra
Figure 2.1: The path from knots and links to the Brauer algebra.
2.1 Links, knots and diagrams
Knot theory is a branch of topology in which mathematics is applied to analyse
pieces of twine whose ends are fused. It is of use in such diverse (cid:12)elds as quantum
physics and molecular biology [28].
The primary objects of study in knot theory are knots and links. A knot is to
be thought as a single piece of twine positioned in an ambient space of which the
ends are fused together. A link is a (cid:12)nite collection of one or more knots lying
in the same space. Continuous deformations of the ambient space can transform
the link, butsuchdeformationsareconsiderednon-essential. Twolinksthatcanbe
transformedintooneanotherbyasequenceofsuchdeformationsarecalledisotopic.
Isotopy de(cid:12)nes an equivalence relation on the collection of all links.
Isotopy of links and knots depends on the ambient space. Di(cid:11)erent choices of
ambientspaceleadtodi(cid:11)erentknottheories. Acommonchoiceisthree-dimensional
Euclideanspace,denotedE3. Euclideanspacesof higherorlowerdimensiondonot
2
(a) Unknot (b) TrefoilKnot
(c) FigureEightKnot (d) GrannyKnot
Figure 2.2: Knots in three-dimensional Euclidean space.
lead to interesting knot theories. Let m > 3. Every link in Em is equivalent to a
collection of non-intertwined unknots. The proof of this fact consists of two parts.
First, the link is repositioned by a deformation of Em such that it is completely
containedinathreedimensionalsubspace. Onepiece ofstrandcanlocallybelifted
into the orthogonal complement of this three dimensional subspace, and as such
be pulled through another piece of strand of the link. In this way the link can be
untangled to form a collection of unknots.
If the ambient space is two-dimensional Euclidean space then every knot is
equivalent to a circle. This follows from a theorem of Schoen(cid:13)ies on extensions of
homeomorphisms from a Jordan curve to the circle. See for example [17].
In mathematical language, a knot is an embedding of the circle in the ambient
space. Similarly,alinkisanembeddingofadisjointunionof circlesinthe ambient
space. These embedding need to be either tame, smooth or piecewise linear. A
tame embedding of a collection of closed curves is an embedding of those curves
3
that extends to small tubes around them. In other words, links are characterised oriented link
by the condition that they can be drawn as in Figure 2.2. Finally, an oriented link unknot
is a tame, smooth or piecewise linear embedding of a collection of oriented circles. trivial knot
knotinvariant
A formal de(cid:12)nition of isotopy would lead us too far a(cid:12)eld, and is not very
invariant
important for our purposes. Instead we refer the reader to [16] or [24]. It su(cid:14)ces
knot
to say that the isotopy class of a link L is denoted [L].
link diagram
The simplest link is called the unknot, which is also known as the trivial knot.
transversal
It is the closed curve S E3 parametrized by over
!
under
x(t) (cost;sint;0);
7!
and isdepicted in Figure2.2(a). Given somecomplicatedknot it is a naturalques-
tion to ask whether it is isotopic to the unknot. In general this question cannot be
easily solved by simply looking at a pictorial representation of the knot. Di(cid:11)erent
methods have been proposed to solve this question and its generalization of deter-
mining the isotopy classes of knots. One such method is by de(cid:12)ning a function on
the set of knots that is constant on the isotopy classes. Such a function is called a
knot invariant or simply an invariant.
Aknotinvariantshouldbeeasytocalculateanddistinguishasmanynon-isotopic
knots as possible. Determining a complete invariant has been a di(cid:14)cult question
which even now is not completely solved. One of the earliest examples of a knot
invariant was de(cid:12)ned by Alexander in [2]. It is now known as the Alexander-
polynomial or Alexander-Conwaypolynomial. Other examples are the Jones poly-
nomial [19], the HOMFLYPT polynomial [11], [25] and the Kau(cid:11)man polynomial
[21]. Forashorthistoricaloverviewofthesubjectseeforexample[4]. Insubsequent
sectionsweshallbeconcernedonlywiththelatterthree,sincetheygivegoodexam-
ples of how to connect algebraic structures to knot theory. For reference purposes
these polynomials and their notations are collected in Table 2.1. The last column
lists the algebra connected with the invariant. It will be explained in Paragraph
2.2.
Notation Invariant Algebra
V (t) Jones polynomial Temperley-Lieb algebra
L
P (l;m) HOMFLYPT polynomial Hecke algebra
L
K (l;m) Kau(cid:11)man polynomial BMW algebra
L
Table 2.1: Knot invariants
A common to way to represent a link or knot is by a planar diagram called a
knotorlinkdiagram. Itistheprojectionofaknotorlinkonasuitablychosenplane.
The projection has only a (cid:12)nite number of double points and these double points
are transversal. A double point is assigned the label over or under depending on
the position of the preimages of the crossing lines. Under- and over-crossing are
depicted by erasing a small piece of the line near the double point. Examples of
link diagrams are given in Figure 2.3.
One link can have many di(cid:11)erent link diagrams. For example both (a) and (d)
4
Reidemeister
moves
equivalent
(a) Unknot (b) Intertwinedunknots (c) Trefoilknot
(d) Millettunknot (e) Whitehead link (f) Anotherknot
Figure 2.3: Link diagrams
of Figure 2.3 are both representatives of the isotopy class of the unknot. Even for
the same embedding of a link a di(cid:11)erent plane of projection results in a di(cid:11)erent
link diagram. However, changing a link diagram according to the rules given in
Figure 2.4 results in a link diagram of an equivalent knot. The rules are called the
Reidemeister moves and are denoted R1, R2 and R3. In a sense it is surprising
that these moves are su(cid:14)cient; diagramsof isotopic knots can be changed into one
another by applying a sequence of them. This is proved for example in [7]. It is
called Reidemeister’s Theorem and is givenbelow. Let two link diagramsbe called
equivalent when there is a sequence of Reidemeister moves transforming one into
the other. Then characterizingisotopyclassesof linksisthe sameascharacterizing
equivalence classes of link diagrams.
Theorem 2.1.1 (Reidemeister 1926) Two links are isotopic if and only if their
link diagrams can be transformed into one another by a sequence of Reidemeister
moves R1, R2 and R3.
Theusualstartingpointfromwhichtocalculatelinkinvariantsarethe oriented
link diagrams. A link diagram is transformed into a weighted sum of other link
diagramsbysocalledskeinrelations,untilonlydiagramsofnon-intertwinedunknots
remain. Specifying the value of the link invariant on the non-intertwined unknot
diagramsthen completelydeterminesit. The skeinrelations depend onthe speci(cid:12)c
invariant and are de(cid:12)ned using the partial diagrams depicted in Figure 2.5. Let
L denote a link (diagram). The interpretations of L , L , L and L are as
+ (cid:0) 0 1
follows. Choose some crossing in L, and change it locally into one of the four
partial diagrams of Figure 2.5. The resulting diagram is called L , L , L or
(cid:0) + 0
5
(cid:31) (cid:31) algebraic crossing
(cid:31) (cid:31)
(cid:31) (cid:31) number
(cid:31) (cid:31)
(cid:31) (cid:31)
(cid:31)
(cid:31)
(cid:31)
(cid:31) (cid:31)
(cid:31) ! (cid:31)
(cid:31) (cid:31)
(cid:31) (cid:31)
(cid:31) (cid:31)
R1
(cid:31) (cid:31)
(cid:31) (cid:31)
(cid:31) (cid:31)
(cid:31) (cid:31)
(cid:31) (cid:31)
(cid:31) (cid:31)
(cid:31) (cid:31)
(cid:31) (cid:31)
(cid:31) (cid:31)
! (cid:31) (cid:31)
(cid:31) (cid:31)
(cid:31) (cid:31)
(cid:31) (cid:31)
R2
!
R3
Figure 2.4: The Reidemeister moves
L , according to the partial diagram chosen. The skein relation for the Jones
1
and HOMFLYPT polynomials are given below. The symbol O denotes the unknot
diagram. See [24, p. 13] for (2.1) and [4] for (2.2). Examples of calculations of the
Jones polynomial using the skein relations below can be found for example in [24,
page 9-12].
1 1
V (t) tV (t) = pt+ V (t);
t L+ (cid:0) L(cid:0) pt L0
V (t) = 1(cid:0): (cid:1) (2.1)
O
l(cid:0)1P (l;m)+lP (l;m) = mP (l;m);
L+ L(cid:0) L0
P (l;m) = 1: (2.2)
O
Theskein relationforthe Kau(cid:11)man polynomialrequiresa short digression. We
follow [4]. Let L be an oriented link represented as an oriented diagram D. The
algebraic crossing number "ofD isde(cid:12)nedasfollows. Consideragainthe diagrams
and+asinFigure2.5. Crossingthatcorrespondto areassignedthevalue 1,
(cid:0) (cid:0) (cid:0)
6
L-polynomial
braid
isotopy of braids
(cid:0) (cid:1)
+ 0
(cid:0) 1
Figure 2.5: Partial link diagrams
whilecrossingthatcorrespondto+areassignedthevalue1. Thealgebraiccrossing
numberofDisthesumofthevaluesassignedtothecrossingsofD. Kau(cid:11)manin[21]
de(cid:12)nes a precursor of the Kau(cid:11)man polynomial K (l;m) called the L-polynomial,
L
here denoted as K^ (l;m). Its relation to the Kau(cid:11)man polynomial is
D
K^ (l;m)=l"K (l;m): (2.3)
D L
The skein relation of the L-polynomial is the following.
K^ (l;m)+K^ (l;m) = m(K^ (l;m)+K^ (l;m)):
D+ D(cid:0) D0 D1
K^ (l;m) = 1 (2.4)
O
Moreover, if two diagrams can be transformed into one another by applying a
sequence of the second and third Reidemeister moves, then their L-polynomials
are equal. Applying the (cid:12)rst Reidemeister move to a diagram multiplies the L-
polynomial by a factor l or l(cid:0)1. These conditions completely determine K^ (l;m)
D
on all oriented diagrams. By (2.3) the K (l;m) is completely determined on all
L
links. Withoutproofthefollowingtheoremisstated. Itsproofcanbefoundin[21].
Theorem 2.1.2 The L-polynomial is an invariant on the collection of all link di-
agrams under sequences of R2 and R3 Reidemeister moves. The Kau(cid:11)man polyno-
mial is an invariant of links.
2.2 Braids
A braid is a mathematical formalization of the everyday concept of a braid. Let
n N. Considerthe unionbofncurvesthatareembeddedinR2 [0;1]andwhose
2 (cid:2)
boundary @b consists of the points
@b= (1;0);(2;0);:::;(n;0) 0;1 :
f g(cid:2)f g
Moreover,suppose that none of the curveshavea critical point with respect to the
third (vertical) coordinate. Then b is called a braid on n strands. Two braids on
the same number of strands are called isotopic if they are related by an isotopy of
R2 [0;1]that preservesthe boundaryand the verticalcoordinate. Isotopyde(cid:12)nes
(cid:2)
7
an equivalence relation on the set of braids, and as was the case with knots, two braid group
braids are considered the same if they are isotopic.
Let B denote the collectionof isotopyclassesof all braidsonn strands. There
n
is a composition de(cid:12)ned on B . Let b and c braids on n strands, and let [b] and
n
[c] denote their isotopy classes. Then the composition of [b] and [c], denoted by
juxtaposition, is the isotopy class of the braid obtained by identifying the lower
boundary points of b with the upper boundary points of c. Without proof it is
stated that this indeed de(cid:12)nes a composition that makes B into a group. It is
n
called the braid group on n strands.
There is an algebraic characterizationof the braid group B in terms of gener-
n
ators and relations. Then B is the group de(cid:12)ned by the relations
n
(cid:27) (cid:27) = (cid:27) (cid:27) ; if i j >1;
i j j i
j (cid:0) j
(cid:27) (cid:27) (cid:27) = (cid:27) (cid:27) (cid:27) ; if i j =1:
i j i j i j
j (cid:0) j
onthe generators(cid:27) ;:::;(cid:27) . Thesegeneratorsandtheirinversesarerepresented
1 n(cid:0)1
diagrammatically in Figure 2.6. In particular, it follows from this presentation
of B that its subgroup generated by (cid:27) ;:::;(cid:27) is isomorphic to B . This
n+1 1 n(cid:0)1 n
subgroup consists of all isotopy classes of braids that have a representative for
which (1;0;n) is connected to (0;0;n) by a straight line that is not intertwined
with any of the other curves. Conversely, the map | that maps B into B by
n n n+1
adding a straight line connecting (1;0;n) and (0;0;n) that is not intertwined with
any of the other strands is a monomorphism. The subscript n is omitted from |
n
when it does not lead to confusion.
(cid:2) (cid:3)
::: ::: ::: :::
1 i i+1 n 1 i i+1 n
(cid:0)1
(a) (cid:27)i (b) (cid:27)i
Figure 2.6: Fundamental generators (cid:27) and their inverses for the braid group B
i n
Remark. The braid group on n strands is connected to the Coxeter diagram
A , throughthe algebraicrepresentationjust given. After all, the generatorsare
n(cid:0)1
indexed by the vertices of A , while the relations depend on the edges of A .
n(cid:0)1 n(cid:0)1
Generators (cid:27) and (cid:27) commute if and only if there is no edge in A between
i j n(cid:0)1
vertex i and j. Similarly, these generators do not commute when there is an edge
between i and j. This observation allows for the construction by generators and
relations of braid-like groups related to the other simply laced Coxeter diagrams.
Such groups are commonly known as Artin groups, in honour of Emil Artin, who
introduced the groups B . The Artin group of type A is isomorphic to B .
n n(cid:0)1 n
8
