An isomorphism between the Hopf algebras A and B of Jacobi diagrams in the theory of knot invariants

36 Pages·2012·0.545 MB·English

by  Janis Lazovskis

An isomorphism between the Hopf algebras A and B of Jacobi
diagrams in the theory of knot invariants
Ja¯nis Lazovskis
December 14, 2012
Abstract
We construct a graded Hopf algebra B from the symmetric algebra of a metrized Lie algebra, and
examinethestructureoflow-dimensionalspacesofthegrading. Withthisweconstructanddiagrammise
an isomorphism to the Hopf algbera A, this algebra arising from the universal enveloping algebra of the
same metrized Lie algebra.
Contents
0 Motivating remarks 2
0.1 Diagrammisation of U(g) through the algebra A . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.2 Diagrammisation of S(g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 The vector space B 4
1.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Connected elements of B of low degree . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 An alternative grading and bi-grading . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 The vector spaces B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
n,m
2 The Hopf algebras induced by B and A 14
2.1 D and B as bi-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 The Hopf algebra B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 The Hopf algebra A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 An isomorphism between A and B 20
3.1 The Poincar´e-Birkhoff-Witt theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Diagrammisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
A Construction of B 26
2
B Construction of B 31
3
C Graph generation 34
C.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
C.2 Complexity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1
0 Motivating remarks
Penrose’s diagrammatic analogue of tensor calculus was applied to generate a diagrammatic construction
of the universal enveloping algebra U(g) of a metrized Lie algebra g. The Poincare-Birkhoff-Witt theorem
gives an isomorphism between U(g) and S(g), the symmetric algebra of the same metrized Lie algebra. It is
therefore of potential interest to construct a diagrammization of S(g).
0.1 Diagrammisation of U(g) through the algebra A
Previously (see [4]) U(g) was diagrammised, for g with a non-degenerate symmetric bilinear form and an
orthonormal basis. The algebra was generated by disjoint union, linear combination, and contraction, from
the basis elements
 
 i i i j a1 a2 
, , , ∀ p∈N (0.1)
 j j k ap 
The third element in this list is termed the diagram of the Lie bracket (or simply the Lie diagram). The
last element in this list describes an anchored tensor with p arrows directed outward. Through isomorphism
in a common vector space, it may be consrtucted by a combination of the second and third diagrams, for
example,
a a
2 1
a a
2 1
a
4
∼
a a
3 4
a
3
Hence the anchors may be disregarded1. The basis of the described algebra was then reduced to four
elements, namely the first three of (0.1) and i . As the next step was to attach a skeleton to the
outward edges of the diagrams, the first two diagrams of (0.1) became irrelevant, as they only reverse
direction of edges.
The algebra A was then constructed (see [3]) using the following basis modulo a skeletal version of the
simplification of the Lie diagram,
 
i j i j i j i j
 
, = −
 k i  k i j i j
In A, directions and labels are ignored, and as there is only one type of vertex not on the skeleton, the
previously used different shapes are no longer used. For example,
1see[4]Sec. 3.6.3foracompleteargument
2
a a
2 1
a
4
(cid:55)→ = −
a
3
= − − +
The link with chord diagrams and knot invariants is now clear. Here we first focus on diagrammising
S(g) and extracting the algebra B, and in Section 3 the algebra A described above will be associated by an
isomorphism to B, through their Hopf algebra structures.
0.2 Diagrammisation of S(g)
Penrose’s tensor calculus was applied to the construction of the symmetric algebra S(g) in [4], the key
properties being no anchors or labels. As in A edges in the diagrammisation of S(g) will be undirected
as well. Hence B may be described as the space of linear combinations of unitriangular diagrams, not
necessarily connected, modulo the two relations:
AS relation: = − Jacobi relation: + + = 0
We rearrange the Jacobi relation and apply the AS relation to get an equivalent statement, which from
now on will be termed the IHX relation.
c b a
b a
→
c d
d
c b a
b a b a b a
→ = − = −
c d c d c d
d
c b a
b a b a b a
→ = − =
c d c d c d
d
3
Theedgeshavebeenmarkedtokeeptrackofthemanipulations. Theaboverotationsandrearrangements
of the pendent edges lead to a parallel diagram equality. Hence the Jacobi relation is equivalent to:
− + = 0 ⇐⇒ = −
This is the IHX relation. We now begin with a description of the space of elements of B.
1 The vector space B
1.1 Derivation
Definition 1.1.1. (The vector space Do ) We define D0 for m ∈ N to be the space of all formal linear
m m
combinations of open Jacobi diagrams of degree m, or equivalently, connected unitrivalent graphs of degree
m with oriented vertices. For an open Jacobi diagram D, deg(D)= 1|V(D)|.
2
Definition 1.1.2. (The vector spaces B and B ) Define the vector spaces B and B by
m m
Do (cid:77)
B = m B = B (1.1)
m (IHX,AS) m
m(cid:62)0
with the degree-preserving IHX, AS relations given by
IHX: = − AS: = − (1.2)
Coefficients are taken over C, the zero element is ∅ (termed the null diagram), and the unit is [1] , the
B
equivalence class of all linear combinations of Jacobi diagrams that evaluate to ∅ (see (2.4) for more on the
unit).
Example 1.1.3. The IHX and AS relations are used to prove properties about Jacobi diagrams, by com-
pleting the diagrams of each term in a consistent manner. For example,
AS
= − = − =⇒ ∼ 0
Graph isomorphism was applied in the last equality. To emphasize graph equality under the application of
AS IHX B
certain relations, the symbols ∼, ∼ and ∼ will be used, the latter meaning that either of IHX or AS have
been applied in constructing the equalities.
Notation. Vertices of diagrams in Do that are oriented in the default direction, counterclockwise, will be
presented without a directional arrow, whereas vertices oriented clockwise will be marked by
This allows us to restate the AS relation as
AS: = −
4
AS
Lemma 1.1.4. If D ∈B contains a trivalent vertex with two univalent neighbors, then D ∼ 0.
Proof:SupposethatDhasatrivalentvertextwithtwounivalentneighborst ,t . DenotebyD(cid:48)thesubgraph
1 2
of D that connects to t via the edge not ending in t or t . Apply the AS relation to t.
1 2
= − = − = − = −
D(cid:48) D(cid:48) D(cid:48) D(cid:48) D(cid:48)
The result follows. (cid:4)
The lemma shows that every trivalent vertex has at most one univalent neighbor in every diagram in B.
Therefore every graph in B is a cubic graph with incident trees consisting of a single edge. This observation
allows us to systematise the diagrams of B.
Definition 1.1.5. (Ciliation) Given a graph G, if G(cid:48) is obtained from G by attaching single edges to edges
of G, in the process creating new trivalent vertices, then G(cid:48) is termed a ciliated graph, obtained by ciliating
G. The sigle edges are termed cilia. If m cilia are used in ciliating G, then G(cid:48) is termed an m-ciliation.
Definition 1.1.6. (m-wheel) If G is an m-ciliation of a single loop, then G is termed an m-wheel. In this
case, G is denoted by w .
m
m
···
w =
m
The result of (1.1.4) may be generalized on the number of trivalent vertices of a given diagram.
Proposition1.1.7. GivenD ∈B,letv (D)bethenumberoftrivalentverticesofD. IfDhasanorientation-
3
reversing automorphism, then D =(−1)v3(D)D.
Proof: Let f be an automorphism of D that reverses the orientation of every vertex of D. Apply the AS
relation to every one of the trivalent vertices of D to get
D A∼S(−1)v3(D)f(D)=(−1)v3(D)D
This completes the proof. (cid:4)
B
Note that if v (D) is odd for D ∈B, then it directly follows that D ∼0. This shows the main use of the
3
AS relation, to reduce diagrams to 0. As will be demonstrated in the following section, the IHX relation is
used more to reduce diagrams to their constituent basis elements.
1.1.1 Connected elements of B of low degree
For D an open Jacobi diagram, m = deg(D) = 1|V(D)|, one-half the number of vertices of D. As
(cid:80) 2
2|E| = deg(v) = 3|V(D)|, it follows that |E| = 3m. We shall only consider cubic graphs with
v∈V(D)
2m vertices and 3m edges.
The space Do of diagrams with no edges is empty.
0
5
Connected diagrams in Do with 3 edges. The set of cubic graphs with 2 vertices and 3 edges is:
1
G G
1 2
Hence in Do, there are at most 2·(22) = 8 diagrams, by applying an ordering to the incident edges of
1
each vertex.
G G G G
1,1 1,2 1,3 1,4
G G G G
2,1 2,2 2,3 2,4
We may identify pairs by smooth maps:
G = = = G
1,1 1,4
G = = = G
1,2 1,3
G = = = G
2,1 2,4
G = = = G
2,2 2,3
Each equality follows after a rotation through 180◦ around the indicated axis. This leaves 4 diagrams
G ,G ,G ,G . Next note that by a reversal of an orientation,
1,1 1,2 2,1 2,2
6
G = = = = = G
1,2 1,1
It follows that there are three distinct open Jacobi diagrams in Do:
1
∈ Do
1
To see how they embed in B , apply the AS and IHX relations. The observation from (1.1.3) of the AS
1
B
relation indicates that G ∼0. By the IHX relation,
1,1
= − = 0 − = − = − G
2,2
Hence G IH∼XG , and Do contributes to B only the following diagram:
2,1 2,2 1 1
∈ B
1
Connected diagrams in Do with 6 edges. The set of cubic graphs with 4 vertices and 6 edges is:
2
G G G G G
1 2 3 4 5
In Do we have at most 5·(24)=80 diagrams. By calculations as above (see Appendix A), there are 11
2
distinct open Jacobi diagrams in Do.
2
7
∈ Do
2
Checking for independence modulo the IHX and AS relations, (see Appendix A), B has only 1 distinct
2
diagram.
= ∈ B
2
Connected diagrams in Do with 9 edges. The number of cubic graphs with 6 vertices and 9 edges in
3
17, hence there are 17·26 =1088 vertex-oriented cubic graphs with 6 vertices and 9 edges, leaving a single
distinct element in B :
3
∈ B
3
The diagrams are reduced fully in Appendix B.
1.1.2 An alternative grading and bi-grading
Given a graph G, it is possible to obtain another bi-grading of the algebra. The rank of the fundamental
group of a graph G gives a homotopy equivalence of graphs2. The fundamental group is computed by con-
tracting a spanning tree of each connected component and counting the loops emanating from the single
vertex.
Definition 1.1.8. (loop-degree) Let G be a planar graph. Define the loop-degree of a graph G to be the
rank of the fundamental group of G, or
loop(G)=rank(π (G))
1
Definition 1.1.9. (The vector space Do ) Define Do to be the freely-generated subspace of Do of
n−loop n−loop
unitrivalent open Jacobi diagrams with loop-degree n.
2See[6],p.30
8
Proposition 1.1.10. The IHX and AS relations preserve loop-degree.
Proof: Since the graphs of the AS relation
= −
are isomorphic, loop-degree does not change. For the IHX relation, let e be the edge with both ends in
eachofthediagramsoftherelation. Bycontractinge(whichisnotaloop,hencecontractiondoesnotaffect
loop-degree), it follows that
(cid:32) (cid:33) (cid:32) (cid:33)
loop = loop
(cid:32) (cid:33) (cid:32) (cid:33)
loop = loop
(cid:32) (cid:33) (cid:32) (cid:33)
loop = loop
Hence the IHX relation does not affect loop-degree. (cid:4)
Using the fact that application of IHX and AS keeps the diagrams in a fixed space, we may formalize the
grading.
Definition 1.1.11. (The vector spaces B and B) Define the vector spaces B and B as
n−loop n−loop
B = Dno−loop B =(cid:77)B
n−loop (IHX,AS) n−loop
n(cid:62)0
Thisgivesabi-gradingofB. DefineB tobethesubspaceofB ofdegreemdiagrams,orequivalently,
n,m n−loop
the subspace of B of diagrams with loop-degree n.
m
(cid:77)
B = B
n,m
n,m(cid:62)0
1.2 The vector spaces B
n,m
The basis of B is given by trivalent graphs found in Section 1.1.1 with total degree (cid:54) m and a single
n,m
edge, which will be the cilia attached to the edges of the connected diagrams.
The vector space B
0,m
If m=0, there are no vertices, and hence no graphs. Therefore the span is ∅.
If m(cid:62)1, there are no loops, and the graph is a tree. Hence the span is a single edge
(cid:40) (cid:41)
9
The vector space B
1,m
If m=0, there are no vertices, and hence no graphs. Thus the span is ∅.
If m=1, there in one edge attached to a loop. Since the edge may be attached in two different ways, there
are two graphs in this space:
 
 
,
 
However, by rotating the first graph about its center axis and reorienting the trivalent edge, we get the
second graph. And a rotation followed by an application of AS gives the original graph back with a negative
sign.
rot. =
=
r=ot. A=S −
Hence the basis for B is null.
1,1
For B , we have 4 different diagrams.
1,2
Applyingasimilarapproachofrotationaroundthecenteraxis,reorientation,andAS,wefindthatthespace
is the span of a single diagram.
rot. =
=
AS −
=
AS −
=
Therefore the space B is
1,2
 
 
 
As m increases, the patterns above generalize by the folowing proposition.
Proposition 1.2.1. When m is odd, B is empty, and when m is even, the basis of B contains only
1,m 1,m
the m-wheel (as defined in (1.1.6)) with all cilia oriented in the same direction.
10