Q-polynomial distance-regular graphs and the q-Onsager algebrausers.wpi.edu/~martin/MEETINGS/LINESTALKS/Terwilliger.pdf · Paul Terwilliger Q-polynomial distance-regular graphs and

Q-polynomial distance-regular graphsand the q-Onsager algebra

Paul Terwilliger

University of Wisconsin-Madison

Paul Terwilliger Q-polynomial distance-regular graphs and the q-Onsager algebra

Contents

Part I: The subconstituent algebra of a graph

• The adjacency algebra

• The dual adjacency algebra

• The subconstituent algebra

• The notion of a dual adjacency matrix

Part II: The tridiagonal relations

• The notion of a tridiagonal pair

• The q-Serre relations and q-Dolan/Grady relations

• The positive part U+q of Uq(sl2)

• The q-Onsager algebra Oq

• How U+q and Oq are related


Part I. The subconstituent algebra of a graph

Let X denote a nonempty finite set.

MatX (C) denotes the C-algebra consisting of the matrices over Cthat have rows and columns indexed by X .

V = CX denotes the vector space over C consisting of columnvectors with rows indexed by X .

MatX (C) acts on V by left multiplication.


Preliminaries, cont.

Endow V with a Hermitean inner product

〈u, v〉 = utv (u, v ∈ V )

For each x ∈ X let x denote the vector in V that has a 1 incoordinate x and 0 in all other coordinates.

Observe {x |x ∈ X} is an orthonormal basis for V .


The graph Γ

Let Γ = (X ,R) denote a finite, undirected, connected graph,without loops or multiple edges, with vertex set X , edge set R, andpath-length distance function ∂.

For an integer i ≥ 0 and x ∈ X let

Γi (x) = {y ∈ X |∂(x , y) = i}

We abbreviate Γ(x) = Γ1(x).


The graphs of interest

Our main case of interest is when Γ is “highly regular” in a certainway.

A good example to keep in mind is the D-dimensionalhypercube, also called the binary Hamming graph H(D, 2).

Note that H(2, 2) is a 4-cycle; this will be used as a runningexample.


The adjacency matrix

Let A ∈ MatX (C) denote the (0, 1)-adjacency matrix of Γ.

For x ∈ X ,

Ax =∑

y∈Γ(x)

y

Example

For H(2, 2),

A =

0 1 1 01 0 0 11 0 0 10 1 1 0


The adjacency algebra M

Let M denote the subalgebra of MatX (C) generated by A.

M is called the adjacency algebra of Γ.

M is commutative and semisimple.

Example

For H(2, 2), M has a basis

I ,A, J

where the matrix J has every entry 1.


The primitive idempotents of Γ

Since M is semisimple it has basis {Ei}di=0 such that

EiEj = δijEi (0 ≤ i , j ≤ d),

I =d∑

i=0

Ei .

We call {Ei}di=0 the primitive idempotents of Γ.Write

A =d∑

i=0

θiEi .

For 0 ≤ i ≤ d the scalar θi is the eigenvalue of A for Ei .


The primitive idempotents, cont.

Example

For H(2, 2) we have θ0 = 2, θ1 = 0, θ2 = −2. Moreover

E0 = 1/4J,

E1 = 1/4

2 0 0 −20 2 −2 00 −2 2 0−2 0 0 2

,

E2 = 1/4

1 −1 −1 1−1 1 1 −1−1 1 1 −11 −1 −1 1

.


The eigenspaces of A

The vector space V decomposes as

V =d∑

i=0

EiV (orthogonal direct sum)

For 0 ≤ i ≤ d the space EiV is the eigenspace of A associatedwith the eigenvalue θi .

the matrix Ei represents the orthogonal projection onto EiV .


The dual primitive idempotents of Γ

Until further noticefix x ∈ X .

We call x the base vertex.

Define D = D(x) by

D = max{∂(x , y) | y ∈ X}

We call D the diameter of Γ with respect to x .

For 0 ≤ i ≤ D let E ∗i = E ∗i (x) denote the diagonal matrix inMatX (C) with (y , y)-entry

(E ∗i )yy =

{1, if ∂(x , y) = i ;

0, if ∂(x , y) 6= i .(y ∈ X ).


The dual primitive idempotents, cont.

Example

For H(2, 2),

E ∗0 =

1 0 0 00 0 0 00 0 0 00 0 0 0

, E ∗1 =

0 0 0 00 1 0 00 0 1 00 0 0 0

,

E ∗2 =

0 0 0 00 0 0 00 0 0 00 0 0 1

.


The dual primitive idempotents, cont.

The {E ∗i }Di=0 satisfy

E ∗i E∗j = δijE

∗i (0 ≤ i , j ≤ D),

I =D∑i=0

E ∗i .

We call {E ∗i }Di=0 the dual primitive idempotents of Γ withrespect to x .


The dual adjacency algebra

The {E ∗i }Di=0 form a basis for a semisimple commutativesubalgebra of MatX (C) denoted M∗ = M∗(x).

We call M∗ the dual adjacency algebra of Γ with respect to x .


The subconstituents of Γ

The vector space V decomposes as

V =D∑i=0

E ∗i V (orthogonal direct sum)

The above summands are the common eigenspaces for M∗.

These eigenspaces have the following combinatorial interpretation.For 0 ≤ i ≤ D,

E ∗i V = Span{y |y ∈ Γi (x)}

We call E ∗i V the ith subconstituent of Γ with respect to x .

The matrix E ∗i represents the orthogonal projection onto E ∗i V .


The subconstituent algebra T

So far we defined the adjacency algebra M and the dual adjacencyalgebra M∗. We now combine M and M∗ to get a larger algebra.

Definition

(Ter 92) Let T = T (x) denote the subalgebra of MatX (C)generated by M and M∗. T is called the subconstituent algebraof Γ with respect to x .

T is finite-dimensional.

T is noncommutative in general.


T is semisimple

T is semi-simple because it is closed under the conjugate-transposemap.

So by the Wedderburn theory the algebra T is isomorphic to adirect sum of matrix algebras.

Example

For H(2, 2),

T ' Mat3(C)⊕ C.

Moreover dim(T ) = 10.


T is semisimple, cont.

Example

(Junie Go 2002) For H(D, 2),

T ' MatD+1(C)⊕MatD−1(C)⊕MatD−3(C)⊕ · · ·

Moreover dim(T ) = (D + 1)(D + 2)(D + 3)/6.


T -modules

We mentioned that T is closed under the conjugate-transposemap.

So for each T -module W ⊆ V , its orthogonal complement W⊥ isalso a T -module.

Therefore V decomposes into an orthogonal direct sum ofirreducible T -modules.


The dual adjacency matrix

We now describe a family of graphs for which the irreducibleT -modules are nice.

These graphs possess a certain matrix called a dual adjacencymatrix.

To motivate this concept we consider some relations in T .


Some relations in T

By the triangle inequality

AE ∗i V ⊆ E ∗i−1V + E ∗i V + E ∗i+1V (0 ≤ i ≤ D),

where E ∗−1 = 0 and E ∗D+1 = 0.

This is reformulated as follows.

Lemma

For 0 ≤ i , j ≤ D,

E ∗i AE∗j = 0 if |i − j ] > 1.



Definition

Referring to the graph Γ, consider a matrix A∗ ∈ MatX (C) thatsatisfies both conditions below:

(i) A∗ generates M∗;

(ii) For 0 ≤ i , j ≤ d ,

EiA∗Ej = 0 if |i − j ] > 1.

We call A∗ a dual adjacency matrix (with respect to x and thegiven ordering {Ei}di=0 of the primitive idempotents).

A dual adjacency matrix A∗ is diagonal.



A dual adjacency matrix A∗ acts on the eigenspaces of A as follows.

A∗EiV ⊆ Ei−1V + EiV + Ei+1V (0 ≤ i ≤ d),

where E−1 = 0 and Ed+1 = 0.


An example

Example

H(2, 2) has a dual adjacency matrix

A∗ =

2 0 0 00 0 0 00 0 0 00 0 0 −2

Example

H(D, 2) has a dual adjacency matrix

A∗ =D∑i=0

(D − 2i)E ∗i .


More examples

Any Q-polynomial distance-regular graph has a dual adjacencymatrix. For instance:

• cycle

• Hamming graph

• Johnson graph

• Grassman graph

• Dual polar spaces

• Bilinear forms graph

• Alternating forms graph

• Hermitean forms graph

• Quadratic forms graph

See the book Distance-Regular Graphs by Brouwer, Cohen,Neumaier.


How A and A∗ are related

To summarize so far, for our graph Γ the adjacency matrix A andany dual adjacency matrix A∗ generate T . Moreover they act oneach other’s eigenspaces in the following way:

AE ∗i V ⊆ E ∗i−1V + E ∗i V + E ∗i+1V (0 ≤ i ≤ D),

where E ∗−1 = 0 and E ∗D+1 = 0.

A∗EiV ⊆ Ei−1V + EiV + Ei+1V (0 ≤ i ≤ d),

where E−1 = 0 and Ed+1 = 0.

To clarify this situation we reformulate it as a problem in linearalgebra.


Part II: The tridiagonal relations

We now recall a linear-algebraic object called a TD pair.

From now on F denotes a field.

V will denote a vector space over F with finite positive dimension.

We consider a pair of linear transformations A : V → V andA∗ : V → V .


Definition of a Tridiagonal pair (Ito, Tanabe, Ter 2001)

We say the pair A,A∗ is a TD pair on V whenever (1)–(4) holdbelow.

1 Each of A,A∗ is diagonalizable on V .

2 There exists an ordering {Vi}di=0 of theeigenspaces of A such that

A∗Vi ⊆ Vi−1 + Vi + Vi+1 (0 ≤ i ≤ d),

where V−1 = 0, Vd+1 = 0.

3 There exists an ordering {V ∗i }Di=0 of the eigenspaces of A∗

such that

AV ∗i ⊆ V ∗i−1 + V ∗i + V ∗i+1 (0 ≤ i ≤ D),

where V ∗−1 = 0, V ∗D+1 = 0.

4 There is no subspace W ⊆ V such that AW ⊆W andA∗W ⊆W and W 6= 0 and W 6= V .


The diameter

Referring to our definition of a TD pair,

it turns out d = D; we call this common value the diameter of thepair.


Each irreducible T -module gives a TD pair

Briefly returning to the graph Γ, the adjacency matrix and any dualadjacency matrix act on each irreducible T -module as a TD pair.

This motivates us to understand TD pairs.


The tridiagonal relations

Theorem (Ito+Tanabe+T, 2001)

For our TD pair A,A∗ there exist scalars β, γ, γ∗, %, %∗ in F suchthat

A3A∗ − (β + 1)A2A∗A + (β + 1)AA∗A2 − A∗A3

= γ(A2A∗ − A∗A2) + %(AA∗ − A∗A),

A∗3A− (β + 1)A∗2AA∗ + (β + 1)A∗AA∗2 − AA∗3

= γ∗(A∗2A− AA∗2) + %∗(A∗A− AA∗).

The sequence β, γ, γ∗, %, %∗ is unique if the diameter is at least 3.

The above equations are called the tridiagonal relations.


The q-Serre relations

For the special case

β = q2 + q−2, γ = γ∗ = 0, % = %∗ = 0

the tridiagonal relations become the q-Serre relations

A3A∗ − [3]qA2A∗A + [3]qAA

∗A2 − A∗A3 = 0,

A∗3A− [3]qA∗2AA∗ + [3]qA

∗AA∗2 − AA∗3 = 0.

[n]q =qn − q−n

q − q−1n = 0, 1, 2, . . .

The q-Serre relations are the defining relations for the positivepart U+

q of the quantum group Uq(sl2).


The q-Dolan/Grady relations

For the special case

β = q2 + q−2, γ = γ∗ = 0, % = %∗ = −(q2 − q−2)2

the tridiagonal relations become the q-Dolan/Grady relations

A3A∗ − [3]qA2A∗A + [3]qAA

∗A2 − A∗A3

= (q2 − q−2)2(A∗A− AA∗),

A∗3A− [3]qA∗2AA∗ + [3]qA

∗AA∗2 − AA∗3

= (q2 − q−2)2(AA∗ − A∗A).

The q-Dolan/Grady relations are the defining relations for theq-Onsager algebra Oq.


The definition of U+q

We now officially define U+q and Oq. For the rest of this talk fix

0 6= q ∈ F such that q2 6= 1.

Definition

Let U+ = U+q denote the F-algebra with generators X ,Y and

relations

X 3Y − [3]qX2YX + [3]qXYX

2 − YX 3 = 0,

Y 3X − [3]qY2XY + [3]qYXY

2 − XY 3 = 0.

The algebra U+ is called the positive part of Uq(sl2).


The definition of Oq

Definition

Let O = Oq denote the F-algebra with generators A,B andrelations

A3B − [3]qA2BA + [3]qABA

2 − BA3 = (q2 − q−2)2(BA− AB),

B3A− [3]qB2AB + [3]qBAB

2 − AB3 = (q2 − q−2)2(AB − BA).

We call O the q-Onsager algebra.


Comparing U+q and Oq

Consider how the algebras U+q and Oq are related.

These algebras have at least a superficial resemblance, since forthe q-Serre relations and q-Dolan/Grady relations their left-handsides match.

We now consider how U+q and Oq are related at an algebraic level.


A filtration of O

For n ∈ N let On denote the subspace of O spanned by theproducts g1g2 · · · gr such that 0 ≤ r ≤ n and gi is among A,B for1 ≤ i ≤ r .

For example, O0 = F1 and O1 = F1 + FA + FB.

For notation convenience define O−1 = 0.


A filtration of O, cont.

By construction,

(i) On−1 ⊆ On for n ∈ N;

(ii) 1 ∈ O0;

(iii) OrOs ⊆ Or+s for r , s ∈ N;

(iv) O = ∪n∈NOn.

By (i)–(iv) the sequence {On}n∈N is a filtration of O.


The algebra O

Using our filtration {On}n∈N we now define an F-algebra O.

For n ∈ N define the quotient F-vector space On = On/On−1.

The subspace O1 is spanned by

A = A +O0, B = B +O0.

Now consider the formal direct sum

O =∑n∈NOn.

By construction O is a vector space over F.


The algebra O, cont.

We next define a product O ×O → O that turns O into anF-algebra.

For r , s ∈ N the product sends Or ×Os → Or+s as follows.

For u ∈ Or and v ∈ Os the product of u +Or−1 and v +Os−1 isuv +Or+s−1.

We have turned O into an F-algebra.


The algebra O, cont.

By construction OrOs ⊆ Or+s for r , s ∈ N.

The F-algebra O is generated by A,B.

The F-algebra O is called the graded algebra associated with thefiltration {On}n∈N.


Comparing U+ and O

The algebras U+ and O are related as follows.

From the q-Dolan/Grady relations we obtain

A3B − [3]qA

2B A + [3]qAB A

2 − B A3

= 0,

B3A− [3]qB

2AB + [3]qB AB

2 − AB3

= 0.

Therefore there exists an F-algebra homomorphism ψ : U+ → Othat sends X 7→ A and Y 7→ B.


Comparing U+ and O, cont.

We now state one of our main results.

Theorem

The map ψ : U+ → O is an isomorphism of F-algebras.


The algebra �q

We have been discussing the algebra U+ = U+q which is the

positive part of Uq(sl2), and the q-Onsager algebra O = Oq.

As we compare these algebras, it is useful to bring in anotheralgebra �q.

We will show how �q is related to both U+ and O.

Let Z4 = Z/4Z denote the cyclic group of order 4.


The algebra �q

The algebra �q is defined as follows.

Definition

Let �q denote the F-algebra with generators {xi}i∈Z4 andrelations

qxixi+1 − q−1xi+1xiq − q−1

= 1,

x3i xi+2 − [3]qx

2i xi+2xi + [3]qxixi+2x

2i − xi+2x

3i = 0.


The algebra �q has Z4 symmetry

The algebra �q has the following Z4 symmetry.

Lemma

There exists an automorphism ρ of �q that sends xi 7→ xi+1 fori ∈ Z4. Moreover ρ4 = 1.


The algebra �q and U+

.The algebra �q is related to U+ in the following way.

Definition

Define the subalgebras �evenq , �odd

q of �q such that

(i) �evenq is generated by x0, x2;

(ii) �oddq is generated by x1, x3.


The algebra �q and U+, cont.

Theorem

The following (i)–(iii) hold:

(i) there exists an F-algebra isomorphism U+ → �evenq that sends

X 7→ x0 and Y 7→ x2;

(ii) there exists an F-algebra isomorphism U+ → �oddq that sends

X 7→ x1 and Y 7→ x3;

(iii) the following is an isomorphism of F-vector spaces:

�evenq ⊗�odd

q → �q

u ⊗ v 7→ uv


The algebra �q and O

We now describe how �q is related to the q-Onsager algebra O.

Theorem

Pick nonzero a, b ∈ F. Then there exists a unique F-algebrahomomorphism ] : O → �q that sends

A 7→ ax0 + a−1x1, B 7→ bx2 + b−1x3.

The homomorphism ] is injective.


Summary

In this talk, we used the notion of a dual adjacency matrix tomotivate the concept of a tridiagonal pair of lineartransformations.

In the theory of tridiagonal pairs, we find the algebra U+q which is

the positive part of Uq(sl2), and also the q-Onsager algebra Oq.We wish to understand these algebras.

We presented several theorems that describe how U+q and Oq are

related.

Thank you for your attention!

THE END


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Q-polynomial distance-regular graphs and the q-Onsager algebrausers.wpi.edu/~martin/MEETINGS/LINESTALKS/Terwilliger.pdf · Paul Terwilliger Q-polynomial distance-regular graphs and