Miraculous cancellations for quantum <span class="mathjax-formula">$\protect \mathrm{SL}_2$</span>

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

FRANCISBONAHON

Miraculous cancellations for quantumSL₂

Tome XXVIII, n^o3 (2019), p. 523-557.

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Miraculous cancellations for quantum SL

Francis Bonahon⁽¹⁾

À Jean-Pierre Otal, en l’honneur de ses3×4×5ans

ABSTRACT. — In earlier work, Helen Wong and the author discovered certain

“miraculous cancellations” for the quantum trace map connecting the Kauffman bracket skein algebra of a surface to its quantum Teichmüller space, occurring when the quantum parameterq= e^2πi~is a root of unity. The current paper is devoted to giving a more representation theoretic interpretation of this phenomenon, in terms of the quantum group Uq(sl2) and its dual Hopf algebra SL^q₂.

RÉSUMÉ. — Des travaux précédents de Helen Wong et de l’auteur ont mis en évidence, quand le paramètre quantique q = e^2πi~ est une racine de l’unité, des

« annulations miraculeuses » pour l’application de trace quantique qui relie l’algèbre d’écheveaux du crochet de Kauffman à l’espace de Teichmüller quantique d’une sur- face. L’article ci-dessous fournit une interprétation plus conceptuelle de ce phéno- mène, en termes de représentations du groupe quantique Uq(sl2) et de son algèbre de Hopf duale SL^q₂.

Introduction

The equation

(X+Y)ⁿ=Xⁿ+Yⁿ (0.1)

is (unfortunately) very familiar to some of our students, who find it conve- nient to “simplify” computations. However, it is also well-known that this relation does hold in some cases, for instance in a ring of characteristicnwith nprime, or when the variablesX andY satisfy theq-commutativity relation thatY X =qXY withq∈Ca primitiven–root of unity; see Section 1.

The structure of Equation (0.1) can be described by considering the two- variable polynomialP(X, Y) =X+Y. Then (0.1) states that the polynomial

(1) Department of Mathematics, University of Southern California, Los Angeles CA 90089-2532, U.S.A. — fbonahon@usc.edu

This work was partially supported by the grants DMS-1406559 and DMS-1711297 from the U.S. National Science Foundation.

P(X, Y)ⁿ, obtained by taking then–th power ofP(X, Y), coincides with the polynomialP(Xⁿ, Yⁿ) obtained by replacing the variablesX,Y with their powersXⁿ,Yⁿ, respectively.

Helen Wong and the author discovered similar identities in their study of the Kauffman bracket skein algebra of a surface [6]. These relations involved a 2–dimensional version of the operation of “taking then–th power”, through the Chebyshev polynomialT_n(t)∈Z[t] defined by the property that

TraceAⁿ=T_n(TraceA)

for every 2-by-2 matrix A ∈ SL2(C) with determinant 1. A typical con- sequence of the miraculous cancellations discovered in [6] is that, when Y X=qXY andqis a primitive n–root of unity,

Tn(X+Y +X⁻¹) =Xⁿ+Yⁿ+X⁻ⁿ, (0.2) which fits the pattern Tn P(X, Y)

= P(Xⁿ, Yⁿ) for the polynomial P(X, Y) = X +Y +X⁻¹. The arguments of [6] provide many examples of such polynomials, involving severalq–commuting variables.

In [6] these “Chebyshev cancellations” were used to connect, when q is a root of unity, irreducible representations of the Kauffman bracket skein algebra S^q(S) of a surface S to group homomorphisms π1(S) → SL2(C).

The skein algebra S^q(S) is a purely topological object whose elements are represented by framed links in the thickened surfaceS×[0,1]. It draws its origin from Witten’s interpretation [17, 18, 22] of the Jones polynomial knot invariant within the framework of a topological quantum field theory, and as a consequence it is closely connected to the quantum group Uq(sl2).

The arguments of [6] were often developed by trial and error. The purpose of the current article is to put these constructions into a more conceptual framework, where the connection with Uq(sl2) and SL2(C) appears more clearly. Another goal is to emphasize the representation theoretic nature of these phenomena, with the long term objective of generalizing them to quantum knot invariants and skein algebras based on other quantum groups U_q(g), such as the U_q(sl_n)–based HOMFLY polynomial and skein algebra.

In addition to the fact that quantum groups are still an acquired taste for many mathematicians, including the author, the connection between U_q(sl₂) and SL₂(C) is more intuitive if we replace U_q(sl₂) with its dual Hopf alge- bra SL^q₂, in the sense of [14, 15, 16, 19, 20]. This will enable us to express our constructions solely in terms of 2-by-2 matrices with coefficients in an arbitrary noncommutative algebraA; in [6], the algebraAwas the quantum Teichmüller space of the surface. This point of view is sufficiently close to SL2(C) that it should be relatively intuitive for those mathematicians who

have a long track record in hyperbolic geometry, since PSL2(C) is the isom- etry group of the hyperbolic space H³. This category includes the author and Jean-Pierre Otal, and it is a pleasure to dedicate this article to him as an acknowledgement of the great influence that he had on the author’s work, either through their joint articles [1, 2, 3, 4] or through many informal conversations.

We now state the main result of this article.

Theorem 0.1. — LetA1, A2, . . . , Ak be2-by-2matrices with coefficients in an algebraAoverC, such that:

(1) eachAi is triangular of the form_a

i b_i 0 a⁻¹_i

or _{ai 0}

b_ia⁻¹_i

withbiai= qaibi for some nonzero number q∈C− {0};

(2) ai andbi commute withaj andbj wheneveri6=j.

Then, ifq² is a primitiven–root of unity,

Tn(TraceA1A2. . . Ak) = TraceA⁽ⁿ⁾₁ A⁽ⁿ⁾₂ . . . A⁽ⁿ⁾_k

where eachA⁽ⁿ⁾_i is obtained fromAi by replacingaiandbi with their powers aⁿ_i andbⁿ_i

This statement is easier to understand if we illustrate it by an example.

Consider the product of five triangular matrices A=

a₁ b₁ 0 a⁻¹₁

a₂ b₂ 0 a⁻¹₂

a₃ 0 b3 a⁻¹₃

a₄ b₄ 0 a⁻¹₄

a₅ 0 b5 a⁻¹₅

whereb_ia_i=qa_ib_i, anda_i andb_i commute witha_j andb_j wheneveri6=j.

Computing the product and taking the trace straightforwardly gives TraceA=a1a2a3a4a5+a1a2a3b4b5+a1b2b3a4a5+a1b2b3b4b5

+a1b2a⁻¹₃ a⁻¹₄ b5+b1a⁻¹₂ b3a4a5+b1a⁻¹₂ b3b4b5

+b1a⁻¹₂ a⁻¹₃ a⁻¹₄ b5+a⁻¹₁ a⁻¹₂ b3b4a⁻¹₅ +a⁻¹₁ a⁻¹₂ a⁻¹₃ a⁻¹₄ a⁻¹₅ . Since TraceA has 10 terms and the Chebyshev polynomial Tn(t) has degreen, one would expectTn(TraceA) to have about 10ⁿ terms. However, when q² is a primitive n–root of unity, many cancellations occur and only 10 terms remain. In fact

Tn(TraceA) =aⁿ₁aⁿ₂aⁿ₃aⁿ₄aⁿ₅+aⁿ₁aⁿ₂aⁿ₃bⁿ₄bⁿ₅ +aⁿ₁bⁿ₂bⁿ₃aⁿ₄aⁿ₅+aⁿ₁bⁿ₂bⁿ₃bⁿ₄bⁿ₅ +aⁿ₁bⁿ₂a⁻ⁿ₃ a⁻ⁿ₄ bⁿ₅+bⁿ₁a⁻ⁿ₂ bⁿ₃aⁿ₄aⁿ₅+bⁿ₁a⁻ⁿ₂ bⁿ₃bⁿ₄bⁿ₅

+bⁿ₁a⁻ⁿ₂ a⁻ⁿ₃ a⁻ⁿ₄ bⁿ₅ +a⁻ⁿ₁ a⁻ⁿ₂ bⁿ₃bⁿ₄a⁻ⁿ₅ +a⁻ⁿ₁ a⁻ⁿ₂ a⁻ⁿ₃ a⁻ⁿ₄ a⁻ⁿ₅

= TraceA⁽ⁿ⁾

where A⁽ⁿ⁾=

aⁿ₁ bⁿ₁ 0 a⁻ⁿ₁

aⁿ₂ bⁿ₂ 0 a⁻ⁿ₂

aⁿ₃ 0 bⁿ₃ a⁻ⁿ₃

aⁿ₄ bⁿ₄ 0 a⁻ⁿ₄

aⁿ₅ 0 bⁿ₅ a⁻ⁿ₅

is obtained fromA by replacing eacha_i,b_i withaⁿ_i, bⁿ_i.

When q is transcendental, there are only very few cancellations and Proposition 6.2 shows that Tn(TraceA) is a sum of exactly (5 +√

24)ⁿ+ (5−√

24)ⁿ>9.89ⁿ monomials. This explicit count is based on a positivity result (Proposition 6.1) which may be of independent interest.

Similarly, the example of Equation (0.2) is provided by applying The- orem 0.1 to the matrix A = _a

1 b₁ 0 a⁻¹₁

a2 0 b₂ a⁻¹₂

, setting X = a1a2 and Y =b₁b₂, and replacingqwith√

q.

The proof of Theorem 0.1 essentially has two parts. The first step is rep- resentation theoretic, and connectsTn(TraceA) to the action of the matrix Aon the spaceA[X, Y]^q of polynomials inq–commuting variablesX andY and with coefficients in the algebraA. This is a relatively easy adaptation to our context of a deep but classical result in the representation theory of the quantum group Uq(sl2), the Clebsch–Gordan Decomposition. The author is here grateful to the referee for pointing out the reference [9], where a differ- ent proof of Corollary 4.12 can be found; see Remark 4.13. The second step is a simple computation of traces for this action ofAonA[X, Y]^q, which is much simpler than the original arguments of [6].

Among the hypotheses of Theorem 0.1, some are more natural than oth- ers. Theq–commutativity relationsbiai=qaibiare essentially mandated by the connection of the objects considered to the quantum group Uq(sl2) and its dual Hopf algebra SL^q₂. Similarly, the requirement that the matrices Ai

be triangular is deeply tied to the structure of the Lie group SL2(C) and the quantum group Uq(sl2) (and their Borel subgroups/subalgebras). The commutativity hypothesis thataiandbi commute withaj andbj whenever i6=jis less critical, and was introduced here to define TraceA₁A₂. . . A_k ∈ A (and the product matrix A₁A₂. . . A_k ∈ SL^q₂(A)) in a straightforward way.

In fact, it is possible to define such a trace without these commutativity properties, but this requires using the cobraiding of the Hopf algebra SL^q₂ as well as additional data that is reminiscent of the original topological con- text. This was implicitly done in [5] for the Kauffman bracket skein algebra of a surface, but a quick comparison between the formulas of [5, Lem. 21]

and [11, Cor. VIII.7.2] should make it clear that these arguments can be expanded to a more representation theoretic framework. The cancellations of Theorem 0.1 then extend to this generalized setup, as in [6].

1. The equation(X+Y)ⁿ=Xⁿ+Yⁿ

In spite of the first sentence of this article, most of our students do know theBinomial Formula, which says that

(X+Y)ⁿ

=Xⁿ+ n

1

Xⁿ⁻¹Y + + n

2

Xⁿ⁻²Y²+· · ·+ n

n−1

XYⁿ⁻¹+Yⁿ

=

n

X

k=0

n k

X^n−kY^k where ⁿ_k

is thebinomial coefficient n

k

= n(n−1)(n−2). . .(n−k+ 2)(n−k+ 1) k(k−1)(k−2). . .2 1 .

If we are working in a ring R with characteristic n and if n is prime (which in particular occurs when R is a field), then n = 0 in R while k(k−1)(k−2). . .2 16= 0 for everyk < n. It follows that ⁿ_k

= 0 whenever 0< k < n, so that the Frobenius relation

(X+Y)ⁿ=Xⁿ+Yⁿ (0.1)

holds in this case.

Note that the hypothesis that the characteristicnis prime is really nec- essary. For instance, in the ringZ/4 of characteristic 4,

(X+Y)⁴=X⁴+ 6X²Y²+Y⁴6=X⁴+Y⁴.

A less well-known occurrence of Equation (0.1) involves variablesX and Y that q–commute, in the sense that Y X = qXY for some q ∈ C. The Quantum Binomial Formula(see for instance [11, §IV.2]) then states that

(X+Y)ⁿ

=Xⁿ+ n

1

q

Xⁿ⁻¹Y + + n

2

q

Xⁿ⁻²Y²+· · ·+ n

n−1

q

XYⁿ⁻¹+Yⁿ

=

n

X

k=0

n k

q

X^n−kY^k (1.1)

where ⁿ_k

q is thequantum binomial coefficient n

k

q

=(n)q(n−1)q(n−2)q. . .(n−k+ 2)q(n−k+ 1)q

(k)q(k−1)q(k−2)q. . .(2)q(1)q

defined by thequantum integers (j)_q =q^j−1

q−1 = 1 +q+q²+· · ·+q^j−1.

We state the following property as a lemma, as we will frequently need to refer to it.

Lemma 1.1. — If q is a primitive n–root of unity, in the sense that qⁿ = 1and q^k 6= 1 whenever 0 < k < n, the quantum binomial coefficient

n k

q is equal to 0for every kwith 0< k < n.

Proof. — Since q is a primitiven–root of unity, (n)q = ^q_q−1ⁿ⁻¹ = 0 while (k)q= ^q_q−1^k−16= 0 whenever 0< k < n. The result follows.

Corollary 1.2. — If Y X = qXY with q ∈ C a primitive n–root of unity, the Frobenius relation

(X+Y)ⁿ=Xⁿ+Yⁿ (0.1)

holds.

As in the characteristicn case, it is necessary thatq be a primitiven–

root of unity. For instance, whenY X =−XY,q=−1 is a (non-primitive) 4–root of unity and (X+Y)⁴=X⁴+ 2X²Y²+Y⁴6=X⁴+Y⁴.

We will now discuss generalizations of Equation (0.1) arising from prop- erties of the quantum group Uq(sl2). As indicated in the introduction, we will express these properties in terms of 2-by-2 matrices with coefficients in an algebraA.

Incidentally, all algebras considered in this article will be overC. Other fields could be used, but the need for primitiven–roots of unity make this convention more natural.

We begin with the classical case of the algebra A =C, and of the Lie group SL₂(C).

2. The classical action of SL2(C)on C[X, Y]

Thespecial linear group of order 2 is the group SL2(C) = _{a b}

c d

;a, b, c, d∈C, ad−bc= 1

of 2-by-2 matrices with determinant 1. It has a left action on the planeC², and therefore a right action by precomposition on the algebra

C[X, Y] ={polynomial functions onC²}

=n

polynomialsPm i=0

Pn

j=0aijXⁱY^j;aij ∈C o

of polynomials in the variablesXandY. More precisely, the action of ^{a b}_{c d}

∈ SL₂(C) onC[X, Y] is such that

P(X, Y) a b

c d

=P(aX+bY, cX+dY) (2.1) for every polynomialP(X, Y)∈C[X, Y].

This defines a map

ρ: SL2(C)→End_C C[X, Y]

from SL2(C) to the algebra ofC–linear mapsC[X, Y]→C[X, Y].

We collect a few elementary properties in the following lemma.

Lemma 2.1.

(1) For every ^{a b}_{c d}

∈ SL2(C), ρ ^{a b}_{c d}

∈ End_C C[X, Y]

is also an algebra endomorphism ofC[X, Y].

(2) The map ρ is valued in the group Aut_C C[X, Y]

of linear au- tomorphisms of C[X, Y], and induces a group antihomomorphism ρ: SL2(C)→Aut_C C[X, Y]

, in the sense that ρ

a b c d

a⁰ b⁰ c⁰ d⁰

=ρ

a⁰ b⁰ c⁰ d⁰

◦ρ a b

c d

for every ^{a b}_{c d} , ^a_c⁰0 ^bd⁰⁰

∈SL₂(C).

(3) IfC[X, Y]_n=P

i+j=na_ijXⁱY^j;a_ij∈C ∼=Cⁿ⁺¹ denotes the vec- tor space of homogeneous polynomials of degreen, the representation ρrestricts to a finite-dimensional representation

ρn: SL2(C)→Aut_C C[X, Y]n∼= Aut_C(Cⁿ⁺¹).

The order reversal in the second conclusion reflects the fact that SL2(C) acts on C[X, Y] on the right. We could easily turn ρ into a group homo- morphism by composing it with any of the standard antiautomorphisms of SL2(C), such as ^{a b}_{c d}

7→ ^{a b}_{c d}^t

= (^{a c}_{b d}) or ^{a b}_{c d}

7→ ^{a b}_{c d}⁻¹

= _{−c a}^{d −b} , and many authors do this. For this reason, we will refer toρand its restrictions ρn as representations of SL2(C).

A classical property is that, up to isomorphism, theρnform the collection of all irreducible representations of SL2(C).

3. The quantum plane and SL^q₂(A)

For a nonzero number q ∈ C− {0}, the quantum plane is the alge- bra C[X, Y]^q defined by two generators X and Y, and by the relation Y X =qXY. Namely, the elements ofC[X, Y]^q are polynomials P(X, Y) = Pm

i=0

Pn

j=0a_ijXⁱX^j that are multiplied using the relationY X=qXY. If we want to keep the property that the matrices ^{a b}_{c d}

and (^{a c}_{b d}) act as algebra homomorphisms onC[X, Y]^q, we need that

(cX+dY)(aX+bY) =q(aX+bY)(cX+dY) and (bX+dY)(aX+cY) =q(aX+cY)(bX+dY)

to preserve the relationY X=qXY. Identifying the coefficients ofX²,XY andY², this would require

ba=qab db=qbd bc=cb

ca=qac dc=qcd ad−q⁻¹bc=da−qbc (3.1) which is clearly impossible ifq6= 1 anda,b,c, dcommute with each other.

This leads us to the following definition (see for instance [11, Chap. IV] for more background).

Given an algebraAoverC, let SL^q₂(A) denote the set of matrices ^{a b}_{c d} wherea,b,c,d∈ Asatisfy the relations of (3.1), as well as

ad−q⁻¹bc= 1. (3.2)

More formally, let SL^q₂be the algebra defined by generatorsa,b,c,dand by the relations of (3.1) and (3.2). Then SL^q₂(A) can be interpreted as the set of all algebra homomorphisms SL^q₂ → A. The elements of SL^q₂(A) are called theA–points ofSL^q₂.

Unlike SL₂(C), the set SL^q₂(A) is far from being a group. It only comes with a partially defined multiplication. Indeed, if ^{a b}_{c d}

and ^a_c0^0bd⁰⁰

∈SL^q₂(A), the usual formula

a b c d

a⁰ b⁰ c⁰ d⁰

=

aa⁰+bc⁰ ab⁰+bd⁰ ca⁰+dc⁰ cb⁰+dd⁰

(3.3) gives an element of SL^q₂(A) only under additional hypotheses on the entries of these matrices, for instance ifa, b, c, dcommute with a⁰,b⁰, c⁰, d⁰. This partially defined multiplication has an identity element (^{1 0}_{0 1}) ∈ SL^q₂(A).

However, the operation of passing to the inverse somewhat misbehaves in the sense that the formal inverse ^{a b}_{c d}−1

= _{d −qb}

−q⁻¹c a

of ^{a b}_{c d}

∈SL^q₂(A) is an element of SL^q₂⁻¹(A) = SL^q₂(A^op) instead of SL^q₂(A); here A^op is the opposite algebraof the algebraA, consisting of the vector spaceAendowed

with the new multiplication ∗op defined by the property that a∗opb = ba for every a, b ∈ A. In general, the formula (3.3) gives a globally defined multiplication SL₂(A)⊗SL₂(B)→SL₂(A⊗B) for any two algebrasAandB.

In order to generalize to SL^q₂(A) the action of SL₂(C) on C[X, Y], we introduce thequantumA–planeas theA–algebraA[X, Y]^q =A ⊗C[X, Y]^q. In practice the elements ofA[X, Y]^q are polynomials

P(X, Y) =

m

X

i=0 n

X

j=0

aijXⁱX^j

with coefficientsaij ∈ A, and are algebraically manipulated using the rela- tionY X = qXY while the variables X and Y commute with all elements ofA.

The relations defining SL^q₂(A) are specially designed that anA–point of SL^q₂ acts as an algebra homomorphism on the quantum A–planeA[X, Y]^q, by the same formula (2.1) as in the commutative plane. More precisely, if EndA(A[X, Y]^q) denotes the space ofA–linear mapsA[X, Y]^q → A[X, Y]^q, we can define a map

ρ: SL^q₂(A)→End_A(A[X, Y]^q) such that

ρ a b

c d



 X

i,j

aijXⁱY^j



=X

i,j

aij(aX+bY)ⁱ(cX+dY)^j for every polynomialP

i,ja_ijXⁱY^j∈ Aq[X, Y].

The mapρsatisfies the following elementary properties.

Lemma 3.1.

(1) For every ^{a b}_{c d}

∈SL^q₂(A), the restriction C[X, Y]^q → A[X, Y]^q of the A–linear mapρ ^{a b}_{c d}

∈EndA(A[X, Y]^q)is a C–algebra homo- morphism.

(2) For each n, the map ρ ^{a b}_{c d}

∈ EndA(A[X, Y]^q) respects the space A[X, Y]^q_n =A ⊗C[X, Y]^q_n of homogeneous polynomials of degree n inX and Y with coefficients in A. As a consequence,ρinduces by restriction a map

ρn: SL^q₂(A)→EndA(A[X, Y]^q_n).

(3) If a, b, c,d commute with a⁰, b⁰, c⁰, d⁰ (so that the product ^{a b}_{c d}

a⁰ b⁰ c⁰ d⁰

=

aa⁰+bc⁰ ab⁰+bd⁰ ca⁰+dc⁰ cb⁰+dd⁰

∈SL^q₂(A)makes sense), ρ

a b c d

a⁰ b⁰ c⁰ d⁰

=ρ a⁰ b⁰

c⁰ d⁰

◦ρ a b

c d

.

Remark 3.2. — The first conclusion of Lemma 3.1 can be rephrased by saying that, for everyP,Q∈C[X, Y]^q,

ρ a b

c d

(P Q) =ρ a b

c d

(P)ρ a b

c d

(Q)

in A[X, Y]^q. However, note that this property does not always hold for P, Q∈ A[X, Y]^q, so that the A–linear map ρ ^{a b}_{c d}

: A[X, Y]^q → A[X, Y]^q is not necessarily anA–algebra homomorphism.

4. Traces and Chebyshev polynomials

4.1. Traces

Traces can misbehave in the noncommutative context. However, we are interested in endomorphisms of A[X, Y]^q_n = A ⊗C[X, Y]^q_n, which have a natural trace.

To emphasize the key property needed, let V be a finite dimensional vector space overC, let A be an algebra overC, and consider an A–linear mapf ∈EndA(A ⊗V). Thetrace off is defined as

Tracef =

n

X

i=0

aii ∈ A

where, if e₀, e₁, . . . , en is a basis for the C-vector space V, the coefficients aij ∈ A are defined by the property that f(ej) =Pn

i=0aijei in A ⊗V for everyj= 0,1, . . . , n.

The usual commutative proof immediately extends to this context to give:

Lemma 4.1. — The trace Tracef ∈ A is independent of the choice of the basise₀, e₁, . . . , e_n for the C-vector space V.

Note that this property would be false if we only required e0, e1, . . . , en

to be a basis for theA–moduleA ⊗V ∼=Aⁿ⁺¹.

For practice, let us carry out a few elementary computations for the representationsρn: SL^q₂(A)→End_A A[X, Y]^q_n

of Lemma 3.1.

For n = 1, consider ^{a b}_{c d}

∈ SL^q₂(A) and its image ρ₁ ^{a b}_{c d}

∈ EndA A[X, Y]^q₁

. The polynomials X, Y form a basis forC[X, Y]^q₁. Since ρ1 a b

c d

(X) = aX +bY and ρ1 a b c d

(Y) = cX+dY we conclude that Traceρ₁ ^{a b}_{c d}

is equal to a+d, namely to what we have implicitly called Trace ^{a b}_{c d}

in the introduction.

Forn= 2, an elementary computation gives ρ₂

a b c d

(X²) = (aX+bY)² =a²X²+ (1 +q²)abXY +b²Y² ρ2

a b c d

(XY) = (aX+bY)(cX+dY) =acX²+ (ad+qbc)XY +bdY² ρ2

a b c d

(Y²) = (cX+dY)² =c²X²+ (1 +q²)cdXY +d²Y² so that

Traceρ₂ a b

c d

=a²+ (ad+qbc) +d²

=a²+ad+da+d²−1

= (a+d)²−1

by remembering thatda−qbc= 1 from the definition of SL^q₂(A).

Forn= 3, a longer but similar first step gives that Traceρ₃

a b c d

=a³+ a²d+q(1 +q²)abc

+ a(1 +q²)bcd+ad² +d³

=a³+a²d+ada+q²ada−a−q²a+dad+q²dad−d−q²d+ad²+d³, using again the property thatda−qbc= 1.

Sinceqbc=da−1 andbca=q²abc, we have thatda²−a=q²ada−q²a.

Similarly, becausedbc =q²bcd, d²a−a=q²dad−q²d. Substituting these values in our first expression for Traceρ₃ ^{a b}_{c d}

gives Traceρ₃

a b c d

=a³+a²d+ada+da²+dad+d²a+ad²+d³−2a−2d

= (a+d)³−2(a+d).

In all three cases, we have been able to express the trace Traceρn a b c d

as a polynomial in Trace ^{a b}_{c d}

=a+d. We will see in Corollary 4.12 that this is a general phenomenon. Part of the purpose in the above calculations was to let the reader experience the fact that this property is not that easy to check by bare-hand computations, which justifies the introduction of the more theoretical constructions of Section 4.3 and Section 4.4 below.

4.2. Chebyshev polynomials

We will encounter two types of Chebyshev polynomials. The(normalized) Chebyshev polynomials of the first kind are the polynomials Tn(t) ∈ Z[t]

recursively defined by

T_n+1(t) =t T_n(t)−T_n−1(t) T1(t) =t

T0(t) = 2.

(4.1)

The(normalized) Chebyshev polynomials of the second kind S_n(t)∈Z[t]

are remarkably similar, and defined by

Sn+1=t Sn(t)−S_n−1(t) S₁(t) =t

S0(t) = 1.

(4.2)

In particular, in addition toT0(t) = 2,S0(t) = 1 andT1(t) =S1(t) =t, T2(t) =t²−2 S2(t) =t²−1

T3(t) =t³−3t S3(t) =t³−2t T4(t) =t⁴−4t²+ 2 S4(t) =t⁴−3t²+ 1 T₅(t) =t⁵−5t³+ 5t S₅(t) =t⁵−4t³+ 3t T₆(t) =t⁶−6t⁴+ 9t²−2 S₆(t) =t⁶−5t⁴+ 6t²−1

Lemma 4.2. — The two types of Chebyshev polynomials are related by the property that

T_n(t) =S_n(t)−S_n−2(t) for everyn>2

Proof. — This is an immediate consequence of the fact that the Tn(t) and Sn(t) satisfy the same linear recurrence relation, and of the initial

conditions.

The following classical properties connect the Chebyshev polynomials Tn(t) andSn(t) to the group SL2(C).

Lemma 4.3. — For every A∈SL2(C), Tn(TraceA) = TraceAⁿ S_n(TraceA) = Traceρ_n(A) whereρn: SL2(C)→End C[X, Y]n

is the(n+ 1)–dimensional representa- tion of Section 2.

Proof. — From the Cayley–Hamilton Theorem (or inspection) A²−(TraceA)A+ Id = 0

for everyA= ^{a b}_{c d}

∈SL2(C). Multiplying both sides byAⁿ⁻¹ and taking the trace, we see that TraceAⁿ satisfies the same recurrence relation

TraceAⁿ⁺¹= (TraceA)(TraceAⁿ)−TraceAⁿ⁻¹

as T_n(TraceA), as well as the same initial values forn = 0 and n = 1. It follows that TraceAⁿ=T_n(TraceA) for everyn.

For the property that Sn(TraceA) = Traceρn(A), which is not needed in this article, we can just refer to the special caseq= 1 of Corollary 4.12

below.

Note the following closed form expression for the Chebyshev polynomial T_n(t)∈Z[t].

Lemma 4.4.

Tn(t) = t+√ t²−4 2

!n

+ t−√ t²−4 2

!n

Proof. — LetA∈SL2(C) be a matrix such that TraceA=t. Ift²−46=

0, the characteristic polynomial λ²−tλ+ 1 of A has two distinct roots

t±√ t²−4

2 , which consequently are the eigenvalues ofA. Then Aⁿ has eigen- values

t±√ t²−4 2

ⁿ

and, by Lemma 4.3, Tn(t) = TraceAⁿ = t+√

t²−4 2

!n

+ t−√ t²−4 2

!n

.

By continuity, the property also holds whent²−4 = 0.

4.3. The Hopf algebras SL^q₂ and Uq(sl2)

For most of the article, we are trying to keep the exposition at an elemen- tary level in order to make the algebra more intuitive and to emphasize its connection with geometry. However, we now need deeper algebraic concepts and constructions, which will enable us to apply the well-known Clebsch–

Gordan Decomposition for Uq(sl2) (Theorem 4.8) to obtain a similar state- ment (Proposition 4.11) for the set SL^q₂(A) of 2-by-2 matrices that we are interested in. We follow here the conventions of [11, Chap. VI–VII].

We already encountered the C–algebra SL^q₂, defined by generators a, b, c, d and by the relations of (3.1)–(3.2). In particular, SL^q₂(A) is the set of homomorphisms from SL^q₂ to the algebraA.

A better known object is the quantum group Uq(sl₂), which is a de- formation of the enveloping algebra of the Lie algebra sl₂(C) of SL₂(C);

see [7, 8, 10, 12]. Recall that U_q(sl₂) is defined by generatorsE,F,K,K⁻¹ and by the relations

KK⁻¹=K⁻¹K= 1 KE=q²EK EF−F E= K−K⁻¹

q−q⁻¹ KF =q⁻²F K (4.3)

This is a Hopf algebra, whose comultiplication ∆ : Uq(sl2) → Uq(sl2)⊗ Uq(sl2), counit ε: Uq(sl2) → C and antipode map S: Uq(sl2) → Uq(sl2) are respectively determined by the properties that

∆(E) =E⊗K+ 1⊗E ε(E) = 0 S(E) =−EK⁻¹

∆(F) =F⊗1 +K⁻¹⊗F ε(F) = 0 S(F) =−KF

∆(K) =K⊗K ε(K) = 1 S(K) =K⁻¹.

Similarly, SL^q₂ is a Hopf algebra with comultiplication ∆ : SL^q₂ →SL^q₂⊗ SL^q₂, counitε: SL^q₂→Cand antipodeS: SL^q₂→SL^q₂ given by

∆(a) =a⊗a+b⊗c ε(a) = 1 S(a) =d

∆(b) =a⊗b+b⊗d ε(b) = 0 S(b) =−qb

∆(c) =c⊗a+d⊗c ε(c) = 0 S(c) =−q⁻¹c

∆(d) =c⊗b+d⊗d ε(d) = 1 S(d) =a.

Whenq= 1, the algebra SL¹₂is just the algebra of regular (= polynomial) functions SL2(C)→Con the algebraic group SL2(C). The comultiplication

∆ : SL¹₂ → SL¹₂⊗SL¹₂ then is the algebra homomorphism induced by the group multiplication SL₂(C)×SL₂(C)→SL₂(C), the counitε: SL¹₂→Cis induced by the map{∗} →SL₂(C) sending∗ to the identity (^{1 0}_{0 1}), and the antipodeS: SL¹₂→SL¹₂ is induced by ^{a b}_{c d}

7→ ^{a b}_{c d}−1

.

Similarly, as q → 1, the quantum group U_q(sl₂) converges to the en- veloping algebra U(sl₂) of the Lie algebra of SL₂(C), by consideration of H= ^K−K_q−q−1⁻¹.

The relationship between SL^q₂ and U_q(sl₂) comes in the form of a linear pairing

h ·,· i: U_q(sl₂)⊗SL^q₂ −→ C

U⊗α 7−→ hU , αi (4.4)

determined by the properties that

hE, ai= 0 hE, bi= 1 hE, ci= 0 hE, di= 0 hF, ai= 0 hF, bi= 0 hF, ci= 1 hF, di= 0 hK, ai=q hK, bi= 0 hK, ci= 0 hK, di=q⁻¹

(4.5)

and

hU, αβi=X

(U)

hU⁰, αihU⁰⁰, βi (4.6) hU V, αi=X

(α)

hU, α⁰ihV, α⁰⁰i (4.7) for everyα,β∈SL^q₂ andU, V ∈U_q(sl₂), using Sweedler’s notation that

∆(U) =X

(U)

U⁰⊗U⁰⁰∈Uq(sl2)⊗Uq(sl2) and ∆(α) =X

(α)

α⁰⊗α⁰⁰∈SL^q₂⊗SL^q₂

for the comultiplications of SL^q₂ and Uq(sl2). See Lemma 4.6 for an inter- pretation of the formulas of (4.5), and see [8, 19, 20] and [11, §VII.4] for details.

In particular, the dualityh ·,· i induces a linear mapδ: SL^q₂ →U_q(sl₂)^∗ from SL^q₂to the dual of U_q(sl₂). We will need the following property.

Lemma 4.5 (Takeuchi [20]). — The above duality map δ: SL^q₂ → Uq(sl2)^∗ is injective.

This is analogous to the property that, because the Lie group SL₂(C) is connected, a regular function on SL2(C) is completely determined by its derivatives and higher derivatives at the identity element.

In general, the representation theory of a Lie algebra is significantly easier to analyze than that of the corresponding Lie group. The same phenomenon in the quantum world is one of the reasons why Uq(sl2) is more popular than SL^q₂.

In particular, there is an action σ: Uq(sl2)⊗C[X, Y]^q → C[X, Y]^q of Uq(sl2) over the quantum planeC[X, Y]^q by “quantum derivation”, defined by the property that

σ(E⊗X^kY^l) = [l]_qX^k+1Y^l−1 σ(F⊗X^kY^l) = [k]_qX^k−1Y^l+1

σ(K⊗X^kY^l) =q^k−lX^kY^l (4.8)

where [k]_q denotes the other type of quantum integer [k]q = q^k−q^−k

q−q⁻¹ =q^k−1+q^k−3+· · ·+q^−k+3+q^−k+1=q^−k+1(k)q². This action restricts to the space of homogeneous polynomials of degree n, and gives an (n+ 1)–dimensional representation

σ_n: U_q(sl₂)→End_C C[X, Y]^q_n for everyn.

Whenq is not a root of unity, the σ_n essentially realize all irreducible finite-dimensional representations of U_q(sl₂), up to isomorphism. To describe all irreducible finite-dimensional representations of U_q(sl₂), one just need one more family of similar representationsσ⁰_n, related toσnby a simple sign twist. See for instance [11, §VI.2].

Similarly, the representationρ: SL^q₂(A)→EndA(A[X, Y]^q) of Section 3 comes from a coaction

τ:C[X, Y]^q →SL^q₂⊗C[X, Y]^q defined by the property that

τ P(X, Y)

=P(a⊗X+b⊗Y, c⊗X+d⊗Y)

for every polynomial P(X, Y) ∈ C[X, Y]^q. Indeed, if A ∈ SL^q₂(A) is considered as an algebra homomorphism A: SL^q₂ → A, then ρ(A) ∈ End_A(A[X, Y]^q) is clearly theA–linear extension of theC–linear map A⊗ Id_C[X,Y]q

◦τ:C[X, Y]^q → A[X, Y]^q.

The following statement relates the duality h ·,· i of (4.4) to the 2–

dimensional representationσ₁: U_q(sl₂)→End_C C[X, Y]^q₁ .

Lemma 4.6. — For every U ∈ Uq(sl₂), the matrix of σ₁(U) ∈ End_C C[X, Y]^q₁

in the basis{X, Y} is σ1(U) =

hU, ai hU, bi hU, ci hU, di

, wherea,b,c,d are the generators ofSL^q₂.

Proof. — The property holds forU =E,F orK^±1by inspection in (4.5), since σ1(E) = (^{0 1}_{0 0}), σ1(F) = (^{0 0}_{1 0}) and σ1(K) = _{q 0}

0q⁻¹

in the basis {X, Y}. It then holds for any product of these generators by combining the fact thatσ1 is an algebra homomorphism, the compatibility of the duality h ·,· i with the multiplications and comultiplications given by (4.7), and the definition of the comultiplication ∆ : SL^q₂→SL^q₂⊗SL^q₂.