Miraculous cancellations for quantum $SL_2$
Francis Bonahon

Fenchel Dual Gradient Methods for Distributed Convex Optimization over Time-varying Networks
    Xuyang Wu and Jie LuX. Wu and J. Lu are with the School of Information Science and Technology, ShanghaiTech University, 201210 Shanghai, China. Email: {wuxy, lujie}@shanghaitech.edu.cn. 
This work has been supported by the National Natural Science Foundation of China under grant 61603254, the Shanghai Pujiang Program under grant 16PJ1406400, and the Natural Science Foundation of Shanghai under grant 16ZR1422500.
    December 7, 2017
===================================================================================================================================================================================================================================================================================================================================================================================================================================




.5in .5in

 Résumé français. Des travaux précédents de Helen Wong et de l'auteur ont mis en évidence, quand le paramètre quantique  q= e^2πiħ 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 surface. L'article ci-dessous fournit une interprétation plus conceptuelle de ce phénomène, en termes de représentations du groupe quantique U_q(𝔰𝔩_2) et de son algèbre de Hopf duale  SL_2^q.
 
 
 
 English 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 parameter q= e^2πiħ 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 U_q(𝔰𝔩_2) and its dual Hopf algebra SL_2^q. 
 
 




The equation 

    (X+Y)^n=X^n + Y^n

is (unfortunately) very familiar to some of our students, who find it convenient to “simplify” computations. However, it is also well-known that this relation does hold in some cases, for instance in a ring of characteristic n with n prime, or when the variables X and Y satisfy the q-commutativity relation that YX=qXY with q∈ a primitive n–root of unity; see <ref>. 

The structure of Equation (<ref>) can be described by considering the two-variable polynomial P(X,Y)=X+Y. Then (<ref>) states that the polynomial P(X,Y)^n, obtained by taking the n–th power of P(X,Y), coincides with the polynomial P(X^n, Y^n) obtained by replacing the variables X, Y with their powers X^n, Y^n, respectively. 

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

    A^n = T_n( A)

for every 2-by-2 matrix A∈() with determinant 1. A typical consequence of the miraculous cancellations discovered in <cit.> is that, when YX=qXY and q is a primitive n–root of unity, 

    T_n(X + Y + X^-1) = X^n + Y^n + X^-n,

which fits the pattern T_n ( P(X,Y) ) = P(X^n, Y^n) for the polynomial P(X,Y)= X + XY + X^-1. The arguments of <cit.> provide many examples of such polynomials, involving several q–commuting variables. 

In <cit.> these “Chebyshev cancellations” were used to connect, when q is a root of unity, irreducible representations of the Kauffman bracket skein algebra (S) of a surface S to group homomorphisms π_1(S) →(). The skein algebra (S) is a purely topological object whose elements are represented by framed links in the thickened surface S× [0,1]. It draws its origin from Witten's interpretation <cit.> 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 . The arguments of <cit.> 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  and () 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  and () is more intuitive if we replace  with its dual Hopf algebra , in the sense of <cit.>. This will enable us to express our constructions solely in terms of 2-by-2 matrices with coefficients in an arbitrary noncommutative algebra ; in <cit.>, the algebra  was the quantum Teichmüller space of the surface. This point of view is sufficiently close to () that it should be relatively intuitive for those mathematicians who have a long track record in hyperbolic geometry, since PSL_2() is the isometry group of the hyperbolic space ℍ^3. 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 <cit.> or through many informal conversations. 

We now state the main result of this article. 



Let A_1, A_2, …, A_k be 2-by-2 matrices with coefficients in an algebra  over , such that:


  * each A_i is triangular of the form 
(
[    a_i    b_i;      0 a_i^-1 ])
or
(
[    a_i      0;    b_i a_i^-1 ])
with b_ia_i = q a_i b_i for some nonzero number q∈-{0};

  * a_i and b_i commute with a_j and b_j whenever i ≠ j. 



Then, if q^2 is a primitive n–root of unity, 

    T_n(  A_1A_2… A_k) =  A_1^(n)A_2^(n)… A_k^(n)

where each A_i^(n) is obtained from A_i by replacing a_i and b_i with their powers a_i^n and b_i^n


This statement is easier to understand if we illustrate it by an example. Consider the product of five triangular matrices

    A= 
    (
    [    a_1    b_1;      0 a_1^-1 ])
    (
    [    a_2    b_2;      0 a_2^-1 ])
    (
    [    a_3      0;    b_3 a_3^-1 ])
    (
    [    a_4    b_4;      0 a_4^-1 ])
    (
    [    a_5      0;    b_5 a_5^-1 ])

where b_ia_i = q a_i b_i, and a_i and b_i commute with a_j and b_j whenever i ≠ j.

Computing the product and taking the trace straightforwardly gives

    A    = a_1a_2a_3a_4a_5 
    +a_1 a_2a_3b_4b_5
    +a_1 b_2b_3a_4a_5
    +a_1 b_2b_3b_4b_5
            +a_1 b_2a_3^-1a_4^-1b_5
    +b_1 a_2^-1b_3a_4a_5
    +b_1 a_2^-1b_3b_4b_5
            +b_1 a_2^-1a_3^-1a_4^-1b_5
    +a_1^-1 a_2^-1b_3b_4a_5^-1
    +a_1^-1 a_2^-1a_3^-1a_4^-1a_5^-1.


Since A has 10 terms and the Chebyshev polynomial T_n(t) has degree n, one would expect T_n( A) to have about 10^n terms. However, when q^2 is a primitive n–root of unity, many cancellations occur and only 10 terms remain. In fact

    T_n( A)    = a_1^n a_2^n a_3^n a_4^n a_5^n  
    +a_1^n  a_2^n a_3^n b_4^n b_5^n 
    +a_1^n  b_2^n b_3^n a_4^n a_5^n 
    +a_1^n  b_2^n b_3^n b_4^n b_5^n 
            +a_1^n  b_2^n a_3^-na_4^-nb_5^n 
    +b_1^n  a_2^-nb_3^n a_4^n a_5^n 
    +b_1^n  a_2^-nb_3^n b_4^n b_5^n 
            +b_1^n  a_2^-na_3^-na_4^-nb_5^n 
    +a_1^-n a_2^-nb_3^n b_4^n a_5^-n
    +a_1^-n a_2^-na_3^-na_4^-na_5^-n
       =  A^(n)

where

    A^(n)= 
    (
    [  a_1^n  b_1^n;      0 a_1^-n ])
    (
    [  a_2^n  b_2^n;      0 a_2^-n ])
    (
    [  a_3^n      0;  b_3^n a_3^-n ])
    (
    [  a_4^n  b_4^n;      0 a_4^-n ])
    (
    [  a_5^n      0;  b_5^n a_5^-n ])

is obtained from A by replacing each a_i, b_i with a_i^n, b_i^n. 



When q is transcendental,  there are only very few  cancellations and Proposition <ref>  shows that T_n( A) is a sum of exactly (5+√(24))^n + (5-√(24))^n >9.89^n monomials.  This explicit count is based on a positivity result (Proposition <ref>) which may be of independent interest. 

Similarly, the example of Equation (<ref>) is provided by applying Theorem <ref> to the matrix A= 
(
[    a_1    b_1;      0 a_1^-1 ])
(
[    a_2      0;    b_2 a_2^-1 ]), setting X=a_1a_2 and Y=b_1b_2, and replacing q with √(q). 

The proof of Theorem <ref> essentially has two parts. The first step is representation theoretic, and connects T_n( A) to the action of the matrix A on the space [X,Y]^q of polynomials in q–commuting variables X and Y and with coefficients in the algebra . This is a relatively easy adaptation to our context of a deep but classical result in the representation theory of the quantum group , the Clebsch-Gordan Decomposition. The second step is a simple computation of traces for this action of A on  [X,Y]^q, which is much simpler than the original arguments of <cit.>.

Among the hypotheses of Theorem <ref>, some are more natural than others. The q–com­mutativity relations b_ia_i = q a_i b_i are essentially mandated by the connection of the objects considered to the quantum group  and its dual Hopf algebra . Similarly, the requirement that the matrices A_i be triangular is deeply tied to the structure of the Lie group () and the quantum group  (and their Borel subgroups/subalgebras). The commutativity hypothesis that a_i and b_i commute with a_j and b_j whenever i ≠ j is less critical, and was introduced here to define A_1A_2… A_k ∈ (and the product matrix A_1A_2… A_k∈()) 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  as well as additional data that is reminiscent of the original topological context. This was implicitly done in <cit.> for the Kauffman bracket skein algebra of a surface, but a quick comparison between the formulas of <cit.> and <cit.> should make it clear that these arguments can  be expanded to a more representation theoretic framework. The cancellations of Theorem <ref> then extend to this generalized setup, as in <cit.>. 





§ THE EQUATION (X+Y)^N=X^N + Y^N



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

    (X+Y)^n    = X^n + n1X^n-1Y+ + n2X^n-2Y^2 + … + nn-1XY^n-1 +Y^n
       = ∑_k=0^n nk X^n-kY^k

where nk is the binomial coefficient

    nk = 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 1≠ 0 for every k<n. It follows that nk=0 whenever 0<k<n, so that the Frobenius relation
holdequation


    (X+Y)^n = X^n + Y^n


holds in this case. 


Note that the hypothesis that the characteristic n is prime is really necessary. For instance, in the ring /4 of characteristic 4, 

    (X+Y)^4 = X^4 + 6X^2Y^2 + Y^4 ≠ X^4 + Y^4.


A less well-known occurrence of Equation (<ref>)  involves variables X and Y that q–commute, in the sense that YX = qXY for some q∈. The Quantum Binomial Formula (see for instance <cit.>) then states that

    (X+Y)^n    = X^n + n1qX^n-1Y+ + n2qX^n-2Y^2 + … + nn-1q XY^n-1 +Y^n
       = ∑_k=0^n nkq X^n-kY^k

where nkq is the quantum binomial coefficient

    nkq = nqn-1qn-2q…n-k+2qn-k+1q/kqk-1qk-2q…2q1q

defined by the quantum integers

    jq = q^j-1/q-1 = 1 +q+q^2 + … +q^j-1.


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


If q is a primitive n–root of unity, in the sense that q^n=1 and q^k≠ 1 whenever 0<k<n, the quantum  binomial coefficient nkq is equal to 0 for every k with 0<k<n. 


Since q is a primitive n–root of unity, nq=q^n-1/q-1=0 while kq=q^k-1/q-1≠0 whenever 0<k<n. The result follows. 



If YX=qXY with q∈ a primitive n–root of unity, the Frobenius relation


    (X+Y)^n = X^n + Y^n


holds. 

As in the characteristic n case, it is necessary that q be a primitive n–root of unity. For instance, when  YX=-XY, q=-1 is a (non-primitive) 4–root of unity and (X+Y)^4 = X^4 + 2X^2Y^2 + Y^4 ≠ X^4 + Y^4. 

We will now discuss generalizations of Equation (<ref>) arising from properties of the quantum group . As indicated in the introduction, we will express these properties in terms of 2-by-2 matrices with coefficients in an algebra . 

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

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




§ THE CLASSICAL ACTION OF () ON [X,Y]



The special linear group of order 2 is the group

    () = {(
    [ a b; c d ]); a, b,c,d ∈, ad-bc=1
    }

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

    [X,Y]    = {polynomial functions on ^2}
       = {polynomials ∑_i=0^m ∑_j=0^n a_ij X^i Y^j; a_ij∈}

of polynomials in the variables X and Y. More precisely, the action of (
[ a b; c d ])∈() on [X,Y] is such that 

    P(X,Y)(
    [ a b; c d ]) = P(aX+bY, cX + dY)

for every polynomial P(X,Y) ∈[X,Y]. 

This defines a map

    ρ() →_( [X,Y] )

from () to the algebra of –linear maps [X,Y] →[X,Y]. 

We collect a few elementary properties in the following lemma. 





  * For every (
[ a b; c d ]) ∈(), ρ(
[ a b; c d ]) ∈_( [X,Y] ) is also an algebra endomorphism of [X,Y].



  * The map  ρ is valued in the group _( [X,Y] ) of linear automorphisms of [X,Y], and induces a group antihomomorphism
ρ() →_( [X,Y] ), in the sense that

    ρ(
    [ a b; c d ][ a' b'; c' d' ])
    =
    ρ[ a' b'; c' d' ]∘ρ[ a b; c d ]

for every 
(
[ a b; c d ]), 
(
[ a' b'; c' d' ])∈(). 



  * If [X,Y]_n = {∑_i+j=n a_ij X^iY^j; a_ij∈}≅^n+1 denotes the vector space of homogeneous polynomials of degree n, the representation ρ restricts to a finite-dimensional representation 

    ρ_n () →_( [X,Y] _n) ≅_(^n+1).





The order reversal in the second conclusion reflects the fact that () acts on [X,Y] on the right. We could easily turn ρ into a group homomorphism by composing it with any of the standard antiautomorphisms of (), such as 
(
[ a b; c d ])
↦(
[ a b; c d ])^t
=
(
[ a c; b d ])
or
(
[ a b; c d ])
↦(
[ a b; c d ])^-1
=
(
[  d -b; -c  a ]), and many authors do  this. For this reason, we will refer to ρ and its restrictions ρ_n as representations of (). 


A classical property is that, up to isomorphism, the ρ_n form the collection of all irreducible representations of ().






§ THE QUANTUM PLANE AND ()



 For a nonzero number q ∈-{0}, the  quantum plane is the algebra [X,Y]^q defined by two generators X and Y, and by the relation YX=qXY. Namely, the elements of [X,Y]^q are polynomials P(X,Y)=∑_i=0^m ∑_j=0^n a_ij X^i X^j that are multiplied using the relation YX=qXY. 

If we want to keep the property that  the matrices (
[ a b; c d ]) and (
[ a c; b d ]) act as algebra homomorphisms on [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 relation YX=qXY. Identifying the coefficients of X^2, XY and Y^2, this would require

    ba    = qab 
            
     db   = qbd 
        
     bc   =cb 
    
     ca    = qac
        
     dc   =qcd
           
    ad- q^-1 bc    = da - qbc

which is clearly impossible if q≠ 1 and a, b, c, d commute with each other. This leads us to the following definition (see for instance <cit.> for more background).



Given an algebra 𝒜 over , let () denote the set of matrices
(
[ a b; c d ])
where a, b, c, d∈𝒜 satisfy the relations of (<ref>), as well as 

    ad- q^-1bc =1.




More formally, let  be the algebra defined by generators a, b, c, d and by the relations of (<ref>) and (<ref>). Then () can be interpreted as the set of all algebra homomorphisms ^q →. 
The elements of () are called the 𝒜–points of . 


Unlike (), the set () is far from being a group. It only comes with a partially defined multiplication. Indeed, if 
(
[ a b; c d ]) and (
[ a' b'; c' d' ]) ∈(), the usual formula

    (
    
     a   b
    c   d
    )
    (
    
     a'   b'
    c'   d'
    )
    =
    (
    
     aa'+bc'   ab'+bd'
    ca'+ dc'   cb' + dd'
    )

gives an element of () only under additional hypotheses on the entries of these matrices, for instance if a, b, c, d commute with a', b', c', d'. This partially defined multiplication has an identity element 
(
[ 1 0; 0 1 ]) ∈(). 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 ]) ∈() is an element of ^q^-1()=(^) instead of (); here ^ is the opposite algebra of the algebra , consisting of the vector space  endowed with the new multiplication *_ defined by the property that a*_ b=ba for every a, b∈. 
In general, the formula (<ref>) gives a globally defined  multiplication () ⊗(ℬ) →(⊗ℬ) for any two algebras  and ℬ. 

In order to generalize to ()  the action of () on [X,Y], we introduce the  quantum –plane as the algebra [X,Y]^q= ⊗[X,Y]^q. In practice the elements of [X,Y]^q are polynomials P(X,Y)=∑_i=0^m ∑_j=0^n a_ij X^i X^j with coefficients a_ij∈, and are algebraically manipulated using the relation YX=qXY while the variables X and Y commute with all elements of .   


The relations defining () are specially designed that an 𝒜–point of  acts as an algebra homomorphism on the quantum –plane [X,Y]^q, by the same formula (<ref>) as in the commutative plane. More precisely, if _ ([X,Y]^q) denotes the space of –linear maps [X,Y]^q →[X,Y]^q, we can define a map 

    ρ() →_ ([X,Y]^q)

such that 

    ρ(
    [ a b; c d ]) (∑_i,ja_ij X^i Y^j ) = ∑_i,ja_ij (aX+bY)^i (cX + dY)^j

for every polynomial ∑_i,ja_ijX^iY^j ∈_q[X,Y]. 

The map ρ satisfies the following elementary properties. 






  * For every ([ a b; c d ])∈(), the –linear map
ρ([ a b; c d ]) ∈_ ([X,Y]^q)
is an algebra homomorphism.


  * For each n, the map ρ([ a b; c d ]) ∈_ ([X,Y]^q) respects the space [X,Y]_n^q = ⊗[X,Y]_n^q of homogeneous polynomials of degree n in X and Y with coefficients in . As a consequence,  ρ induces by restriction  a map 

    ρ_n() →_ ([X,Y]^q_n).




  * If a, b, c, d commute with  a', b', c', d' (so that the product ([ a b; c d ]) ([ a' b'; c' d' ]) = ([   aa'+bc' ab' + bd'; ca' + dc'   cb'+dd' ]) ∈() makes sense), 

    ρ(
    [ a b; c d ][ a' b'; c' d' ])
    =
    ρ(
    
     a'   b'
    c'   d'
    )
    ∘ρ(
    
     a   b
    c   d
    ).











§ TRACES AND CHEBYSHEV POLYNOMIALS




 §.§ Traces


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

To emphasize the key property needed, let V be a finite dimensional vector space over , let  be an algebra over , and consider an –linear map f ∈_(⊗ V). The trace of f is defined as

    f = ∑_i=0^n a_ii∈

where e_0, e_1, …, e_n is a basis for the -vector space V, and where the coefficients a_ij∈ are defined by the property that f(e_j) = ∑_i=0^n a_ije_i in ⊗ V for every j=0, 1, …, n. 

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


The trace f  ∈ is independent of the choice of the basis  e_0, e_1, …, e_n  for the -vector space V. 

Note that this property would be false if we only required e_0, e_1, …, e_n to be a basis for the –module ⊗ V ≅^n+1. 


For practice, let us carry out a few elementary computations for the representations ρ_n () →_([X,Y]_n^q) of Lemma <ref>. 

For n=1, consider ([ a b; c d ])∈() and its image ρ_1 ([ a b; c d ])∈_([X,Y]_1^q). The polynomials X, Y form a basis for [X,Y]_1^q. Since ρ_1([ a b; c d ])(X) = aX + bY and  ρ_1([ a b; c d ])(Y) = cX + dY we conclude that  ρ_1([ a b; c d ]) is equal to a+d, namely to what we have implicitly called ([ a b; c d ]) in the introduction. 

For n=2,  an elementary computation gives

    ρ_2 ([ a b; c d ]) (X^2)
       = (aX+bY)^2
          = a^2 X^2 + (1+q^2) abXY + b^2 Y^2
    ρ_2 ([ a b; c d ]) (XY)
       = (aX+bY)(cX+dY)
     -50pt
          =ac X^2 + (ad+qbc) XY + bd Y^2
    ρ_2 ([ a b; c d ]) (Y^2)
       = (cX+dY)^2 
          =c^2 X^2 + (1+q^2) cdXY + d^2 Y^2

so that

    ρ_2 ([ a b; c d ]) 
       = a^2 + (ad+qbc) + d^2
       = a^2 + ad + da +d^2 -1
       = (a+d)^2-1

by remembering that da-qbc=1 from the definition of (). 

For n=3, a longer but similar first step gives that

    ρ_3 ([ a b; c d ]) 
       = a^3 + ( a^2d+q(1+q^2) abc ) + ( a(1+q^2) bcd + ad^2 ) + d^3
       = a^3 + a^2d + ada + q^2 ada -a -q^2a + dad + q^2 dad - d -q^2d + ad^2 +d^3,

using again the property that da-qbc=1.

Since qbc=da-1 and bca=q^2 abc, we have that da^2-a=q^2ada-q^2a. Similarly, because dbc=q^2 bcd, d^2a-a = q^2dad-q^2d. Substituting these values in our first expression for ρ_3 ([ a b; c d ]) gives

    ρ_3 ([ a b; c d ]) 
       = a^3 + a^2d + ada +  da^2 + dad +  d^2a + ad^2 +d^3-2a-2d
       = (a+d)^3-2(a+d).


In all three cases, we have been able to express the trace ρ_n ([ a b; c d ]) as a polynomial in ([ a b; c d ]) =a+d. We will see in Corollary <ref> 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 <ref> and <ref> below. 



 §.§ Chebyshev polynomials
 


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

    T_n+1(t)    = t T_n(t) - T_n-1(t)
    
    T_1(t)    =t  
    
    T_0(t)   =2.


The (normalized) Chebyshev polynomials of the second kind S_n(t) ∈ℤ[t] are remarkably similar, and defined by 

    S_n+1   = t S_n(t) - S_n-1(t)
    
    S_1(t)    =t  
    
    S_0(t)   =1.


In particular, in addition to T_0(t)=2, S_0(t) =1 and T_1(t) = S_1(t)=t, 

    T_2(t)    = t^2-2     S_2(t)    = t^2-1
    
    T_3(t)    = t^3 -3t     S_3(t)    = t^3-2t
    
    T_4(t)    = t^4-4t^2+2     S_4(t)    = t^4-3t^2 +1
    
    T_5(t)    = t^5-5t^3 +5t     S_5(t)    = t^5-4t^3 +3t
    
    T_6(t)    = t^6-6t^4 +9t^2 -2     S_6(t)    = t^6-5t^4 + 6t^2 -1




The two types of Chebyshev polynomials are related by the property that

    T_n(t) = S_n (t) - S_n-2(t)

for every n≥ 2


This is an immediate consequence of the fact that the T_n(t) and S_n(t) satisfy the same linear recurrence relation, and of the initial conditions. 



The following classical properties connect the Chebyshev polynomials T_n(t) and S_n(t) to the group (). 


For every A∈(),

    T_n ( A)    =  A^n 
    
    S_n ( A)   = ρ_n(A)

where  ρ_n () →( [X,Y]_n ) is the (n+1)–dimensional representation of <ref>. 



From the Cayley-Hamilton Theorem (or inspection)

    A^2 - ( A )A+Id =0

for every A=(
[ a b; c d ]) ∈(). Multiplying both sides by A^n-1 and taking the trace, we see that A^n satisfies the same recurrence relation

    A^n+1 = ( A) ( A^n) -  A^n-1

as T_n( A), as well as the same initial values for n=0 and n=1. It follows that A^n = T_n( A) for every n.

For the property that S_n ( A)= ρ_n(A), which is not needed in this article, we can just refer to the special case q=1 of Corollary <ref> below. 



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




    T_n(t) =( t+√(t^2-4)/2)^n + ( t-√(t^2-4)/2)^n



Let A ∈() be a matrix such that A=t. 
If t^2-4 ≠ 0, the characteristic polynomial λ^2 -tλ+1 of A has two distinct roots t±√(t^2-4)/2, which consequently are the eigenvalues of A. Then A^n has eigenvalues (t±√(t^2-4)/2)^n and, by Lemma <ref>,

    T_n(t) = A^n = ( t+√(t^2-4)/2)^n + ( t-√(t^2-4)/2)^n.


By continuity, the property also holds when  t^2-4 = 0. 






 §.§ The Hopf algebras  and 




 For most of the article, we are trying to keep the exposition at an elementary 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  (Theorem <ref>) to obtain a similar statement (Proposition <ref>) for the set () of  2-by-2 matrices that we are interested in. 
 We follow here the conventions of <cit.>. 

We already encountered the –algebra , defined by generators a, b, c, d and by the relations of (<ref>–<ref>). In particular, () is the set of  homomorphisms from  to the algebra . 

A better known object is the quantum group , which is a deformation of the enveloping algebra of the Lie algebra 𝔰𝔩_2() of (); see <cit.>. Recall that  is defined by generators E, F, K, K^-1 and by the relations

    KK^-1   =K^-1K=1       KE   =q^2EK
    
     EF-FE   =K-K^-1/q-q^-1   KF   =q^-2FK

This is a Hopf algebra, whose comultiplication Δ→⊗, counit ϵ→ and antipode map S → are respectively determined by the properties that

    Δ(E)    = E⊗ K + 1⊗ E    ϵ(E)    =0    
      S(E)   = -EK^-1
    Δ(F)    = F⊗ 1 + K^-1⊗ F    ϵ(F)    =0    
       S(F)   = -KF
     
    Δ(K)    = K ⊗ K   ϵ(K)    =1    
     S(K)   = K^-1.


Similarly,   is a Hopf algebra with comultiplication Δ→⊗, counit ϵ→ and antipode  S → given by

    Δ(a)    = a ⊗ a + b⊗ c    ϵ(a)   = 1    
      S(a)   = d 
     
    Δ(b)    = a⊗ b + b⊗ d    ϵ(b)   = 0    
       S(b)   = -qb
    
    Δ(c)    = c ⊗ a + d ⊗ c    ϵ(c)   = 0    
        S(c)   = -q^-1c 
    
    Δ(d)    = c ⊗ b + d ⊗ d    ϵ(d)   = 1    
        S(d)   =a.


When q=1, the algebra ^1 is just the algebra of regular (= polynomial) functions () → on the algebraic group (). The comultiplication Δ^1 →^1 ⊗^1 then is the algebra homomorphism induced by the group multiplication () ×() →(), the counit ϵ^1 → is induced by the map {*}→() sending * to the identity ( 
[ 1 0; 0 1 ]),
and the  antipode  S ^1 →^1 is induced by ( 
[ a b; c d ])
↦( 
[ a b; c d ])^-1. 

Similarly, as q→ 1, the quantum group  converges to the enveloping algebra U(𝔰𝔩_2) of the Lie algebra of (), by consideration of H=K-K^-1/q-q^-1. 




The relationship between  and  comes in the form of a linear pairing

    ⟨  , ⟩   ⊗      ⟶      
    
     U    ⊗α      ⟼   ⟨ U   , α⟩

determined by the properties that

    ⟨ E,a ⟩   =0      ⟨ E,b ⟩   =1      ⟨ E,c ⟩   =0      ⟨ E,d ⟩   =0   
    ⟨ F,a ⟩   =0      ⟨ F,b ⟩   =0      ⟨ F,c ⟩   =1      ⟨ F,d ⟩   =0   
    ⟨ K,a ⟩   =q      ⟨ K,b ⟩   =0      ⟨ K,c ⟩   =0      ⟨ K,d ⟩   =q^-1

and 

    ⟨ U, αβ⟩   = ∑_(U)⟨ U',α⟩⟨ U” ,β⟩
    ⟨ UV, α⟩   = ∑_(α)⟨ U, α' ⟩⟨ V, α”⟩

for every α, β∈ and U, V∈, using Sweedler's notation that

    Δ(U)    = ∑_(U) U' ⊗ U”∈⊗
    and Δ(α)    = ∑_(α)α' ⊗α”∈⊗

for the comultiplications of  and . See Lemma <ref> for an interpretation of the formulas of (<ref>), and see <cit.>  and <cit.> for  details. 

In particular, the duality ⟨ ,  ⟩ induces a linear map δ→^* from  to the dual of . We will need the following property. 


 
 The above duality map δ→^* is injective. 

This is analogous to the property that, because the Lie group () is connected, a regular function on () 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  is more popular than . 

In particular, there is an action 
σ⊗[X,Y]^q →[X,Y]^q of  over the quantum plane [X,Y]^q by “quantum derivation”, defined by the property that 

    σ(E ⊗ X^kY^l)    = lq X^k+1 Y^l-1    σ(F ⊗ X^kY^l) = kq X^k-1 Y^l+1
    σ(K ⊗ X^kY^l)    = q^k-l X^k Y^l

where kq denotes the other type of quantum integer 

    kq = q^k - q^-k/q-q^-1 = q^k-1 + q^k-3 + … + q^-k+3 + q ^-k+1= q^-k+1kq^2.


This action restricts to the space of homogeneous polynomials of degree n, and gives an (n+1)–dimensional representation

    σ_n →_( [X,Y]_n^q )

for every n. 

When q is not a root of unity, the σ_n essentially realize all irreducible finite-dimensional representations of , up to isomorphism. To describe all irreducible finite-dimensional representations of , one just need one more  family of similar representations σ_n', related to σ_n by a simple sign twist. See for instance <cit.>. 

Similarly, the representation ρ() →_ ([X,Y]^q) of <ref> comes from a coaction

    τ[X,Y]^q →⊗[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) ∈[X,Y]^q. Indeed, if A ∈() is considered as an algebra homomorphism A →, then ρ(A) ∈_ ([X,Y]^q) is clearly the –linear extension of the –linear map (A ⊗_[X,Y]^q) ∘τ[X,Y]^q →[X,Y]^q. 

The following statement relates the duality ⟨ , ⟩ of (<ref>) to the 2–dimensional representation σ_1→_( [X,Y]_1^q). 



 For every U∈, the matrix of σ_1(U) ∈_( [X,Y]_1^q) in the basis { X, Y} is 
 
    σ_1(U) = 
    [ ⟨ U,a⟩ ⟨ U,b⟩; ⟨ U,c⟩ ⟨ U,d⟩ ],

 where a, b, c, d are the generators of . 



 The property holds for U=E, F or K^±1 by inspection in (<ref>), since σ_1(E) = (
[ 0 1; 0 0 ]), σ_1(F) = (
[ 0 0; 1 0 ]) and σ_1(K) = (
[    q    0;    0 q^-1 ]) in the basis {X,Y}. It then holds for any product of these generators by combining the fact that σ_1 is an algebra homomorphism, the compatibility of the duality ⟨ , ⟩ with the multiplications and comultiplications given by (<ref>), and the definition of the comultiplication Δ→⊗.


We now show that the duality ⟨ , ⟩ connects the action of  on [X,Y]^q to the coaction of . 

To see this, we first rewrite the action σ⊗[X,Y]^q →[X,Y]^q as a linear map Σ[X,Y]^q →^* ⊗[X,Y]^q. 



 The action Σ[X,Y]^q →^* ⊗[X,Y]^q, coaction τ[X,Y]^q →⊗[X,Y]^q and duality map δ→^* are related by the property that the diagram
 
    [X,Y]^q  
      @/^20pt/[rrr]^Σ[r]_τ       ⊗[X,Y]^q [rr]_δ⊗_[X,Y]^q      ^* ⊗[X,Y]^q

is commutative. Namely,
Σ=(δ⊗_[X,Y]^q) ∘τ.



The property is equivalent to the commutativity of the diagram

    60pt⊗[X,Y]^q  
      @/^20pt/[rr]^σ[r]__⊗τ       ⊗⊗[X,Y]^q
       [r]_        ⟨  , ⟩⊗_[X,Y]^q   [X,Y]^q

 
 To simplify the formulas, write
ϕ = _⊗τ and ψ= ⟨  , ⟩⊗_[X,Y]^q.
 We need to prove that 

    ψ∘ϕ (U⊗ P(X,Y) ) = σ (U⊗ P(X,Y) )

for every U∈ and P(X,Y)∈[X,Y]^q. 

We first consider the case where P(X,Y)=X. Then,

    ψ∘ϕ  (U⊗ X)
         = ψ (U ⊗ a ⊗ X + U ⊗ b ⊗ Y)
       =   ⟨ U , a ⟩  X + ⟨ U , b ⟩  Y
       = σ_1(U)(X) = σ(U⊗ X)

by Lemma <ref>. So, the property of (<ref>) holds for P(X,Y)=X and every U∈. 

An almost identical argument shows that (<ref>) holds for P(X,Y)=Y and every U∈. As a consequence,  (<ref>) holds for every P(X,Y) ∈[X,Y]_1^q and  U∈.

We now claim that, if  (<ref>) holds for P(X,Y) and Q(X,Y) and every U∈, then it holds for the product P(X,Y)Q(X,Y) and every U∈. 

A fundamental property of the action σ is that, in the terminology of <cit.>,  it makes [X,Y]^q a module algebra over . This means that, in addition to making [X,Y]^q a module over the algebra ,  σ satisfies the “quantum product rule” that

    σ (U⊗ P(X,Y)Q(X,Y) ) 
    = ∑_(U)σ (U'⊗ P(X,Y) ) σ (U”⊗ Q(X,Y) ) ,

using Sweedler's notation that Δ(U) = ∑_(U) U' ⊗ U”. See  <cit.>.

Now, 

    ψ∘ϕ (U⊗ P(X,Y)Q(X,Y) )
         = ψ(U ⊗ P(a ⊗ X +  b ⊗ Y ) Q(a ⊗ X +  b ⊗ Y ) )
       =  ∑_(U)ψ(U' ⊗ P(a ⊗ X +  b ⊗ Y ) ) ψ(U”⊗  Q(a ⊗ X +  b ⊗ Y ) )
       = ∑_(U)σ (U'⊗ P(X,Y) ) σ (U”⊗ Q(X,Y) ) 
       = σ (U⊗ P(X,Y)Q(X,Y) ) ,

where the second equality comes from (<ref>), the third equality reflects our hypothesis that P(X,Y) and Q(X,Y) satisfy (<ref>) for every U”'∈, and the fourth equality results from (<ref>). This proves our claim that (<ref>) holds for P(X,Y)Q(X,Y) and for every U∈. 


This inductive step proves that (<ref>) holds in all cases, and concludes the proof of Lemma <ref>. 




 §.§ The Clebsch-Gordan Decomposition for  and ()



 A great feature of Hopf algebras is that their comultiplication Δ enables one to take the tensor product of two representations. 

For , the Quantum Clebsch-Gordan Decomposition expresses the action of  on the tensor product [X,Y]^q_m ⊗[X,Y]^q_n as a direct sum of (irreducible) representations over [X,Y]^q_m+n-2k with 0≤ k ≤inf{m,n}. See <cit.>, and <cit.>. 

We state here the result for the case we need, when m=1.  Recall that the action of  on [X,Y]^q restricts to an algebra homomorphism σ_n →_( [X,Y]_n^q ) for every n.  Also, the tensor product 
 
    σ_1 ⊗σ_n →_(  [X,Y]^q_1 ⊗[X,Y]^q_n ) = _(  [X,Y]^q_1)  ⊗_(  [X,Y]^q_n )

  is defined by the property that 
  
    σ_1 ⊗σ_n(U) = ∑_(U)σ_1(U') ⊗σ_n(U”)

  for every U∈, using Sweedler's notation that Δ(U) = ∑_(U) U' ⊗ U”. 


 
 When q is not a k–root of unity with k≤ n, there exists a –linear isomorphism ϕ[X,Y]^q_1 ⊗[X,Y]^q_n →[X,Y]^q_n+1⊕[X,Y]^q_n-1 such that the diagram
 
    [X,Y]^q_1 ⊗[X,Y]^q_n 
     [rrr]^σ_1 ⊗σ_n(U)[d]_ϕ^≅         [X,Y]^q_1 ⊗[X,Y]^q_n 
     [d]^ϕ_≅
    [X,Y]^q_n+1⊕[X,Y]^q_n-1[rrr]_σ_n+1(U) ⊕σ_n-1(U)         [X,Y]^q_n+1⊕[X,Y]^q_n-1

 commutes for every U∈. 
 

 
 We now consider the coaction τ[X,Y]^q →⊗[X,Y]^q, and more precisely its restriction τ_n [X,Y]^q_n →⊗[X,Y]^q_n. 
 
 Tensor products of coactions are much simpler to define, and
 
    τ_1 ⊗τ_n  (P⊗ Q ) =  τ_1 (P) ⊗τ_n(Q) ∈⊗[X,Y]^q_1 ⊗[X,Y]^q_n

for every P ∈[X,Y]^q_1 and Q∈[X,Y]^q_n.

We now use the duality ⟨ , ⟩ to deduce the following result from Theorem <ref>. 


  
  Suppose that  q is not a k–root of unity with k≤ n, and consider 
 the –linear isomorphism ϕ[X,Y]^q_1 ⊗[X,Y]^q_n →[X,Y]^q_n+1⊕[X,Y]^q_n-1 of Theorem <ref>. Then,  the diagram
 
    [X,Y]^q_1 ⊗[X,Y]^q_n 
     [rr]^τ_1 ⊗τ_n  [d]_ϕ^≅      ⊗[X,Y]^q_1 ⊗[X,Y]^q_n 
     [d]^_⊗ϕ_≅
    [X,Y]^q_n+1⊕[X,Y]^q_n-1[rr]_τ_n+1⊕τ_n-1        ⊗[X,Y]^q_n+1⊕[X,Y]^q_n-1

 commutes, in the sense that (_⊗ϕ) ∘ (τ_1 ⊗τ_n) = (τ_n+1⊕τ_n-1) ∘ϕ


 To simplify the notation, set V_k = [X,Y]^q_k. Then the commutative diagram of Lemma <ref> restricts for each k to a commutative diagram 

    V_k  
      @/^20pt/[rrrr]^Σ_k[rr]_τ_k      ⊗  V_k [rr]_δ⊗_V_k      ^*⊗ V_k

 where Σ_k is related to the action σ_k →_(V_k)  by the property that, for each U∈ and P ∈ V_k, the element σ_k(U)(P) ∈ V_k is obtained by evaluating Σ_k(P) ∈^* ⊗ V_k at U. 

In the diagram 

    V_1 ⊗ V_n 
    @/^20pt/[rrrr]^Σ_1 ⊗Σ_n[rr]_τ_1 ⊗τ_n  [d]_ϕ      ⊗ V_1 ⊗ V_n 
    [rr]_δ⊗_V_1 ⊗ V_n  [d]^_⊗ϕ      ^* ⊗ V_1 ⊗ V_n 
    [d]^_^*⊗ϕ
    
    V_n+1⊕ V_n-1@/_20pt/[rrrr]_Σ_n+1⊕Σ_n-1[rr]^τ_n+1⊕τ_n-1        ⊗ (V_n+1⊕ V_n-1 )
    [rr]^δ⊗_V_n+1⊕ V_n-1        ^* ⊗ (V_n+1⊕ V_n-1 )

we want to show that the left-hand square commutes. Here,  Σ_1 ⊗Σ_n is defined  to be related to the tensor product σ_1 ⊗σ_n →_(V_1 ⊗ V_n) by the property that, for each U∈, P_1 ∈ V_1 and P_n ∈ V_n, the element σ_1 ⊗σ_n(U)(P_1 ⊗ P_n) ∈ V_1 ⊗ V_n is obtained by evaluating Σ_1 ⊗Σ_n(P_1 ⊗ P_n) ∈^* ⊗ V_1 ⊗ V_n at U. The map Σ_n+1⊕Σ_n-1 is similarly associated to the direct sum  σ_n+1⊕σ_n-1→_(V_n+1⊕ V_n-1).

Because of the way Σ_1 ⊗Σ_n and Σ_n+1⊕Σ_n-1 are respectively associated to σ_1 ⊗σ_n and   σ_n+1⊕σ_n-1, Theorem <ref> shows that the outer rectangle of  (<ref>) commutes, in the sense that

    (_^*⊗ϕ) ∘ (Σ_1 ⊗Σ_n) = (Σ_n+1⊕Σ_n-1) ∘ϕ.



The lower triangle 

    V_n+1⊕ V_n-1@/_20pt/[rrrr]_Σ_n+1⊕Σ_n-1[rr]^τ_n+1⊕τ_n-1          ⊗ (V_n+1⊕ V_n-1 )
    [rr]^δ⊗_V_n+1⊕ V_n-1        ^* ⊗ (V_n+1⊕ V_n-1 )

of (<ref>) commutes by an immediate application of (<ref>). 

The commutativity of the upper triangle requires more thought, because  tensor products of actions of algebras are more complicated than direct sums. 




 The diagram
 
    V_1 ⊗ V_n 
    @/^20pt/[rrrr]^Σ_1 ⊗Σ_n[rr]_τ_1 ⊗τ_n        ⊗ V_1 ⊗ V_n 
    [rr]_δ⊗_V_1 ⊗ V_n        ^* ⊗ V_1 ⊗ V_n

 commutes.


 Because of the relationships between Σ_1 ⊗Σ_n and the action σ_1 ⊗σ_n, and between the map δ→^* and the duality ⟨ ,  ⟩⊗→, it suffices to show that
 
    V_1 ⊗ V_n 
    @/^20pt/[rrrr]^σ_1 ⊗σ_n(U)[rr]_τ_1 ⊗τ_n        ⊗ V_1 ⊗ V_n 
    [rr]_    ⟨ U,  ⟩           V_1 ⊗ V_n

commutes for every U∈. Here, for a vector space V, we shorten the notation and  write ⟨ U,  ⟩⊗ V → V for the map that we previously denoted by ⟨ U,  ⟩⊗_V. 

This property is an immediate consequence of the fact (<ref>–<ref>) that ⟨ ,  ⟩ establishes a duality between multiplications and comultiplications. 
Indeed, given two polynomials P_1 ∈ V_1 and P_n ∈ V_n,

    σ_1 ⊗σ_n(U)(P_1 ⊗ P_n)    = ∑_(U)σ_1(U')(P_1) ⊗σ_n(U”) (P_n)
       =  ∑_(U)⟨ U', τ_1(P_1)⟩⊗⟨ U”, τ_n(P_n) ⟩
       = ⟨ U, τ_1(P_1) ⊗τ_n(P_n) ⟩

 where the first equality reflects the definition of σ_1 ⊗σ_2, the second equality comes from Lemma <ref> (or (<ref>)), and the third equality follows from (<ref>). 
 
 The proves the commutativity of (<ref>), and therefore Lemma <ref>. 

 
 We are now ready to conclude the proof of Proposition <ref>. We proved that, in the diagram (<ref>), the outer rectangle and the two upper and lower triangles commute. By Lemma <ref>, the map δ→^* is injective. It easily follows that the left-hand square
 
    V_1 ⊗ V_n 
    [rr]_τ_1 ⊗τ_n  [d]^ϕ      ⊗ V_1 ⊗ V_n 
    [d]^_⊗ϕ
    
    V_n+1⊕ V_n-1[rr]^τ_n+1⊕τ_n-1        ⊗ (V_n+1⊕ V_n-1 )

 commutes. This is exactly what we needed to prove. 



After this long digression through the Hopf algebras  and , we now return to our original topic of interest, namely the set () of –points of  for some algebra . 

The representation ρ_n() →_ ([X,Y]^q_n) is related to the coaction
τ_n [X,Y]_n^q →⊗[X,Y]_n^q 
by the property that, if an –point A ∈() is considered as an algebra homomorphism A →, then ρ_n(A) ∈_ ([X,Y]_n^q) is  the –linear extension of the –linear map (A ⊗_[X,Y]_n^q) ∘τ[X,Y]_n^q →[X,Y]_n^q. 



 When q is not a k–root of unity with k≤ n, the representation 
    ρ_1 ⊗_ρ_n () →_( [X,Y]_1^q ⊗_[X,Y]_n^q )
 is isomorphic  over  to the direct sum ρ_n+1⊕ρ_n-1 of the representations ρ_n+1() →(  [X,Y]_n+1^q ) and ρ_n-1() →(  [X,Y]_n-1^q ). Namely, there exists a –linear isomorphism

    ϕ[X,Y]_1^q ⊗_[X,Y]_n^q  →[X,Y]_n+1^q  ⊕[X,Y]_n-1^q  ,

inducing an –linear isomorphism

    _⊗_ϕ[X,Y]_2^q ⊗_[X,Y]_n^q →[X,Y]_n+1^q  ⊕[X,Y]_n-1^q   ,

such that the diagram

    [X,Y]_1^q ⊗_[X,Y]_n^q 
     [d]__⊗_ϕ^≅[rrr]^ρ_1(A) ⊗_ρ_n(A)         [X,Y]_1^q ⊗_[X,Y]_n^q 
     [d]^_⊗_ϕ_≅
    [X,Y]_n+1^q  ⊕[X,Y]_n-1^q
     [rrr]_ρ_n+1(A) ⊕ρ_n-1(A)         [X,Y]_n+1^q  ⊕[X,Y]_n-1^q

commutes for every A∈(). 



 This is an immediate consequence of Proposition <ref> and of the relationship between the coactions τ_k [X,Y]_k^q →⊗_[X,Y]_k^q and the –linear maps  ρ_k(A) ∈_ ([X,Y]_k^q). 


The fact that the isomorphism [X,Y]_1^q ⊗_[X,Y]_n^q  →[X,Y]_n+1^q  ⊕[X,Y]_n-1^q between ρ_1 ⊗_ρ_n and ρ_n+1⊕ρ_n-1  comes from a –linear isomorphism  [X,Y]_1^q ⊗_[X,Y]_n^q  →[X,Y]_n+1^q  ⊕[X,Y]_n-1^q  will be crucial for our consideration of traces in the next section. 



 §.§ The trace of ρ_n(A) for A∈()
 After the hard work of <ref> and <ref>, we now have appropriate tools to compute for A∈() the trace of ρ_n(A) in terms of the trace of A. The few computations that we did at the end of <ref> should convince the reader that this result would be hard to obtain without the heavy machinery of <ref>–<ref>. 



For every (
[ a b; c d ]) 
∈(),

    ρ_n
    (
    [ a b; c d ]) 
    =
    S_n (a+d)

where S_n is the n–th Chebyshev polynomial of the second kind. 

 The property makes sense for all q, but we first restrict attention to the case where q is not a root of unity in order to apply Proposition <ref>. 

Consider A= (
[ a b; c d ]) ∈(). By Proposition <ref>, the –linear maps  ρ_1(A) ⊗_ρ_n(A) and ρ_n+1(A) ⊕ρ_n-1(A) are isomorphic over . By Lemma <ref>, they consequently have the same trace. 
Therefore,

    ( ρ_1(A))( ρ_n(A))    = (ρ_1(A) ⊗_ρ_n(A))
       = (ρ_n+1(A) ⊕ρ_n-1(A) )
       = (ρ_n+1(A)) + ( ρ_n-1(A))

and ρ_n(A) therefore satisfies the same recurrence relation (<ref>–<ref>) as the Chebyshev polynomials. 

For n=0, ρ_0(A)=_ and ρ_0(A)=1. By definition of the Chebyshev polynomial S_n(t) in (<ref>), it follows that 

    ρ_n(A) = S_n( ρ_1(A) ).


We already computed ρ_1(A) in <ref>. If A= (
[ a b; c d ]), then ρ_1(A) has matrix (
[ a c; b d ]) in the –basis {X,Y} for [X,Y]_1^q = ⊗[X,Y]_1^q. (Note that this is the transpose matrix.) It follows that ρ_1(A) = a+d, which concludes our computation when q is not a root of unity. 

The case where q is a root of unity follows from this generic case by continuity. To justify this continuity argument for an arbitrary algebra , first consider the case when =, where we can make sense of continuity with respect to q (for instance by considering  as an algebra over [q, q^-1]). The algebra  admits a tautological –point I ∈(), defined by the identity algebra homomorphism I →. Then, ρ_n(I) = S_n(  I ) ∈ for every q by continuity from the case where q is not a root of unity. 

For a general algebra  and an –point A ∈(), a little thought will convince the reader that  ρ_n(A)∈ is the image of ρ_n(I)  ∈ under the algebra homomorphism A →; in particular, A = A(  I) by specialization to the case n=1. Then, 

    ρ_n(A)    = A ( ρ_n(I) ) = A ( S_n(  I ) )
       = S_n ( A(  I ) ) = S_n ( A)

for every q, using the fact that A is an algebra homomorphism for the third equality. 






§ MIRACULOUS CANCELLATIONS



We now prove the main result of this article, namely Theorem <ref> which we rephrase a Theorem <ref> below. Although we just encountered Chebyshev polynomials S_n(t) of the second kind, the property involves the Chebyshev polynomials T_n(t) of the first kind. 

Note that, if an –point A=(
[ a b; c d ]) ∈() of  is upper triangular, namely is such that c=0, then necessarily d=a^-1 by the quantum determinant relation ad-q^-1bc=1 of Relation (<ref>). As a consequence, A can be written as A=(
[    a    b;    0 a^-1 ]) with ba=qab. Similarly, any lower triangular element of () is of the form (
[    a    0;    b a^-1 ]) with ba=qab. 



Let A_1, A_2, …, A_k∈() be 𝒜–points of  such that:


  * each A_i is triangular of the form 
(
[    a_i    b_i;      0 a_i^-1 ])
or
(
[    a_i      0;    b_i a_i^-1 ]) for some a_i, b_i∈ 
(with b_ia_i = q a_i b_i);

  * a_i and b_i commute with a_j and b_j whenever i ≠ j, so that we can make sense of  the product A_1A_2 … A_n ∈(). 

Then, if q^2 is a primitive n–root of unity,

    T_n (  A_1A_2… A_k-1A_k  ) =  A_1^(n)A_2^(n)… A_k-1^(n) A_k^(n)

where, for each i,  A_i^(n) =
(
[  a_i^n  b_i^n;      0 a_i^-n ])
or
(
[  a_i^n      0;  b_i^n a_i^-n ]) is the 𝒜–point of  SL_2^q^n^2 obtained from A_i =(
[    a_i    b_i;      0 a_i^-1 ])
or
(
[    a_i      0;    b_i a_i^-1 ]) by replacing  a_i and b_i with their powers a_i^n and b_i^n, respectively.



Note that b_i^n a_i^n = q^n^2 a_i^n b_i^n since b_ia_i = qa_ib_i. Also, q^n^2 is equal to ±1 since q^2n=1, and  is always +1 when n is even. 


 For notational convenience, we will reverse the indexing and prove the equivalent statement that

    T_n (  A_kA_k-1… A_2A_1  ) =  A_k^(n)A_k-1^(n)… A_2^(n) A_1^(n).

For this, we will use Lemma <ref> and Corollary <ref>, so that 

    T_n (  A_kA_k-1… A_2A_1  )    = S_n (  A_kA_k-1… A_2A_1  ) - S_n-2(  A_kA_k-1… A_2A_1  )
       =ρ_n ( A_kA_k-1… A_2A_1  ) - ρ_n-2 (  A_kA_k-1… A_2A_1 )

for the representations ρ_m() →_([X,Y]^q_m ) of <ref>. 

We first compute these traces. 


When A_i is lower triangular,  the image of X^n-uY^u ∈[X,Y]^q_n under ρ_n(A_i) is 

    ρ_n(A_i)(X^n-u Y^u )   = 
    ρ_n([    a_i      0;    b_i a_i^-1 ])
    (X^n-u Y^u)
    =(a_iX)^n-u (b_iX + a_i^-1Y)^u
       =
    a_i^n-u X^n-u∑_v=0^uuvq^2  b_i^u-v X^u-va_i^-vY^v
       =
    ∑_v=0^uuvq^2  q^-v(u-v)
    a_i^n-u-v b_i^u-v X^n-vY^v,

using the Quantum Binomial Formula (<ref>) of <ref>. 
In particular, if we express ρ_n(A_i) in the basis {X^n-uY^u; u=0, 1, …, n} for [X,Y]^q_n, the entries of the corresponding matrix are

    ρ_n(A_i)_vu = 
    uvq^2  q^-v(u-v)
    a_i^n-u-v b_i^u-v    if  v≤ u
    
    0     if v>u.

Note that this matrix is upper triangular. 



Similarly, when A_i is upper triangular, 

    ρ_n(A_i)(X^n-u Y^u )   = 
    ρ_n([    a_i    b_i;      0 a_i^-1 ])
    (X^n-u Y^u)
    =(a_iX+ b_iY)^n-u ( a_i^-1Y)^u
       =
    ∑_v=u^nn-un-vq^2  q^-u(v-u)
    a_i^n-u-v b_i^v-u X^n-vY^v

and

    ρ_n(A_i)_vu = 
    
    0     if  v< u
    n-un-vq^2  q^-u(v-u)
    a_i^n-u-v b_i^v-u    if v≥ u.



In particular,

    ρ_n(A_kA_k-1… A_2A_1)   = ρ_n(A_1) ∘ρ_n(A_2) ∘…∘ρ_n(A_k-1) ∘ρ_n(A_k) 
       =  -10pt
    ∑_u_1,u_2, …, u_k ∈{0, …, n} -10pt
    ρ_n(A_1)_u_1u_2 ρ_n(A_2)_u_2u_3…
                           …ρ_n(A_k-1)_u_k-1u_k ρ_n(A_k)_u_ku_1

and

    ρ_n-2(A_kA_k-1… A_2A_1)   =  -10pt
    ∑_v_1,v_2, …, v_k ∈{0, …, n-2} -10pt
    ρ_n-2(A_1)_v_1v_2 ρ_n-2(A_2)_v_2v_3…
                           …ρ_n-2(A_k-1)_v_k-1v_k ρ_n-2(A_k)_v_kv_1,

where the terms ρ_n(A_i)_vu are given by Equations (<ref>) and (<ref>). 

We distinguish three types of terms in the sum of Equation (<ref>), according to the corresponding indices u_1, u_2, …, u_k ∈{0, …, n}:


  (i) no u_i is equal to 0 or n;

  (ii) some but not all u_i are equal to 0 or n;

  (iii) all u_i are equal to 0 or n;


We begin with the first type. 


If no u_i is equal to 0 or n, the term

    U_n(u_1, …, u_k)=ρ_n(A_1)_u_1u_2 ρ_n(A_2)_u_2u_3…ρ_n(A_k-1)_u_k-1u_k ρ_n(A_k)_u_ku_1

of Equation (<ref>) corresponding to  u_1, u_2, …, u_k ∈{1, …, n-1} is equal to the term

    U_n-2(v_1, …, v_k) =ρ_n-2(A_1)_v_1v_2 ρ_n-2(A_2)_v_2v_3…ρ_n(A_k-1)_v_k-1v_k ρ_n-2(A_k)_v_kv_1

of Equation (<ref>) corresponding to the indices v_1, v_2, …, v_k ∈{0, …, n-2} with v_i=u_i-1. 


 Set u_k+1=u_1 and v_k+1=v_1 to introduce uniformity in the notation.

If A_i is lower triangular, Equation (<ref>) gives

    ρ_n(A_i)_u_iu_i+1 = 
    u_i+1u_iq^2  q^-u_i(u_i+1-u_i)
    a_i^n-u_i+1-u_i b_i^u_i+1-u_i    if  u_i≤ u_i+1
    
    0     if u_i>u_i+1

while, using the property that v_i=u_i-1,

    ρ_n-2(A_i)_v_iv_i+1 = 
    u_i+1-1u_i-1q^2  q^-(u_i-1)(u_i+1-u_i)
    a_i^n-u_i+1-u_i b_i^u_i+1-u_i    if  u_i≤ u_i+1
    
    0     if u_i>u_i+1.

Since  uvq^2= uq^2/vq^2u-1v-1q^2, it follows that

    ρ_n(A_i)_u_iu_i+1 = 
    u_i+1q^2/u_iq^2q^u_i-u_i+1ρ_n-2(A_i)_v_iv_i+1

when A_i is lower triangular. 

Similarly, when A_i is upper triangular,

    ρ_n(A_i)_u_iu_i+1 = 
    
    0     if  u_i< u_i+1
    n-u_i+1n-u_iq^2  q^-u_i+1(u_i-u_i+1)
    a_i^n-u_i+1-u_i b_i^u_i-u_i+1    if u_i≥ u_i+1.

and

    ρ_n-2(A_i)_v_iv_i+1 = 
    
    0     if  u_i< u_i+1
    n-u_i+1-1n-u_i-1q^2  q^-(u_i+1-1)(u_i-u_i+1)
    a_i^n-u_i+1-u_i b_i^u_i-u_i+1    if u_i≥ u_i+1,

Therefore

    ρ_n(A_i)_u_iu_i+1   = 
    n-u_i+1q^2/n-u_iq^2q^u_i+1-u_iρ_n-2(A_i)_v_iv_i+1
       = -q^-2u_i+1u_i+1q^2/-q^-2u_iu_iq^2q^u_i+1-u_iρ_n-2(A_i)_v_iv_i+1
       = u_i+1q^2/u_iq^2q^u_i-u_i+1ρ_n-2(A_i)_v_iv_i+1

using the property that

    n-uq^2 = q^2n-2u-1/q^2-1 =  -q^-2uq^2u-1/q^2-1 = -q^-2uuq^2

since q^2n=1.

As a consequence, we get the same formula whether A_i is upper or lower triangular. Taking the product over all i,

    U_n(u_1, …, u_k)    = U_n-2(v_1, …, v_k) ∏_i=1^k u_i+1q^2/u_iq^2q^u_i-u_i+1  =U_n-2(v_1, …, v_k) ,

where the second equality comes from the fact that u_k+1=u_1. This proves Lemma <ref>. 




If some but not all indices u_i are equal to 0 or n, the term

    U_n(u_1, …, u_k)=ρ_n(A_1)_u_1u_2 ρ_n(A_2)_u_2u_3…ρ_n(A_k-1)_u_k-1u_k ρ_n(A_k)_u_ku_1

of Equation (<ref>) corresponding to  u_1, u_2, …, u_k ∈{0, …, n} is equal to 0. 

 This is a consequence of Lemma <ref>, which says that, because q^2 is a primitive n–root of unity, the quantum binomial coefficient nuq^2 is equal to 0 for 0<u<n. 

For convenience, set u_k+1=u_1 as in the proof of Lemma <ref>. By hypothesis, there is then an index i such that 0<u_i<n and u_i+1=0 or n. 

Consider first the case when 0<u_i<n and u_i+1=0. If A_i is lower triangular, then ρ_n(A_i)_u_iu_i+1=0 by Equation (<ref>), and consequently U_n(u_1, …, u_k)=0. 
Otherwise, Equation (<ref>) gives

    ρ_n(A_i)_u_iu_i+1 =  nn-u_iq^2  a_i^n-u_i b_i^u_i =0

by Lemma <ref>. This proves that U_n(u_1, …, u_k)=0 in this case. 

Similarly, if  0<u_i<n and u_i+1=n, Equation (<ref>) immediately shows that U_n(u_1, …, u_k)=0 when A_i is upper triangular, and otherwise gives 

    ρ_n(A_i)_u_iu_i+1 =  nu_iq^2  q^-u_i(n-u_i)
    a_i^-u_i b_i^n-u_i =0

by Lemma <ref>, again proving that U_n(u_1, …, u_k)=0. 


Lemmas <ref> and <ref> show that, when computing

    T_n( A_kA_k-1… A_2A_1) = ρ_n(A_kA_k-1… A_2A_1) - ρ_n-2(A_kA_k-1… A_2A_1)

using the expressions of Equations(<ref>–<ref>), the only terms left are 

    ∑_u_1,u_2, …, u_k ∈{0, n} -10pt
    ρ_n(A_1)_u_1u_2 ρ_n(A_2)_u_2u_3…ρ_n(A_k-1)_u_k-1u_k ρ_n(A_k)_u_ku_1

where

    ρ_n(A_i)_u_iu_i+1 = 
    
    a_i^n    if  u_i= u_i+1=0
    
    b_i^n    if  u_i=0  and  u_i+1=n
    
    0     if u_i =n  and  u_i+1=0
    
    a_i^-n    if  u_i= u_i+1=n

if A_i is lower triangular, and 

    ρ_n(A_i)_u_iu_i+1 = 
    
    a_i^n    if u_i= u_i+1=0
    
    0     if  u_i=0  and  u_i+1=n
    
     b_i^n    if u_i =n  and  u_i+1=0
    
    a_i^-n    if u_i= u_i+1=n

if A_i is upper triangular. 

As a consequence, comparing the general case to the case n=1,

    T_n( A_kA_k-1… A_2A_1)    =  -10pt∑_u_1,u_2, …, u_k ∈{0, 1} -10pt
    ρ_1(A_1^(n))_u_1u_2 ρ_1(A_2^(n))_u_2u_3…
                           …ρ_1(A_k-1^(n))_u_k-1u_k ρ_1(A_k^(n))_u_ku_1
       = ρ_1(A_k^(n) A_k-1^(n)… A_2^(n) A_1^(n))

where A_i^(n) = ( 
[ a_i^n b_i^n; b_i^n d_i^n ]) is obtained from A_i = ( 
[ a_i b_i; b_i d_i ])
 by replacing a_i, b_i, b_i, d_i with a_i^n, b_i^n, b_i^n, d_i^n, respectively. 
 
 We already observed that, for an –point A = ( 
[ a b; c d ]) ∈(), the matrix of ρ_1(A) in the basis {X,Y} for [X,Y]_1^q is the transpose ( 
[ a c; b d ]), so that ρ_1(A) = a+d =  A. It follows that
 
    T_n( A_kA_k-1… A_2A_1)    =  ρ_1(A_k^(n) A_k-1^(n)… A_2^(n) A_1^(n))
       =   A_k^(n) A_k-1^(n)… A_2^(n) A_1^(n).

This is exactly the relation (<ref>) that we wanted to prove, which concludes the proof of Theorem <ref>. 






§ A POSITIVITY PROPERTY


Many fewer cancellations occur when q is not a root of unity. This can be precisely quantified by using  a certain positivity property for T_n (  A_1A_2… A_k  ).

Let A_1A_2 … A_k be a product of triangular matrices A_i=  ( 
[    a_i    b_i;      0 a_i^-1 ]) or 
([    a_i      0;    b_i a_i^-1 ]) ∈() satisfying the hypotheses of  Theorems <ref> or <ref>, except that we are not requiring q to be a root of unity. Then, A_1A_2… A_k can be written as a sum of monomials ∏_i=1^k c_i with c_i∈{a_i, b_i, a_i^-1} and therefore, for every polynomial P(t) ∈[t] with integer coefficients,   P (  A_1A_2… A_k  ) can be written as a sum of monomials of the form ± q^ξ∏_i=1^k a_i^α_i b_i^β_i with integer powers ξ, α_i, β_i ∈ (with β_i≥ 0).
 
 The following result states that, when P(t) is one of the Chebyshev polynomials S_n(t) or T_n(t),  the signs ± can always be taken to be +. 
 
 


 Under the hypotheses of  Theorems <ref> or <ref> but without any assumption on the parameter q∈-{0},  the evaluations  S_n (  A_1A_2… A_k  ) and T_n (  A_1A_2… A_k  ) ∈ of the Chebyshev polynomials can be written as a sum of positive monomials of the form + q^ξ∏_i=1^k a_i^α_i b_i^β_i with integer powers ξ, α_i, β_i ∈.


 The case of S_n(t) is  relatively simple. We computed

    S_n (  A_1A_2… A_k  ) = ρ_n ( A_1A_2… A_k )

in the course of the proof of Theorem <ref>. In particular, Equations (<ref>–<ref>) show that, with no assumption on q, ρ_n ( A_1A_2… A_k ) is a sum of positive monomials + q^ξ∏_i=1^k a_i^α_i b_i^β_i. Indeed, it is well-known (and also follows from the Quantum Binomial Formula (<ref>))  that the quantum binomial coefficients uvq^2 are polynomials in q^2 with nonnegative integer coefficients. 

The proof for T_n(t) is more elaborate. We want to show that, when computing 

    T_n (  A_1A_2… A_k  ) = ρ_n ( A_1A_2… A_k ) - ρ_n-2 ( A_1A_2… A_k ),

each monomial of ρ_n-2 ( A_1A_2… A_k ) cancels out with a monomial of ρ_n ( A_1A_2… A_k ); this can be seen as a weaker form of Lemma <ref>. For this, we will give a different computation of ρ_n ( A_1A_2… A_k ). 

This computation goes back to the principles underlying the Quantum Binomial Formula. Let ⟨ X,Y ⟩ be the free algebra generated by the set {X,Y}. Namely, ⟨ X,Y ⟩ consists of all formal polynomials P(X,Y) in noncommutating variables X and Y, and these polynomials are multiplied without simplifications. In particular, the quantum plane [X,Y]^q is the quotient of ⟨ X,Y ⟩ by the ideal generated by YX-qXY, which gives a natural projection π⟨ X,Y ⟩→[X,Y]^q. 

Similarly, consider ⟨ X,Y ⟩ = ⊗⟨ X,Y ⟩, and the projection _⊗π which we will also denote as π⟨ X,Y ⟩→[X,Y]^q for short.

Let ⟨ X,Y ⟩_n be the linear subspace of ⟨ X,Y ⟩ consisting of all homogeneous polynomials of degree n. Namely,  ⟨ X,Y ⟩ _n consists of all finite sums

    P(X,Y) = ∑_u α_u Z_u1 Z_u2… Z_un

where  the coefficients α_u are in  and where each variable Z_uv is equal to X or to Y. In particular, ⟨ X,Y ⟩_n is isomorphic to ^2^n as an –module. For comparison, remember that [X,Y]^q_n is isomorphic to ^n+1. 

The representation ρ_n () →_( [X,Y]^q_n ) lifts to a representation ρ_n () →_( ⟨ X,Y⟩_n ) defined by the property that

    ρ_n ( [ a b; c d ]) (P(X,Y) ) = P(aX+bY, cX + dY )

for every homogeneous polynomial P(X,Y) ∈⟨ X,Y ⟩  of degree n in the noncommuting variables X and Y (and with coefficients in ). 

To give a more combinatorial description of this action, note that every element of ⟨ X,Y⟩_n can be uniquely written as a sum of monomials  α Z_1Z_2 … Z_n where α∈ and  each Z_u ∈{X,Y}. Then, ρ_n( 
[ a b; c d ]) (α Z_1Z_2 … Z_n) ∈⟨ X,Y ⟩_n is the sum of all monomials α' Z_1'Z_2' … Z_n' obtained from α Z_1Z_2 … Z_n by replacing each Z_u with:

 
  • either aX or bY, if Z_u=X;
 
  •  either cX or dY, if Z_u=Y,

(and pushing all coefficients of  to the front). 

As a consequence, 

    ρ_n(A_1A_2 … A_k) (α Z_1Z_2 … Z_n)
     =  ρ_n(A_k) ∘ρ_n(A_k-1) ∘…∘ρ_n(A_1) (α Z_1Z_2 … Z_n)

can be described as follows. Let ℳ(α Z_1Z_2 … Z_n) be the set of all  sequences M =(M_i)_i=0, 1, …, k of monomials M_i=α_i Z_i1Z_i2… Z_in∈⟨ X,Y ⟩_n such that

 
  * M_0 = α Z_1Z_2 … Z_n;
 
  * M_i is obtained from M_i-1 by replacing each Z_(i-1)u with

 
  •   a_i X or b_iY if Z_(i-1)u=X and A_i = ( 
[    a_i    b_i;      0 a_i^-1 ]);
 
  
  •   a_i^-1 Y if Z_(i-1)u=Y and A_i = ( 
[    a_i    b_i;      0 a_i^-1 ]);
 
  
  •   a_i X  if Z_(i-1)u=X and A_i = ( 
[    a_i      0;    b_i a_i^-1 ]);
 
  
  •   b_i X  or a_i^-1 Y if Z_(i-1)u=Y and A_i = ( 
[    a_i      0;    b_i a_i^-1 ]).



Then, as we expand ρ_n(A_1A_2 … A_k) (α Z_1Z_2 … Z_n)∈⟨ X,Y⟩_n, we see that its  decomposition  into monomials is given by

    ρ_n(A_1A_2 … A_k) (α Z_1Z_2 … Z_n) = ∑_M ∈ℳ(α Z_1Z_2 … Z_n) M_k

where M_k is the last term of the sequence M =(M_i)_i=0, 1, …, k∈ℳ(α Z_1Z_2 … Z_n). 

As a consequence, if we use the same notation for the monomial X^n-uY^u=X … XY … Y ∈⟨ X,Y⟩_n and for its image X^n-uY^u ∈[X,Y]^q_n under the projection π⟨ X,Y⟩_n →[X,Y]^q_n, 

    ρ_n(A_1A_2 … A_k) ( X^n-uY^u ) = ∑_M ∈ℳ(X^n-uY^u)π(M_k).


Finally, let ℳ'(X^n-uY^u) be the set of monomial sequences M ∈ℳ(X^n-uY^u) whose contribution  π(M_k) belongs to X^n-uY^u. For such a monomial sequence M =(M_i)_i=0, 1, …, k, let α(M_k)∈ be the coefficient such that π(M_k) = α(M_k) X^n-uY^u. We can then compute the trace of ρ_n(A_1A_2 … A_k) by using  the basis { X^n-uY^u; u=0,1,…,n} for [X,Y]^q_n and [X,Y]^q_n = ⊗[X,Y]^q_n, which gives

    ρ_n(A_1A_2 … A_k) = ∑ _u=0^n ∑_M ∈ℳ'(X^n-uY^u)α(M_k).


Similarly

    ρ_n-2(A_1A_2 … A_k) = ∑ _v=0^n-2 ∑_M ∈ℳ'(X^n-v-2Y^v)α(M_k).


The expression (<ref>) for ρ_n(A_1A_2 … A_k) was obtained by using the basis { X^n-uY^u; u=0,1,…,n} for [X,Y]^q_n. For comparison with (<ref>), it is more convenient to use the other basis {X^n, Y^n}∪{ X^n-v-2Y^vXY; v=0,1,…,n-2} for [X,Y]^q_n. This gives the expression 

    ρ_n(A_1A_2 … A_k)    =  ∑_M' ∈ℳ'(X^n)α(M_k') +  ∑_M' ∈ℳ'(Y^n)α(M_k') 
                    +∑ _v=0^n-2  ∑_M' ∈ℳ'(X^n-v-2Y^vXY)β(M_k')

where, for M' =(M_i')_i=0, 1, …, k∈ℳ'(X^n-v-2Y^vXY), the coefficient β(M_k') ∈ is defined by the property that π(M_k') = β(M_k') X^n-v-2Y^vXY.

Note that, because of the commutativity and q–commutativity properties of the quantities a_i, b_i, X, Y, all terms α(M_k) and β(M_k') in (<ref>–<ref>)  are positive monomials of the form + q^ξ∏_i=1^k a_i^α_i b_i^β_i with ξ, α_i, β_i ∈ and β_i≥ 0. .

We now compare (<ref>) and (<ref>).  Every monomial sequence  M =(M_i)_i=0, 1, …, k∈ℳ'(X^n-v-2Y^v) gives rise to a monomial sequence M'  ∈ℳ'(X^n-v-2Y^vXY) defined by the property that M_i' = M_iXY for every i. Indeed, rewriting

    M_i' = M_i (a_iX)(a_i^-1Y)

shows that M' =(M_i')_i=0, 1, …, k really satisfies the inductive property defining ℳ'(X^n-v-2Y^vXY). In addition, when M'  ∈ℳ'(X^n-v-2Y^vXY) is thus associated to M ∈ℳ'(X^n-v-2Y^v), 

    π(M_k') = π(M_kXY) =π(M_k)π(X)π(Y) = α(M_k) X^n-v-2Y^vXY

 so that β(M_k')= α(M_k). 
 
 
 Therefore, when computing 
 
    T_n (  A_1A_2… A_k  ) = ρ_n ( A_1A_2… A_k ) - ρ_n-2 ( A_1A_2… A_k ),

every monomial α(M_k) occurring in (<ref>) cancels out with a monomial  β(M_k') of (<ref>). It follows that T_n (  A_1A_2… A_k  ) is the sum of the remaining coefficients α(M_k) and β(M_k') of (<ref>). We already observed that these monomials are positive, which concludes the proof of Proposition <ref>. 

 
 Proposition <ref> enables us to  precisely determine the number of monomials in T_n (  A_1A_2… A_k  ) under the hypothesis that there are no extraneous simplifications. This means that q is transcendental and, since we can always assume that the algebra  is generated by the entries a_i, b_i of the matrices A_i, that  is the algebra defined by the generators a_i^± 1, b_i and by the relations that b_ia_i = qa_i b_i and that a_i, b_i commute with a_j, b_j whenever i ≠ j. In other words,  is the algebra ⊗_i=1^k [a_i^±1, b_i]^q. 
 
 In this case, every element of =⊗_i=1^k [a_i^±1, b_i]^q has a unique decomposition as a sum of monomials ξ∏_i=1^k a_i^α_i b_i^β_i with  ξ∈, α_i∈, β_i ∈ and β_i≥ 0. 
  
 
 
Suppose  that q is transcendental, and that =⊗_i=1^k [a_i^±1, b_i]^q. Let triangular matrices A_i=  ( 
[    a_i    b_i;      0 a_i^-1 ]) or 
([    a_i      0;    b_i a_i^-1 ]) ∈() be given for i=1, 2, …, k, and consider the positive integer t_0=  A_1^(0)A_2^(0)… A_k^(0)  where A_i^(0) = ( 
[ 1 1; 0 1 ]) if A_i=  ( 
[    a_i    b_i;      0 a_i^-1 ]) and A_i^(0)=([ 1 0; 1 1 ]) if A_i=([    a_i      0;    b_i a_i^-1 ]). 
Then,  for every n, T_n (  A_1A_2… A_k  ) is the sum of exactly
 
    T_n(t_0) =( t_0+√(t_0^2-4)/2)^n + ( t_0-√(t_0^2-4)/2)^n

positive monomials of the form + q^ξ∏_i=1^k a_i^α_i b_i^β_i with ξ, α_i, β_i ∈ and β_i≥0.



 We already proved in Proposition <ref> that T_n (  A_1A_2… A_k  ) is a sum of monomials of the type indicated. The only issue  is to count their number. 
 
 Because of the positive signs, the number of these monomials can be computed by letting q and the a_i, b_i tend to 1. Under this limiting process, A_1A_2… A_k approaches t_0, and the number of monomials in the expansion for T_n (  A_1A_2… A_k  ) is therefore  equal to T_n(t_0). 
 
 The formula T_n(t_0) =( t_0+√(t_0^2-4)/2)^n + ( t_0-√(t_0^2-4)/2)^n is provided by Lemma <ref>. 





amsalpha
  



