Points of quantum $\mathrm{SL}_n$ coming from quantum snakes

We show that the quantized Fock-Goncharov monodromy matrices satisfy the relations of the quantum special linear group $\mathrm{SL}_n^q$. The proof employs a quantum version of the technology invented by Fock-Goncharov called snakes. This relationship between higher Teichm\"uller theory and quantum group theory is integral to the construction of a $\mathrm{SL}_n$-quantum trace map for knots in thickened surfaces, developed in a companion paper (arXiv:2101.06817).

For a finitely generated group Γ and a suitable Lie group G, a primary object of study in low-dimensional geometry and topology is the G-character variety R G (Γ) = {ρ : Γ − G} / / G consisting of group homomorphisms ρ from Γ to G, considered up to conjugation.Here, the quotient is taken in the algebraic geometric sense of Geometric Invariant Theory [MFK94].Character varieties can be explored using a wide variety of mathematical skill sets.Some examples include the Higgs bundle approach of Hitchin [Hit92], the dynamics approach of Labourie [Lab06], and the representation theory approach of Fock-Goncharov [FG06b].
In the case where the group Γ = π 1 (S) is the fundamental group of a punctured surface S of finite topological type, and where the Lie group G = SL n (C) is the special linear group, we are interested in studying a relationship between two competing deformation quantizations of the character variety R SLn(C) (S) := R SLn(C) (π 1 (S)).Here, a deformation quantization {R q } q of a Poisson space R is a family of non-commutative algebras R q parametrized by a nonzero complex parameter q = e 2πi , such that the lack of commutativity in R q is infinitesimally measured in the semi-classical limit 0 by the Poisson bracket of the space R. In the case where R = R SLn(C) (S) is the character variety, the bracket is provided by the Goldman Poisson structure on R SLn(C) (S) [Gol84,Gol86].
The first quantization of the character variety is the SL n (C)-skein algebra S q n (S) of the surface S; see [Tur89,Wit89,Prz91,BFKB99,Kup96,Sik05].The skein algebra is motivated by the classical algebraic geometric approach to studying the character variety R SLn(C) (S) via its commutative algebra of regular functions C[R SLn(C) (S)].An example of a regular function is the trace function Tr γ : R SLn(C) (S) C associated to a closed curve γ ∈ π 1 (S) sending a representation ρ : π 1 (S) SL n (C) to the trace Tr(ρ(γ)) ∈ C of the matrix ρ(γ) ∈ SL n (C).A theorem of Classical Invariant Theory, due to Procesi [Pro76], says that the trace functions Tr γ generate the algebra of functions C[R SLn(C) (S)] as an algebra.According to the philosophy of Turaev and Witten, quantizations of the character variety should be of a 3-dimensional nature.Indeed, elements of the skein algebra S q n (S) are represented by knots (or links) K in the thickened surface S × (0, 1).The skein algebra S q n (S) has the advantage of being natural, but is difficult to work with in practice.
The second quantization is the Fock-Goncharov quantum SL n (C)-character variety T q n (S); see [FC99,Kas98,FG09].At the classical level, Fock-Goncharov [FG06b] introduced a framed version R PSLn(C) (S) FG (often called the X -space) of the PSL n (C)-character variety, which, roughly speaking, consists of points of the character variety R PSLn(C) (S) equipped with additional linear algebraic data attached to the punctures of S. Associated to each ideal triangulation λ of the punctured surface S is a λ-coordinate chart U λ ∼ = (C − {0}) N for R PSLn(C) (S) FG parametrized by N nonzero complex coordinates X 1 , X 2 , . . ., X N where the integer N depends only on the topology of the surface S and the rank of the Lie group SL n (C).More precisely, the coordinates X i are computed by taking various generalized cross-ratios of configurations of n-dimensional flags attached to the punctures of S. When written in terms of these coordinates X i the trace functions Tr γ = Tr γ (X ±1/n i ) associated to closed curves γ take the form of Laurent polynomials in n-roots of the variables X i .At the quantum level, Fock-Goncharov defined q-deformed versions X q i of their coordinates, which no longer commute but q-commute with each other.A quantum λ-coordinate chart U q λ = T q n (λ) for the quantized character variety T q n (S) is obtained by taking rational fractions in these q-deformed parameters X q i .The quantum character variety T q n (S) has the advantage of being easier to work with than the skein algebra S q n (S), however it is less intrinsic.We seek q-deformed versions Tr q γ of the trace functions Tr γ associating to a closed curve γ a Laurent polynomial in the quantized Fock-Goncharov coordinates X q i .Turaev and Witten's philosophy leads us from the 2-dimensional setting of curves γ on the surface S to the 3dimensional setting of knots K in the thickened surface S × (0, 1).In the case of SL 2 (C), such a quantum trace map was constructed in [BW11] as an injective algebra homomorphism Tr q (λ) : S q 2 (S) − T q 2 (λ) from the SL 2 (C)-skein algebra to (the quantum λ-coordinate chart of) the quantized SL 2 (C)character variety.Their construction is "by hand", however it is implicitly related to the theory of the quantum group U q (sl 2 ) or, more precisely, of its Hopf dual SL q 2 ; see [Kas95].Developing a quantum trace map for SL n (C) requires a more conceptual approach that makes more explicit this connection between higher Teichmüller theory and quantum group theory.In a companion paper [Dou21], we make significant progress in this direction.The goal of the present work is to establish a local building block result that is used in [Dou21].
Loosely speaking, whereas the classical trace Tr γ (ρ) ∈ C is a number obtained by evaluating the trace of a SL n (C)-monodromy ρ taken along a curve γ in S, the quantum trace Tr K (X q i ) ∈ T q n (λ) is a Laurent polynomial obtained by evaluating the trace of a quantum monodromy matrix M q = M q (X q i ) associated to a knot K in S × (0, 1).This quantum matrix M q , with coefficients in the q-deformed fraction algebra T q n (λ), is constructed more or less by taking the product along the knot K of certain local quantum monodromy matrices M q k ∈ M n ( T q n (λ k )) associated to the triangles λ k of the ideal triangulation λ.Theorem 1.The Fock-Goncharov quantum matrices M q k ∈ M n ( T q n (λ k )) are T q n (λ k )-points of the dual quantum group SL q n .Namely, these matrices define algebra homomorphisms ϕ(M q k ) : SL q n − T q n (λ k ) by sending the n 2 -many generators of SL q n to the n 2 -many entries of the matrix M q k .See Theorem 32; compare [Dou20, Theorem 3.10] and [Dou21,Theorem 14].The proof uses a quantum version of the technology invented by Fock-Goncharov called snakes.For a

Fock-Goncharov snakes
We recall some of the classical (as opposed to the quantum) geometric theory of Fock-Goncharov [FG06b], underlying the quantum theory discussed later on; see also [FG07a,FG07b].This section is a condensed version of [Dou20, Chapter 2].For other references on Fock-Goncharov coordinates and snakes, see [HN16,GMN14,Mar19].When n = 2, these coordinates date back to Thurston's shearing coordinates for Teichmüller space [Thu97].
Throughout, let n ∈ Z, n 2, and let V be a n-dimensional complex vector space.
1.1.1.Vectors, covectors, and dual subspaces.The dual space V * is the vector space of linear maps V C.An element v ∈ V is called a vector, and an element u ∈ V * is called a covector.Given a linear subspace W ⊆ V , the dual subspace W ⊥ ⊆ V * is the linear subspace Fact 2. The dual subspace operation satisfies the following elementary properties: 1.1.2.Change of basis matrices and projective bases.We will deal with linear bases U = {u 1 , u 2 , . . ., u n } of the dual space V * .We always assume that bases are ordered.
Given a basis U = {u 1 , u 2 , . . ., u n } of covectors in V * , and given a covector u in V * , the coordinate covector [[u]] U of the covector u with respect to the basis U is the unique row matrix If in addition we are given another basis U = {u 1 , u 2 , . . ., u n } for V * , then the change of basis matrix B U U going from the basis U to the basis U is the unique invertible matrix in GL n (C) ⊆ M n (C) satisfying An important elementary property for change of basis matrices is The nonzero complex numbers C − {0} act on the set of linear bases U for V * by scalar multiplication.A projective basis [U] for V * is an equivalence class for this action.
1.1.3.Complete flags and dual flags.A complete flag, or just flag, E in V is a collection of linear subspaces E (a) ⊆ V indexed by 0 a n, satisfying the property that each subspace E (a) is properly contained in the subspace E (a+1) .In particular, E (a) is a-dimensional, E (0) = {0}, and E (n) = V .Denote the space of flags by Flag(V ).
A basis V = {v 1 , v 2 , . . ., v n } for V determines a standard ascending flag E 0 (V) and a standard descending flag G 0 (V) defined by Given a flag E in V , its dual flag E ⊥ is the flag in the dual space V * defined by 1.1.4.Linear groups.The general linear group GL(V ) is the group of invertible linear maps V V .The special linear group SL(V ) is the subgroup of GL(V ) consisting of the linear maps ϕ preserving a nontrivial top exterior form ω ∈ Λ n (V ) − {0} ∼ = C − {0} on V , which is independent of the choice of form ω. Given a flag E in V , the Borel subgroup B(E) associated to E is the subgroup of GL(V ) consisting of the invertible linear maps preserving the flag E.
The nonzero complex numbers C − {0} act on GL(V ) and B(E) by scalar multiplication, and similarly the n-roots of unity Z/nZ ⊆ C − {0} act on SL(V ).The respective quotients are the projective linear groups PGL(V ), PB(E), and PSL(V ).Since every complex number admits a n-root, the natural inclusion SL(V ) GL(V ) induces a group isomorphism PSL(V ) ∼ = PGL(V ).Note PGL(V ) acts transitively on the space of flags Flag(V ), thereby inducing a bijection Flag(V ) ∼ = PGL(V )/PB(E) of Flag(V ) with the left cosets of PB(E).1.2.Generic configurations of flags and Fock-Goncharov invariants.
1.2.1.Generic pairs of flags.For two flags, the notion of genericity is straightforward.Definition 3. A pair of flags (E, G) ∈ Flag(V ) × Flag(V ) = Flag(V ) 2 is generic if any of the following equivalent properties are satisfied: for every 0 a, c n, Note that the equivalence of these properties can be deduced from the classical relation Proposition 4. The diagonal action of PGL(V ) on the space Flag(V ) 2 restricts to a transitive action on the subset of generic flag pairs.
Proof.Let (E, G) ∈ Flag(V ) 2 be a generic flag pair.By genericity, for every 1 a n, the subspace L a = E (a) ∩ G (n−a+1) is a line in V .It follows by genericity that the lines L a form a line decomposition of V , namely V = ⊕ n a=1 L a .Fix a basis V = {v 1 , v 2 , . . ., v n } of V .Let ϕ : V V be a linear isomorphism sending the line L a to the a-th basis vector v a .Then ϕ maps the flag pair (E, G) to the standard ascending-descending flag pair (E 0 (V), G 0 (V)).1.2.2.Generic triples and quadruples of flags.For three or four flags, there are at least two possible notions of genericity.Here we discuss one of them, the Maximum Span Property; for a complementary notion, the Minimum Intersection Property, see [Dou20, §2.10].
Definition 5. A flag triple (E, F, G) ∈ Flag(V ) 3 satisfies the Maximum Span Property if either of the following equivalent conditions are satisfied: for every 0 a, b, c, n, (1) (a) the sum In this case, the flag triple (E, F, G) ∈ Flag(V ) 3 is called a maximum span flag triple.
Maximum span flag quadruples (E, F, G, H) are defined analogously.
Here, e (a ) is a choice of generator for the a -th exterior power Λ ) are lines, the triangle invariant τ abc (E, F, G) is independent of the choices of generators e (a ) , f (b ) , g (c ) .The Maximum Span Property ensures that each wedge product e (a ) ∧ f The six numerators and denominators appearing in the expression defining τ abc (E, F, G) can be visualized as the vertices of a hexagon in Θ n centered at (a, b, c); see Figure 1.Fact 6.The triangle invariants τ abc (E, F, G) satisfy the following symmetries: (1) Similarly, for a maximum span quadruple of flags (E, G, F, F ) ∈ Flag(V ) 4 , Fock and Goncharov assigned to each integer 1 j n − 1 an edge invariant j (E, G, F, F ) by The four numerators and denominators appearing in the expression defining j (E, G, F, F ) can be visualized as the vertices of a square, which crosses the "common edge" between two "adjacent" discrete triangles Θ n (G, F, E) and Θ n (E, F , G); see Figure 2.
Figure 2. Edge invariants for a generic flag quadruple 1.2.5.Action of PGL(V ) on generic flag triples.We saw earlier that the diagonal action of PGL(V ) on the space of generic flag pairs has a single orbit.The situation is more interesting when PGL(V ) acts on the space of generic flag triples.Note in particular that the triangle invariants τ abc (E, F, G) ∈ C − {0} are preserved under this action.
Theorem 7 (Fock-Goncharov).Two maximum span flag triples (E, F, G) and (E , F , G ) have the same triangle invariants, namely Conversely, for each choice of nonzero complex numbers Proof.See [FG06b,§9].The proof uses the concept of snakes, due to Fock and Goncharov.For a sketch of the proof and some examples, see [Dou20, §2.19].

Snakes and projective bases.
1.3.1.Snakes.Snakes are combinatorial objects associated to the (n − 1)-discrete triangle Θ n−1 ( §1.2.3).In contrast to Θ n , we denote the coordinates of a vertex ν ∈ Θ n−1 by ν = (α, β, γ) corresponding to solutions α Remark 9.In a moment, we will define a snake.The most general definition involves choosing a snake-head η ∈ Γ(Θ n−1 ).For simplicity, we define a snake only in the case η = (n − 1, 0, 0).The definition for other choices of snake-heads follows by triangular symmetry.We will usually take η = (n − 1, 0, 0) and will alert the reader if otherwise.Definition 10.A left n-snake (for the snake-head η = (n−1, 0, 0) ∈ Γ(Θ n−1 )), or just snake, σ is an ordered sequence See Figure 3. On the right hand side, we show a snake σ = (σ k ) k in the case n = 5.On the left hand side, we show how the snake-vertices σ k ∈ Θ n−1 can be pictured as small upward-facing triangles ∆ in the n-discrete triangle Θ n .
by the Maximum Span Property, since α + β + γ = n − 1.Here, we have used the dual subspace construction ( §1.1.1).Consequently, the subspace If in addition we are given a snake σ = (σ k ) k , then we may consider the n-many lines where the snake-vertex By genericity, we obtain a direct sum In summary, a maximum span flag triple (E, F, G) and a snake σ provide a line decomposition of the dual space V * .In fact, as we will see in a moment, this data provides in addition a projective basis ( §1.1.2) of V * compatible with the line decomposition.
1.3.3.Projective basis of V * associated to a triple of flags and a snake.More precisely, we will associate to this data a projective basis [U] of V * , where U = {u 1 , u 2 , . . ., u n } is a linear basis of V * , satisfying the property that the k-th basis element u k ∈ V * is an element of the line L σ k ⊆ V * associated to the k-th snake-vertex σ k ∈ Θ n−1 .

Figure 4. Three coplanar lines involved in the definition of a projective basis
As usual, put σ k = (α k , β k , γ k ).We begin by choosing a covector u n in the line L σn ⊆ V * , called a normalization.Having defined covectors u n , u n−1 , . . ., u k+1 , we will define a covector By the definition of snakes, we see that given σ k+1 there are only two possibilities for σ k , denoted either σ left k+1 or σ right k+1 depending on its coordinates: 4, where σ k = σ right k+1 .Thus, the lines L σ left k+1 and L σ right k+1 can be written It follows by the Maximum Span Property that the three lines L σ k+1 , L σ left k+1 , L σ right k+1 in V * are distinct and coplanar.Specifically, they lie in the plane , respectively, such that See Figure 4, which falls into the second case.Note if the initial choice of normalization u n is replaced by λu n for some scalar λ, then u k is replaced by λu k for all 1 k n.We gather this process produces a projective basis [U] = [{u 1 , u 2 , . . ., u n }] of V * , as desired.
1.4.1.Shearing (and U-turn) matrices.Let A be a commutative algebra, such as A = C.Let X 1/n , Z 1/n ∈ A, and put X = (X 1/n ) n and Z = (Z 1/n ) n .Let M n (A) (resp.SL n (A)) denote the usual ring of n × n matrices (resp.having determinant equal to 1) over A (see §2.1.2).
For k = 1, 2, . . ., n − 1 define the k-th left-shearing matrix S left k (X) ∈ SL n (A) by and define the k-th right-shearing matrix Note that S left 1 (X) and S right 1 (X) do not, in fact, involve the variable X, and so we will denote these matrices simply by S left 1 and S right 1 , respectively.For j = 1, 2, . . ., n − 1 define the j-th edge-shearing matrix S edge j (Z) ∈ SL n (A) by Lastly, define the clockwise U-turn matrix U in SL n (C) by 1.4.2.Adjacent snake pairs.Definition 12.We say that an ordered pair (σ, σ ) of snakes σ and σ forms an adjacent pair of snakes if the pair (σ, σ ) satisfies either of the following conditions: (1) for some 2 k n − 1, (a) ), and σ k = σ left k+1 (= σ left k+1 ), in which case (σ, σ ) is called an adjacent pair of diamond-type, see Figure 5; (2) (a) ), and σ 1 = σ left 2 (= σ left 2 ), in which case (σ, σ ) is called an adjacent pair of tail-type, see Figure 6.1.4.3.Diamond and tail moves.Until we arrive at the next proposition, let (σ, σ ) be an adjacent pair of snakes of diamond-type, as shown in Figure 5.
Consider the snake-vertices σ k+1 (= σ k+1 ), σ k , σ k , and σ k−1 (= σ k−1 ).One checks that Taken together, these three coordinates form a vertex in the interior of the n-discrete triangle Θ n (not Θ n−1 ).The coordinates of this internal vertex (a, b, c) can also be thought of as delineating the boundary of a small downward-facing triangle ∇ in the discrete triangle Θ n−1 , whose three vertices are σ k , σ k , σ k−1 (Figure 5).Put  The following result is the main ingredient going into the proof of Theorem 7.
Proposition 13.Let (E, F, G) be a maximum span flag triple, (σ, σ ) an adjacent pair of snakes, and U, U the corresponding normalized projective bases of V * so that We say this case expresses a diamond move from the snake σ to the adjacent snake σ .
If (σ, σ ) is of tail-type, then the change of basis matrix B U U equals We say this case expresses a tail move from the snake σ to the adjacent snake σ .
To adjust for using right snakes, the definitions of §1.3.3, 1.4.2,1.4.3 need to be modified.Given σ k−1 , there are two possibilities for σ k : In particular, the algorithm starts by making a choice of covector u 1 ∈ L σ 1 = L (n−1,0,0) .Notice that, compared to the setting of left snakes (Definition 11 and Figure 4), the signs defining the projective basis have been swapped.An ordered pair (σ, σ ) of right snakes forms an adjacent pair if either: (1) for some 2 k n − 1, (a) , and σ n = σ right n−1 (= σ right n−1 ), in which case (σ, σ ) is called an adjacent pair of tail-type.Given an adjacent pair (σ, σ ) of right snakes of diamond-type, there is naturally associated a vertex (a, b, c) ∈ Θ n to which is assigned a Fock-Goncharov triangle invariant X abc .
Proposition 15.Let (E, F, G) be a maximum span triple, (σ, σ ) an adjacent pair of right snakes, and U, U the corresponding normalized projective bases of V * so that If (σ, σ ) is of tail-type, then the change of basis matrix B U U equals Proof.See [FG06b,§9].Similar to the proof of Proposition 13.
Consider two copies of the discrete triangle; Figure 7.The bottom triangle Θ n−1 (G, F, E) has a maximum span flag triple (G, F, E) assigned to the corner vertices Γ(Θ n−1 ), and the top triangle Θ n−1 (E, F , G) has assigned to Γ(Θ n−1 ) a maximum span flag triple (E, F , G).
Define two snakes σ and σ in Θ n−1 (G, F, E) and Θ n−1 (E, F , G), respectively, as follows: Notice that the line decompositions associated to the snakes σ and σ are the same:

Normalize the two associated projective bases [U] and [U ] by choosing
Proposition 17.The change of basis matrix expressing the snake move σ σ is Next, define snakes σ and σ in a single discrete triangle Θ n−1 (E, F, G) by (see Figure 8) Notice that the lines

Normalize the two associated projective bases [U] and [U ] by choosing
Proposition 18.The change of basis matrix expressing the snake move σ σ is Proof.See [FG06b,§9].Similar to the proof of Proposition 13; see also [Dou20,§2.22].
Remark 19.This last U-turn move will not be needed in this paper, but appears in [Dou21].
Given one of these small upward-facing triangles, say ∆ 1 , the crucial property we used to define projective bases in §1.3.3 is that the three lines L ν 1 , L ν 3 , L ν 2 in V * attached to the vertices of ∆ 1 are coplanar.Consequently, to the triangle ∆ 1 there are associated six shears: L ν 1 and their inverses.
For instance, the shear S ∆ 1 ν 1 ν 3 sends a point p in L ν 1 to the unique point p in L ν 3 such that for some (unique) point p ∈ L ν 2 .And S ∆ 1 ν 1 ν 2 (p) = p .Similarly for the other triangles ∆ 2 , ∆ 3 .Let X abc = τ abc (E, F, G) be the triangle invariant associated to the vertex (a, b, c).
Proposition 21.Fix a point p 0 in the line L ν 1 .Let p 1 be the point in the line L ν 3 resulting from the shear S ∆ 2 ν 1 ν 3 associated to the triangle ∆ 2 applied to the point p 0 , let p 2 be the point in the line L ν 2 resulting from the shear S ∆ 1 ν 3 ν 2 associated to the triangle ∆ 1 applied to the point p 1 , and let p 3 be the point in the line L ν 1 resulting from the shear S ∆ 3 ν 2 ν 1 associated to the triangle ∆ 3 applied to the point p 2 .It follows that p 3 = +X abc p 0 .This was the case going counterclockwise around the (a, b, c)-downward-facing triangle ∇; see Figure 9.If instead one goes clockwise around ∇, then the total shearing is +X −1 abc .Proof.See [FG06b,GMN14].Similar to that of Proposition 13; see also [Dou20,§2.21].
Proposition 22. Fix a point p 0 in the line L ν 0 (E, F , G).Let p 1 be the point in the line F, E) resulting from the shear S ∆ ν 0 ν 1 associated to the triangle ∆ applied to the point p 0 , and let p 2 be the point in the line L ν 0 (G, F, E) = L ν 0 (E, F , G) resulting from the shear S ∇ ν 1 ν 0 associated to the triangle ∇ applied to the point p 1 .Then This was the case going counterclockwise around the j-th diamond; see Figure 10.If instead one goes clockwise around the diamond, then the total shearing is −Z −1 j .Proof.See [FG06b,GMN14].Similar to that of Proposition 13; see also [Dou20,§2.23].Warning: In this subsection, we will consider snake-heads η in the set of corner vertices {(n − 1, 0, 0), (0, n − 1, 0), (0, 0, n − 1)} other than η = (n − 1, 0, 0); see Remark 9. We will also consider both (left) snakes and right snakes; see Remark 16. 1.6.1.Snake sequences.For the left setting: define a snake-head η ∈ Γ(Θ n−1 ) and two (left) snakes σ bot , σ top , called the bottom and top snakes, respectively, by See Figure 11.Similarly, for the right setting: define η and right snakes σ bot , σ top by In either left or right setting, consider a sequence of snakes having the same snake-head η as do σ bot and σ top , such that (σ , σ +1 ) is an adjacent pair; see Figure 11.Note that this sequence of snakes is not in general unique.For the Nmany projective bases [U ] = [ u 1 , u 2 , . . ., u n ] associated to the snakes σ , choose a common normalization u n = u n ∈ L η (resp.u 1 = u 1 ∈ L η ) when working in the left (resp.right) setting.Then, the change of basis matrix B U bot U top can be decomposed as (see Here, the matrices B U U +1 are computed in Proposition 13 (resp.Proposition 15) in the left (resp.right) setting, and in particular are completely determined by the triangle invariants Of course, the matrix B U bot U top is by definition independent of the choice of snake sequence (σ ) .For concreteness, we will make a preferred choice of such sequence, depending on whether we are in the left or right setting; these two choices are illustrated in Figure 11.1.6.2.Algebraization.Let A be a commutative algebra ( §1.4.1).For i = 1, 2, . . ., and n − 1 is the number of edge invariants Z j = j (E, G, F, F ) on a single edge.
As a notational convention, given a family M ∈ M n (A) of n × n matrices, put Definition 23.The left matrix M left (X i 's) in SL n (A) is defined by where the matrix S left (X abc ) is the -th left-shearing matrix; see §1.4.1.
Similarly, the right matrix M right (X i 's) in SL n (A) is defined by where the matrix S right (X abc ) is the -th right-shearing matrix; see §1.4.1.
Lastly, the edge matrix M edge (Z j 's) in SL n (A) is defined by where the matrix S edge (Z ) is the -th edge-shearing matrix; see §1.4.1.See Figure 12.
Remark 24.In the case where A = C and the X i = τ abc (E, F, G) and Z j = j (E, G, F, F ) in C−{0} are the triangle and edge invariants (as in §1.4.3, 1.4.4,1.4.5):then, the left and right matrices M left (X i 's) and M right (X i 's) recover the change of basis matrix B U bot U top /Det 1/n of Eq. ( * ) in the left and right setting, respectively, normalized to have determinant 1, and decomposed in terms of our preferred snake sequence (Figure 11); and, the edge matrix M edge (Z j 's) is the normalization B U U /Det 1/n of the change of basis matrix from Proposition 17. Crucially, these normalizations require choosing n-roots of the invariants X i and Z j .2. Quantum matrices Although we will not use explicitly the geometric results of the previous section, those results motivate the algebraic objects that are the main focus of the present work.
Definition 25.The quantum torus (with n-roots) T ω (P) associated to P is the quotient of the free algebra by the two-sided ideal generated by the relations ) n .We refer to the X ±1/n i as generators, and the X i as quantum coordinates, or just coordinates.Define the subset of fractions Written in terms of the coordinates X i and the fractions r ∈ Z/n, the relations above become 2.1.2.Matrix algebras.
Definition 26.Let T be a, possibly non-commutative, complex algebra, and let n be a positive integer.The matrix algebra with coefficients in T, denoted M n (T), is the complex vector space of n × n matrices, equipped with the usual "left-to-right" multiplicative structure.Namely, the product MN of two matrices M and N is defined entry-wise by Here, we use the usual convention that the entry M ij of a matrix M is the entry in the i-th row and j-th column.Note that, crucially, the order of M ik and N kj in the above equation matters since these elements might not commute.
2.1.3.Weyl quantum ordering.If T is a quantum torus, then there is a linear map called the Weyl quantum ordering, defined on words and extended linearly.The Weyl ordering is specially designed to satisfies the symmetry for every permutation σ of {1, . . ., k}.Consequently, there is induced a linear map on the associated commutative Laurent polynomial algebra.The Weyl ordering induces a linear map of matrix algebras Note the Weyl ordering [−] depends on the choice of ω 1/2 ; see the beginning of §2.
2.2.Fock-Goncharov quantum torus for a triangle.Let Γ(Θ n ) denote the set of corner vertices Γ(Θ n ) = {(n, 0, 0), (0, n, 0), (0, 0, n)} of the discrete triangle Θ n ; see §1.2.3.Define a set function using the quiver with vertex set Θ n − Γ(Θ n ) illustrated in Figure 13.The function P is defined by sending the ordered tuple (ν 1 , ν 2 ) of vertices of Θ n − Γ(Θ n ) to 2 (resp.−2) if there is a solid arrow pointing from ν 1 to ν 2 (resp.ν 2 to ν 1 ), to 1 (resp.−1) if there is a dotted arrow pointing from ν 1 to ν 2 (resp.ν 2 to ν 1 ), and to 0 if there is no arrow connecting ν 1 and ν 2 .Note that all of the small downward-facing triangles are oriented clockwise, and all of the small upward-facing triangles are oriented counterclockwise.By labeling the vertices of Θ n − Γ(Θ n ) by their coordinates (a, b, c) we may think of the function P as a N × N anti-symmetric matrix P = (P abc,a b c ) called the Poisson matrix associated to the quiver.
Definition 27.The Fock-Goncharov quantum torus associated to the discrete n-triangle Θ n is the quantum torus T ω (P) defined by the N × N Poisson matrix P, with generators ] is the commutative algebra of Laurent polynomials in the variables X 1/n i .As a notational convention, for j = 1, 2, . . ., n−1 we write Z ±1/n j (resp.Z ±1/n j and Z ±1/n j ) in place of X ±1/n j0(n−j) (resp.X ±1/n j(n−j)0 and X ±1/n 0j(n−j) ); see Figure 14.So, triangle-coordinates will be denoted X i = X abc for (a, b, c) ∈ int(Θ n ) while edge-coordinates will be denoted Z j , Z j , Z j .2.3.Quantum left and right matrices.
where we have used the identification ] ∼ = T 1 n , we now use these matrices to define the primary objects of study.Definition 28.The quantum left matrix L ω in M n (T ω n ) is defined by ∈ M n (T ω n ) where we have applied the Weyl quantum ordering [−] discussed in §2.3.1 to the product M edge (Z j 's)M left (X i 's)M edge (Z j 's) of classical matrices in M n (T 1 n ); see Figure 14.Similarly, the quantum right matrix R ω in M n (T ω n ) is defined by 2.4.1.Quantum SL n and its points.Let T be a, possibly non-commutative, algebra.
Definition 29.We say that a 2 We say that a matrix M ∈ M 2 (T) is a T-point of the quantum special linear group SL q 2 , denoted M ∈ SL q 2 (T) ⊆ M q 2 (T) ⊆ M 2 (T), if M ∈ M q 2 (T) and the quantum determinant Det q (M) = da − qbc = ad − q −1 bc = 1 ∈ T.
These notions are also defined for n × n matrices, as follows.
for all i < j and k < m, where 1 i, j, k, m n.
There is a notion of the quantum determinant Det q (M) ∈ T of a T-point M ∈ M q n (T).A matrix M ∈ M n (T) is a T-point of the quantum special linear group SL q n , denoted M ∈ SL q n (T) ⊆ M q n (T) ⊆ M n (T), if both M ∈ M q n (T) and Det q (M) = 1.The definitions satisfy the property that if a T-point M ∈ M q n (T) ⊆ M n (T) is a triangular matrix, then the diagonal entries M ii ∈ T commute, and Det q (M) = i M ii ∈ T.
Remark 31.The subsets M q n (T) ⊆ M n (T) and SL q n (T) ⊆ M n (T) are generally not closed under matrix multiplication (see, however, the sketch of proof below for a relaxed property).
Sketch of proof (see §3 for more details).In the case n = 2, this is an enjoyable calculation.When n 3, the argument hinges on the following well-known fact: If T is an algebra with subalgebras T , T ⊆ T that commute in the sense that a a = a a for all a ∈ T and a ∈ T , and if n .Put M FG := L ω , the Fock-Goncharov quantum left matrix, say.The proof will go the same for the quantum right matrix.The strategy is to see M FG ∈ M n (T ω n ) as the product of simpler matrices, over mutually-commuting subalgebras, that are themselves points of SL q n .More precisely, for a fixed sequence of adjacent snakes σ bot = σ 1 , σ 2 , . . ., σ N = σ top moving left across the triangle from the bottom edge to the top-left edge, we will define for each i = 1, . . ., N − 1 an auxiliary algebra S ω j i called a snake-move algebra, j i ∈ {1, . . ., n − 1}, corresponding to the adjacent snake pair (σ i , σ i+1 ).As a technical step, there is a distinguished subalgebra We construct an algebra embedding T L i S ω j i .Through this embedding, we may view M FG ∈ M n (T L ) ⊆ M n ( i S ω j i ).Following, we construct, for each i, a matrix Since by definition the subalgebras S ω j i , S ω j i ⊆ i S ω j i commute if i = i , as they constitute different tensor factors of i S ω j i , it follows from the essential fact mentioned above that M := in other words M ∈ SL q n ( i S ω j i ).Now, since this matrix product M, as well as the quantum left matrix M FG , are being viewed as elements of M n ( i S ω j i ), it makes sense to ask whether M FG ?
= M ∈ M n ( i S ω j i ).Indeed, this turns out to be true, implying that M FG ∈ SL q n ( i S ω j i ).Since, as we know, M FG ∈ M n (T L ) ⊆ M n ( i S ω j i ), we conclude that M FG is in SL q n (T L ) ⊆ SL q n (T ω n ).2.5.Example.Consider the case n = 4; see Figure 15.On the right hand side we show the commutation relations in the quantum torus T ω 4 , recalling Figure 13, but viewed in Θ n−1 (compare Figures 9 and 10).For instance, some sample commutation relations are: Remark 35.The quiver of Figure 17 for the tail-move quantum torus is divided into a bottom and top side.Similarly, the quiver of Figure 16 for a diamond-move quantum torus has a bottom and top side, connected by a diagonal.Conceptually speaking, as illustrated in the figures, we think of the bottom side (with un-primed generators z j ) as the top "snakeskin" σ 1/2 of a snake σ that has been "split in half down its length".Similarly, we think of the top side (with primed generators z j ) as the bottom snake-skin σ 1/2 of a split snake σ .Compare Figures 3 and 8, illustrating snakes in the classical setting.This snake splitting can be seen more clearly in the quantum snake sweep (see §3.3 and Figure 18 below) determined by the sequence of adjacent snakes σ bot = σ 1 , σ 2 , . . ., σ N = σ top , where each snake σ i is split in half, so that each half's snake-skin forms a side in one of two adjacent snake-move quantum tori.In the figure, the other halves of the bottom-most and top-most quantum snakes (colored grey) can be thought of as either living in other triangles or not existing at all.Prior to splitting a snake σ in half, the snake consists of n − 1 "vertebrae" connecting the n snake-vertices σ k ∈ Θ n−1 .Upon splitting the snake, the j-th vertebra splits into two generators z j and z j living in adjacent snake-move quantum tori.
3.2.Quantum snake-move matrices.We turn to the key observation for the proof.
Proposition 36.For j = 1, . . ., n − 1, the j-th quantum snake-move matrix is a S ω j -point of the quantum special linear group SL q n .That is, M j ∈ SL q n (S ω j ) ⊆ M n (S ω j ).Note the use of the Weyl quantum ordering; see §2.1.3.Here, the matrices S edge j (z) and S left j (x) for z, x in the commutative algebra S 1 j are defined as in §1.4.1; see also §2.3.1-2.3.2.Note when j = 1, the matrix S left 1 (x 0 ) = S left 1 is well-defined, despite x 0 not being defined.
Proof.This is a direct calculation, checking that the entries of the matrix M j satisfy the relations of the dual quantum group SL q n in the j-th snake-move quantum torus S ω j .For example, in the case n = 4, j = 3, the lemma says that the matrix is in SL q 4 (S ω 3 ).The Weyl ordering is needed to satisfy the quantum determinant relation.
Figure 18.Quantum snake sweep (n = 4); compare Figure 11a 3.3.Technical step: embedding a distinguished subalgebra T L of T ω n into a tensor product of snake-move quantum tori.For the snake-sequence (σ i ) i=1,...,N , to each pair (σ i , σ i+1 ) of adjacent snakes we associate a snake-move quantum torus S ω j i , recalling Remark 35 and Figure 18.Recall the Fock-Goncharov quantum torus T ω n ; see Figures 13 and 15.We now take a technical step.Define T L ⊆ T ω n ("L" for "Left") to be the subalgebra generated by all the generators (and their inverses) of T ω n except for Z ±1/n 1 , . . ., Z ±1/n n−1 ; see Figures 14 and 15.We claim that the snake-sequence (σ i ) i (Figure 11) induces an embedding of algebras, realizing T L ⊆ T ω n as a subalgebra of the tensor product of the snake-move quantum tori S ω j i (tensored from left to right) associated to the adjacent snake pairs (σ i , σ i+1 ).We explain the embedding through an example, in the case n = 4; see Figure 19 (compare Figure 18).There, the generator X 2 (emphasized in the figure), for instance, is mapped to Similarly, the generators Z 1 and Z 3 , say, are mapped to The remaining generators Z 2 , Z 3 , X 1 , X 1 , Z 2 , Z 1 are mapped to The desired algorithm grouping the terms in A, where there is one grouping per term in A L , is defined by selecting an ungrouped term a ∈ A and commuting it left or right until it is next to r(a) ∈ A L .This commutation is possible by part (2) of Lemma 38, that is, ( †).
3.5.Setup for the quantum right matrix.We end with a few words about the proof for the quantum right matrix M FG = R ω , which essentially goes the same as for the left matrix.
(i) The right version of the j-th snake algebra S ω j for j = 1, 2, . . ., n−1 is given by replacing the quivers of

Figure 1 .
Figure 1.Discrete triangle, and triangle invariants for a generic flag triple

Figure 3 .
Figure 3. Snake then there exist unique nonzero covectors u left k+1 and u right k+1 in the lines L σ left k+1 and L σ right k+1 namely X abc is the Fock-Goncharov triangle invariant ( §1.2.4) associated to the generic flag triple (E, F, G) and the internal vertex (a, b, c) ∈ int(Θ n ).

2.4. 2 .
Main result.Take T = T ω n = T ω n (Θ n ) to be the Fock-Goncharov quantum torus for the discrete n-triangle Θ n ; see §2.2.For what follows, recall Definition Theorem 32.The quantum left and right matrices

Figure 19 .
Figure 19.Embedding T L in the tensor product of snake-move quantum tori Figures 16 and 17 by the quivers shown in Figures 20 and 21, respectively.(ii)The j-th quantum snake-move matrix M j of Proposition 36 is replaced byM j := j = 1, the matrix S right 1 (x n ) = S right 1 is well-defined, despite x n not being defined.(iii) The subalgebra T R ⊆ T ωn is generated by all but the Z ±1/n j 's; see Figures14 and 15.