AA TALE OF TWO SHUFFLE ALGEBRAS
ANDREI NEGUT , Abstract.
As a quantum affinization, the quantum toroidal algebra U q,q ( ¨ gl n )is defined in terms of its “left” and “right” halves, which both admit shufflealgebra presentations ([6], [11]). In the present paper, we take an orthogonalviewpoint, and give shuffle algebra presentations for the “top” and “bottom”halves instead, starting from the evaluation representation U q ( ˙ gl n ) (cid:121) C n ( z )and its usual R –matrix R ( z ) ∈ End( C n ⊗ C n )( z ) (see [8]). An upshot of thisconstruction is a new topological coproduct on U q,q ( ¨ gl n ) which extends theDrinfeld-Jimbo coproduct on the horizontal subalgebra U q ( ˙ gl n ) ⊂ U q,q ( ¨ gl n ). Introduction U q ( ˙ sl n ) = U q ( (cid:98) sl n ) (hats will be replaced by pointsin the present paper) has the following two presentations: • it is the quantum affinization of U q ( sl n ) • it is the Drinfeld-Jimbo quantum group whose Dynkin diagram is an n -cycleHowever, the two presentations above yield different bialgebra structures on U q ( ˙ sl n ),which is evidenced by the fact that the coproduct in the first bullet is only topo-logical (i.e. ∆ is an infinite sum, which only makes sense in a certain completion).Moreover, the two bullets above yield different triangular decompositions of U q ( ˙ sl n )into positive, Cartan, and negative halves:(1.1) U q ( ˙ sl n ) ∼ = U ← q ( ˙ sl n ) ⊗ (Cartan subalgebra) ⊗ U → q ( ˙ sl n )(1.2) U q ( ˙ sl n ) ∼ = U ↑ q ( ˙ sl n ) ⊗ (Cartan subalgebra) ⊗ U ↓ q ( ˙ sl n )The two decompositions above are quite different: the positive subalgebra U → q ( ˙ sl n )of (1.1) is generated by Drinfeld’s elements e i,k over all 1 ≤ i < n and k ∈ Z , whilethe positive subalgebra U ↑ q ( ˙ sl n ) of (1.2) is generated by the Drinfeld-Jimbo ele-ments { e i } i ∈ Z /n Z . The connection between these two presentations was given in [1].1.2. The main purpose of the present paper is to extend the description above tothe quantum toroidal algebra U q,q ( ¨ gl n ), which is defined as in the first bullet: U q,q ( ¨ gl n ) := affinization of U q ( ˙ gl n )This construction naturally comes with a triangular decomposition (see Subsection3.11 for an overview of the quantum toroidal algebra, as well as of our conventions):(1.3) U q,q ( ¨ gl n ) ∼ = (cid:101) U ← q,q ( ¨ gl n ) ⊗ (cid:101) U → q,q ( ¨ gl n ) a r X i v : . [ m a t h . QA ] F e b ANDREI NEGUT , Our (cid:101) U ← q,q ( ¨ gl n ) and (cid:101) U → q,q ( ¨ gl n ) are the Borel subalgebras of the quantum toroidalalgebra, and they explicitly arise as a unipotent part tensored with a Cartan part: (cid:101) U → q,q ( ¨ gl n ) ∼ = U → q,q ( ¨ gl n ) ⊗ U ≥ q ( ˙ gl ) n (1.4) (cid:101) U ← q,q ( ¨ gl n ) ∼ = U ← q,q ( ¨ gl n ) ⊗ U ≤ q ( ˙ gl ) n (1.5)There is a well-known topological coproduct of U q,q ( ¨ gl n ), which preserves the sub-algebras (1.4) and (1.5), and extends the (almost) cocommutative coproduct on the“vertical” subalgebra:(1.6) U ≥ q ( ˙ gl ) n ⊗ U ≤ q ( ˙ gl ) n = U q ( ˙ gl ) n ⊂ U q,q ( ¨ gl n )The main goal of this paper is to define another decomposition into subalgebras:(1.7) U q,q ( ¨ gl n ) ∼ = (cid:101) U ↑ q,q ( ¨ gl n ) ⊗ (cid:101) U ↓ q,q ( ¨ gl n )(see Corollary 3.28). We will explicitly construct the tensor factors of (1.7) as: (cid:101) U ↑ q,q ( ¨ gl n ) ∼ = U ↑ q,q ( ¨ gl n ) ⊗ U ≥ q ( ˙ gl n )(1.8) (cid:101) U ↓ q,q ( ¨ gl n ) ∼ = U ↓ q,q ( ¨ gl n ) ⊗ U ≤ q ( ˙ gl n )(1.9)where the “horizontal” subalgebra:(1.10) U ≥ q ( ˙ gl n ) ⊗ U ≤ q ( ˙ gl n ) = U q ( ˙ gl n ) ⊂ U q,q ( ¨ gl n )will be the quantum group in the RTT presentation ([8]). Moreover, we endow U q,q ( ¨ gl n ) with a new topological coproduct which preserves the subalgebras (1.8),(1.9), and extends the usual (Drinfeld-Jimbo) coproduct on U q ( ˙ gl n ) ⊂ U q,q ( ¨ gl n ).1.3. To represent the aforementioned decompositions pictorially, we will recall thatthe quantum toroidal algebra is graded by Z n × Z , where Z n is the root lattice of U q ( ˙ sl n ) and Z is the affinization direction. Then the following picture indicates thevarious subalgebras of U q,q ( ¨ gl n ), by displaying which degrees they live in: (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:54) (cid:45) { U ↓ q,q ( ¨ gl n ) U ≥ q ( ˙ gl n ) { U ↑ q,q ( ¨ gl n ) U ≤ q ( ˙ gl n ) } U ← q,q ( ¨ gl n ) } U → q,q ( ¨ gl n ) Z Z n U ≤ q ( ˙ gl ) n U ≥ q ( ˙ gl ) n F IGURE
0. The grading of U q,q ( ¨ gl n ) and its various subalgebras TALE OF TWO SHUFFLE ALGEBRAS 3
In the particular case n = 1, the quantum toroidal algebra is isomorphic to thewell-known Ding-Iohara-Miki ([4], [17]) a.k.a. elliptic Hall algebra ([2], [25]), whichhas an action of SL ( Z ) by automorphisms. In particular, the decomposition(1.7) is obtained from the decomposition (1.3) by applying the automorphismcorresponding to rotation by 90 degrees. However, in the general n case, thealgebras featuring in the two decompositions are not isomorphic to each other,which is sensible given the fact that the grading axes Z n and Z are quite different.1.4. To describe U ← q,q ( ¨ gl n ) and U → q,q ( ¨ gl n ) of (1.3), let us consider the vector space:(1.11) S + ⊂ (cid:77) ( d ,...,d n ) ∈ N n Q ( q, q n )( z , ..., z d , ..., z n , ..., z nd n ) Sym of rational functions which satisfy the wheel conditions (as in [9], [11]): namely thatsuch rational functions have at most simple poles at z ia q − z i +1 ,b (for all i, a, b )and that the residue at such a pole is divisible by z ia (cid:48) − z i +1 ,b and z ia − z i +1 ,b (cid:48) forall a (cid:48) (cid:54) = a and b (cid:48) (cid:54) = b . The vector space (1.11) is called a shuffle algebra, akin tothe classical construction of Feigin and Odesskii concerning certain elliptic algebras([11]). Explicitly, the product on (1.11) is constructed using the function (3.37),see Definition 3.15. An algebra homomorphism was constructed in [6]: U → q,q ( ¨ gl n ) → S + and it was shown to be an isomorphism in [21]. Similarly, U ← q,q ( ¨ gl n ) ∼ = S − = ( S + ) op .To describe the subalgebras U ↑ q, ¯ q ( ¨ gl n ) and U ↓ q, ¯ q ( ¨ gl n ) which appear in (1.7), we willintroduce a new kind of shuffle algebra (let V be an n –dimensional vector space):(1.12) A + ⊂ ∞ (cid:77) k =0 End Q ( q,q n ) ( V ⊗ ... ⊗ V (cid:124) (cid:123)(cid:122) (cid:125) k factors )( z , ..., z k )and the algebra structure on the RHS is constructed using the R –matrix (4.1), seePropositions 4.5 and 4.6. By definition, the subspace (1.12) precisely consists ofEnd( V ⊗ k )–valued rational functions which have at most simple poles at z a q − z b (for all a, b ) and whose residue at such a pole satisfies the conditions outlined inDefinition 4.8. The subalgebra A − is defined similarly, but with q − q − n instead of q . Theorem 1.5.
There exist injective algebra homomorphisms: A + , A − , op (cid:44) → U q,q ( ¨ gl n ) Denoting the images of these maps by U ↑ q,q ( ¨ gl n ) and U ↓ q,q ( ¨ gl n ) yields the decompo-sition (1.7) . Moreover, there exist topological coproducts on the subalgebras: (cid:101) A + = A + ⊗ U ≥ q ( ˙ gl n ) (cid:44) → U q,q ( ¨ gl n )(1.13) (cid:101) A − , op = ( A − ⊗ U ≤ q ( ˙ gl n )) op (cid:44) → U q,q ( ¨ gl n )(1.14) which extend the Drinfeld-Jimbo coproduct on the horizontal subalgebra (1.10) ,and realize U q,q ( ¨ gl n ) as the Drinfeld double of its subalgebras (1.13) and (1.14) . ANDREI NEGUT , In [12], the authors claim that U q,q ( ¨ gl n ) admits a Drinfeld double structure viagenerators and relations, but there seem to be fundamental differences between thesubalgebra called B in loc. cit. and our U ↑ q,q ( ¨ gl n ) ⊂ U q,q ( ¨ gl n ). For example, in de-gree Z n ×{ } , the former algebra has elements parametrized by the positive roots of U q ( ˙ sl n ), while the latter has elements parametrized by all roots of U q ( ˙ sl n ). There-fore, we do not make any claims concerning the connection of our work with loc. cit. We emphasize the fact that U ↑ q,q ( ¨ gl n ) is not the same as the “vertical subalgebra”that was studied in [10] and numerous other works. The latter construction hasto do with U q ( ˙ sl n ) presented as the affinization of U q ( sl n ) and thus implicitlybreaks the symmetry among the vertices of the cyclic quiver. Meanwhile, ourconstruction takes the “horizontal subalgebra” U q ( ˙ gl n ) and its evaluation repre-sentation V = C n ( z ) as an input, and outputs half of the quantum toroidal algebra.More generally, starting from a quantum group U q ( g ) and a representation V en-dowed with a unitary R –matrix, one may ask if the double shuffle algebra:(1.15) D (cid:32) an appropriate subalgebra of ∞ (cid:77) k =0 End( V ⊗ k ) (cid:33) (defined as in Section 2) is related to the quantum group U q ( ˙ g ). Theorem 1.5 dealswith the case g = ˙ gl n and V = C n ( z ). If something along these lines is true inaffine types other than A , we venture to speculate that the algebra (1.15) might berelated to the extended Yangians of [28], appropriately q –deformed and doubled.Such a realization of quantum affinizations is to be expected from the work [16],[23], who have realized Yangians and their q –deformations inside endomorphismrings of tensor products of certain geometrically defined representations of g .1.6. The structure of the present paper is the following: • In Section 2, we construct a shuffle algebra A + starting from a vector space V and a unitary R –matrix ∈ End( V ⊗ ) (see also [19]). By adding certain elements,we construct the extended shuffle algebra (cid:101) A + , which admits a coproduct.From two such extended shuffle algebras, we construct their Drinfeld double A . • In Section 3, we recall the quantum group U q,q ( ¨ gl n ) and its PBW presenta-tion from [22]. This will allow us to construct the decomposition (1.7) as algebras. • In Section 4, we construct a version of the shuffle algebra of Section 2 thatcorresponds to the R –matrix with spectral parameter (4.1), thus yielding (1.12). • In Section 5, we construct the extended version of the shuffle algebra ofSection 4, endow it with a topological coproduct, and construct a PBW basis of it. • In Section 6, we construct a bialgebra pairing between two copiesof the extended shuffle algebras of Section 5. The corresponding Drinfeld double will precisely match U q,q ( ¨ gl n ), thus completing the proof of Theorem 1.5. TALE OF TWO SHUFFLE ALGEBRAS 5
I would like to thank Pavel Etingof, Sachin Gautam, Victor Kac, Andrei Okounkov,and Alexander Tsymbaliuk for many valuable conversations, and all their helpalong the years. I gratefully acknowledge the NSF grants DMS–1600375, DMS–1760264 and DMS–1845034, as well as support from the Alfred P. Sloan Foundation.1.7. Given a finite-dimensional vector space V , we will often write elements X ∈ End( V ⊗ k ) as X ...k in order to point out the set of indices of X . If V = C n , then:(1.16) X = (cid:88) i ,...,i k ,j ,...,j k coefficient · E i j ⊗ ... ⊗ E i k j k for certain coefficients, where E ij ∈ End( V ) denotes the matrix with entry 1 on row i and column j , and 0 everywhere else. For any permutation σ ∈ S ( k ), we write:(1.17) σXσ − = X σ (1) ...σ ( k ) where σ (cid:121) V ⊗ k by permuting the factors (therefore, the effect of conjugating (1.16)by σ is to replace the indices i , ..., j k by i σ (1) , ..., j σ ( k ) ). Moreover, we will write:(1.18) X ...k = X ...i ⊗ X i +1 ...k ∈ End( V ⊗ i ) ⊗ End( V ⊗ k − i ) ∼ = End( V ⊗ k )if we wish to set apart the first i tensor factors from the last k − i tensor factors of X . There is an implicit summation in the right-hand side of (1.18) which we willnot write down, much alike Sweedler notation. For any a ∈ N , we will write: E ( a ) ij = 1 ⊗ ... ⊗ E ij (cid:124)(cid:123)(cid:122)(cid:125) a –th position ⊗ ... ⊗ ∈ End( V ⊗ k )(the number k ≥ a will always be clear from context). More generally, for any X ∈ End( V ⊗ k ) and any collection of distinct natural numbers a , ..., a k , write: X a ...a k ∈ End( V ⊗ N )(the number N ≥ a , ..., a k will always be clear from context) for the image of X under the map End( V ⊗ k ) → End( V ⊗ N ) that sends the i –th factor of the domainto the a i –th factor of the codomain, and maps to the unit in all factors (cid:54) = { a , ..., a k } .2. Shuffle algebras and R –matrices • a vector space V , assumed finite-dimensional for simplicity • an element ( R –matrix) R ∈ Aut( V ⊗ ) satisfying the Yang-Baxter equation:(2.1) R R R = R R R • an element (cid:101) R ∈ Aut( V ⊗ ) satisfying:(2.2) (cid:101) R (cid:101) R R = R (cid:101) R (cid:101) R (2.3) R (cid:101) R (cid:101) R = (cid:101) R (cid:101) R R
12 ANDREI NEGUT , • a scalar f so that:(2.4) R R = f · Id V ⊗ V = R R The present Section will be concerned with generalities in the context above, whileSection 4 deals with a particular setting, namely that of:(2.5) R ( x ) = RHS of (4.1) ∈ End( C n ⊗ C n )( x )and (cid:101) R ( x ) = R (cid:0) x − q − (cid:1) , for a parameter q . Many Propositions in the currentSection have counterparts in Section 4, and we will only prove such statements once.2.2. We will represent the tensor product V ⊗ k as k labeled dots on a vertical line,and certain elements of End( V ⊗ k ) will be represented as braids between two col-lections of labeled dots situated on parallel vertical lines. Specifically, the crossingsbelow represent either the automorphisms R or (cid:101) R , with indices given by the labelsof the strands (which are inherited from the labels of their endpoints): Figure 1.
Various crossingsThe strands are represented either as straight or squiggly, because we wish toindicate whether the picture in question refers to either R or (cid:101) R . Compositions arealways read left-to-right, for example the following equivalence of braids underliesthe Yang-Baxter relations (2.1): Figure 2.
Reidemeister III move - version 1while the following equivalences underlie equations (2.2) and (2.3), respectively:
TALE OF TWO SHUFFLE ALGEBRAS 7
Figure 3.
Reidemeister III move - version 2
Figure 4.
Reidemeister III move - version 3We will equivalate braids connected by the Reidemeister III type moves above.2.3. We will now recall the construction of Section 5.2 of [19] (itself a dualversion of the construction of [8]) and present it in the language of shufflealgebras. We will then construct an extended shuffle algebra which admits acoproduct and bialgebra pairing, and then define the corresponding Drinfeld double.
Proposition 2.4.
For
This procedure is simply a succession of the Reidemeister III moves of Figures 2,3and 4, which in the end produces the bottom picture.2.5. The symmetrization of a tensor X ∈ End( V ⊗ k ) is defined as:(2.8) Sym X = (cid:88) σ ∈ S ( k ) R σ · ( σXσ − ) · R − σ where σXσ − refers to the permutation of the tensor factors of X in accordancewith σ , and R σ ∈ End( V ⊗ k ) is any braid connecting the i –th endpoint on the rightwith the σ ( i )–th endpoint on the left. Figure 8.
A braid representation of R σ · ( σXσ − ) · R − σ Choosing one braid lift of σ over another is just the ambiguity of choosing R ab over R − ba for any crossing between the strands labeled a and b . Since (2.4) saysthat these two endomorphisms differ by a scalar, the ambiguity does not affect (2.8).A tensor Y ∈ End( V ⊗ k ) is called symmetric if:(2.9) Y = R σ · ( σY σ − ) · R − σ ∀ σ ∈ S ( k ). It is easy to see that any symmetrization (2.8) is a symmetric tensor. Proposition 2.6.
The shuffle product of Proposition 2.4 preserves the vector space: (2.10) A + ⊂ ∞ (cid:77) k =0 End( V ⊗ k ) consisting of symmetric tensors. We will call A + the shuffle algebra.Proof. Let A ...k ∈ End( V ⊗ k ) and B ...l ∈ End( V ⊗ l ) be any symmetric tensors. Apermutation µ ∈ S ( k + l ) is called a ( k, l )–shuffle if:(2.11) a := µ (1) < ... < a k := µ ( k ) b := µ ( k + 1) < ... < b l := µ ( k + l )Because of the diagram (depicted for k = 2, l = 2, ( a , a ) = (1 , b , b ) = (2 , , Figure 9. R µ · ( µ Φ µ − ) · R − µ it is easy to see that the definition (2.6) can be restated as:(2.12) A ∗ B = µ goes over (cid:88) ( k,l )–shuffles R µ · ( µ Φ µ − ) · R − µ where:(2.13) Φ = (cid:104) R k,k +1 ...R ,k + l (cid:105) A ...k (cid:104) (cid:101) R ,k + l ... (cid:101) R k,k +1 (cid:105) B k +1 ...k + l For any τ ∈ S ( k ) × S ( l ) ⊂ S ( k + l ), we have: R τ · ( τ Φ τ − ) · R − τ = Φwhich can be seen from the fact that the braid below: Figure 10. R τ · ( τ Φ τ − ) · R − τ is equivalent to Φ (since we can cancel the braids representing R τ and R − τ bypulling them through the symmetric tensors A and B ). Then (2.12) implies: A ∗ B = 1 k ! l ! µ goes over (cid:88) ( k,l )–shuffles (cid:88) τ ∈ S ( k ) × S ( l ) R µ · µ (cid:0) R τ · τ Φ τ − · R − τ (cid:1) µ − · R − µ Since any σ ∈ S ( k + l ) can be written uniquely as µ ◦ τ , where µ is a ( k, l )–shuffleand τ ∈ S ( k ) × S ( l ) ⊂ S ( k + l ), the formula above yields:(2.14) A ∗ B = 1 k ! l ! (cid:88) σ ∈ S ( k + l ) R σ · ( σ Φ σ − ) · R − σ = 1 k ! l ! · Sym Φ
TALE OF TWO SHUFFLE ALGEBRAS 11 (we have used the fact that R µτ = R µ · µR τ µ − , times a product of scalars (2.4)).Since the symmetrization of any tensor is symmetric, this concludes the proof. (cid:3) By analogy with formula (2.14), one has the following:(2.15) A ∗ B = 1 k ! l ! · Sym Ψwhere Ψ = A l +1 ...l + k (cid:104) (cid:101) R l +1 ,l ... (cid:101) R l + k, (cid:105) B ...l (cid:104) R l + k, ...R l +1 ,l (cid:105) .2.7. Let us fix a basis v , ..., v n of V and write E ij for the elementary symmet-ric matrix with a single 1 at the intersection of row i and column j , and 0 otherwise. Definition 2.8.
Consider the extended shuffle algebra: (cid:101) A + = (cid:68) A + , s ij , t ij (cid:69) ≤ i,j ≤ n (cid:46) relations (2.16) – (2.20) In order to concisely state the relations, it makes sense to package the new genera-tors s ij , t ij into generating functions: S = n (cid:88) i,j =1 s ij ⊗ E ij ∈ (cid:101) A + ⊗ End( V ) T = n (cid:88) i,j =1 t ij ⊗ E ij ∈ (cid:101) A + ⊗ End( V ) and the required relations take the form: (2.16) RS S = S S R (2.17) T T R = RT T (2.18) T (cid:101) RS = S (cid:101) RT as well as: (2.19) X ...k · S = S · R k ...R f k ...f X ...k (cid:101) R ... (cid:101) R k (2.20) T · X ...k = (cid:101) R k ... (cid:101) R X ...k R ...R k f ...f k · T ∀ X ...k ∈ End( V ⊗ k ) . The latter two formulas should be interpreted as identi-ties in (cid:101) A + ⊗ End( V ) , where the latter copy of V is the one represented by index 0. We write f ij = f ji for the scalar f ∈ Aut( V ⊗ V ), interpreted as an element of Aut( V ⊗ k ) viathe inclusion of the i –th and j –th tensor factors. This notation will come in handy in Section 5,when f ij will be a scalar-valued rational functions of variables z i /z j . , Proposition 2.9.
The following assignments make (cid:101) A + into a bialgebra: (2.21) ∆( S ) = (1 ⊗ S )( S ⊗ , ε ( S ) = Id(2.22) ∆( T ) = ( T ⊗ ⊗ T ) , ε ( T ) = Id while for all X = X ...k ∈ A + ⊂ (cid:101) A + for k ≥ , we set ε ( X ) = 0 and: (2.23) ∆( X ) = k (cid:88) i =0 ( S k ...S i +1 ⊗ (cid:32) X ...i ⊗ X i +1 ...k (cid:81) ≤ u ≤ i 1) == R (1 ⊗ S S )( S S ⊗ (2.16) = (1 ⊗ S S ) R ( S S ⊗ (2.16) = (1 ⊗ S S )( S S ⊗ R == (1 ⊗ S )( S ⊗ ⊗ S )( S ⊗ R (2.21) = ∆( S S R )An analogous argument shows that ∆( T T R ) = ∆( RT T ). As for (2.18):∆( T (cid:101) RS ) = ( T ⊗ ⊗ T ) (cid:101) R (1 ⊗ S )( S ⊗ (2.18) = ( T ⊗ S ) (cid:101) R ( S ⊗ T ) == (1 ⊗ S )( T ⊗ (cid:101) R ( S ⊗ ⊗ T ) (2.18) = (1 ⊗ S )( S ⊗ (cid:101) R ( T ⊗ ⊗ T ) = ∆( S (cid:101) RT )Let us now apply the coproduct to the left-hand side of (2.19):∆( X ...k S ) = k (cid:88) i =0 ( S k ...S i +1 ⊗ X ...i ⊗ X i +1 ...k )( T i +1 ...T k ⊗ ⊗ S )( S ⊗ (2.19) = k (cid:88) i =0 (1 ⊗ S ) R k ...R i +1 , f k ...f i +1 , ( S k ...S i +1 ⊗ X ...i ⊗ X i +1 ...k )( T i +1 ...T k ⊗ (cid:101) R i +1 , ... (cid:101) R k ( S ⊗ (2.18) = k (cid:88) i =0 (1 ⊗ S ) R k ...R i +1 , f k ...f i +1 , ( S k ...S i +1 ⊗ X ...i ⊗ X i +1 ...k )( S ⊗ (cid:101) R i +1 , ... (cid:101) R k ( T i +1 ...T k ⊗ (2.19) = k (cid:88) i =0 (1 ⊗ S ) R k ...R i +1 , f k ...f i +1 , ( S k ...S i +1 S ⊗ R i ...R f i ...f ( X ...i ⊗ X i +1 ...k ) (cid:101) R ... (cid:101) R k ( T i +1 ...T k ⊗ (2.16) = k (cid:88) i =0 (1 ⊗ S )( S ⊗ S k ...S i +1 ⊗ R k ...R f k ...f ( X ...i ⊗ X i +1 ...k ) (cid:101) R ... (cid:101) R k ( T i +1 ...T k ⊗ TALE OF TWO SHUFFLE ALGEBRAS 13 The last line above equals ∆(RHS of (2.19)), so we are done. The computationshowing that ∆ respects relation (2.20) is analogous, and therefore left to the inter-ested reader. As for the shuffle product itself, we must show that ∆( A ...k ∗ B ...l ) =∆( A ...k ) ∗ ∆( B ...l ). Applying formulas (2.6) and (2.23) implies:∆( A ...k ) ∗ ∆( B ...l ) = k (cid:88) d =0 l (cid:88) e =0 ( S k ...S d +1 ⊗ (cid:32) A ...d ⊗ A d +1 ,...k (cid:81) ≤ u ≤ d 1) = k (cid:88) d =0 l (cid:88) e =0 a <...b j R a k b e +1 ...R a d +1 b l (cid:124) (cid:123)(cid:122) (cid:125) if a i >b j In the next-to-last row of the expression above, we may apply (2.18) in order tomove the T ’s to the right and the S ’s to the left. Afterwards, we apply (2.19) and(2.20) to move the S ’s to the left of A a ...a d and the T ’s to the right of B b ...b e :∆( A ...k ) ∗ ∆( B ...l ) = k (cid:88) d =0 l (cid:88) e =0 a <...b j R a k b e +1 ...R a d +1 b l (cid:124) (cid:123)(cid:122) (cid:125) if a i >b j Finally, we may use (2.16) and (2.17) to move the outermost products of R a i b j pastthe S ’s and the T ’s, at the cost of re-ordering the latter:∆( A ...k ) ∗ ∆( B ...l ) = k (cid:88) d =0 l (cid:88) e =0 a <...b j R a d b ...R a b e (cid:124) (cid:123)(cid:122) (cid:125) if a i >b j increasing x (cid:89) x ∈{ a d +1 ,...,a k ,b e +1 ,...,b l } T x ⊗ The right-hand side is simply ∆ applied to the RHS of (2.6), as we needed to prove. (cid:3) (cid:101) A + , (cid:101) A − ,defined as in the previous Subsections with respect to the same R , but with: (cid:101) R + = (cid:101) R, (2.24) (cid:101) R − = (cid:16) ( (cid:101) R † ) − (cid:17) † (2.25)where End( V ⊗ k ) † s → End( V ⊗ k ) denotes the transposition of the s –tensor factor:(2.26) ( E i j ⊗ ... ⊗ E i s j s ⊗ ... ⊗ E i k j k ) † s = E i j ⊗ ... ⊗ E j s i s ⊗ ... ⊗ E i k j k It is an elementary exercise to show that if properties (2.2)–(2.3) hold for (cid:101) R + = (cid:101) R ,then they also hold for (cid:101) R − given by formula (2.25). We will now define a pairing:(2.27) (cid:101) A + ⊗ (cid:101) A − (cid:104)· , ·(cid:105) −→ ground fieldwhich respects the bialgebra structure in the following sense:(2.28) (cid:104) ab, c (cid:105) = (cid:104) b ⊗ a, ∆( c ) (cid:105) ∀ a, b ∈ (cid:101) A + , c ∈ (cid:101) A − (2.29) (cid:104) a, bc (cid:105) = (cid:104) ∆( a ) , b ⊗ c (cid:105) ∀ a ∈ (cid:101) A + , b, c ∈ (cid:101) A − We will henceforth write X ± for the copy of X ∈ End( V ⊗ k ) in (cid:101) A ± . The analogousnotation will apply to S ± , T ± ∈ (cid:101) A ± ⊗ End( V ). Proposition 2.11. The assignments: (2.30) (cid:104) S +2 , S − (cid:105) = (cid:101) R + , (cid:104) T +2 , T − (cid:105) = (cid:101) R − (2.31) (cid:104) S +2 , T − (cid:105) = Rf , (cid:104) T +2 , S − (cid:105) = R and for all X, Y ∈ End( V ⊗ k ) : (2.32) (cid:104) X + , Y − (cid:105) = 1 k ! Tr V ⊗ k XY (cid:89) ≤ i 32 (2.3) == (cid:101) R (cid:101) R R = (cid:104) S +2 ⊗ S +1 , S − ⊗ S − (cid:105) R 12 (2.28) = (cid:104) S +2 S +1 R , S − (cid:105) TALE OF TWO SHUFFLE ALGEBRAS 15 and: (cid:104) R S +1 S +2 , T − (cid:105) (2.28) = R (cid:104) S +2 ⊗ S +1 , T − ⊗ T − (cid:105) = R R R f f 31 (2.1) == R R R f f = (cid:104) S +1 ⊗ S +2 , T − ⊗ T − (cid:105) R 12 (2.28) = (cid:104) S +2 S +1 R , T − (cid:105) We leave the analogous formulas when (2.16) is replaced by (2.17), or when the rolesof S and T are switched, or when the roles of the two arguments of the pairing areswitched, as exercises to the interested reader. As for (2.18), we have: (cid:104) T +1 (cid:101) R S +2 , S − (cid:105) (2.28) = (cid:104) T +1 , S − (cid:105) (cid:101) R (cid:104) S +2 , S − (cid:105) = R (cid:101) R (cid:101) R 32 (2.2) == (cid:101) R (cid:101) R R = (cid:104) S +2 , S − (cid:105) (cid:101) R (cid:104) T +1 , S − (cid:105) (2.28) = (cid:104) S +2 (cid:101) R T +1 , S − (cid:105) The analogous formulas when S − is replaced by T − , or when the roles of thearguments of the pairing are switched, are left as exercises to the interested reader.To prove that (2.19) pairs correctly with elements of A − , note that (2.23) implies:∆( Y − ...k ) = Y − ...k ⊗ S − k ...S − ⊗ ⊗ Y − ...k )( T − ...T − k ⊗ 1) + ... where ... stands for summands in which Y − ...k has a non-zero number of indices oneither side of the ⊗ sign. Then we have: (cid:104) X +1 ...k S +0 , Y − ...k (cid:105) (2.28) = (cid:68) S +0 ⊗ X +1 ...k , ( S − k ...S − ⊗ ⊗ Y − ...k )( T − ...T − k ⊗ (cid:69) =(2.33) = 1 k ! Tr V ⊗ k X ...k (cid:32) (cid:101) R † k ... (cid:101) R † Y ...k R † ...R † k f ...f k (cid:33) † (cid:89) ≤ i 1) + ... where the ellipsis denotes summands whose second tensor factor has a non-zeronumber of indices on either side of the ⊗ sign. Because of this, formula (2.29) whenone of b and c is either S − or T − (which we have already checked yields a consistentbialgebra pairing) implies that: (cid:104) B +1 ...l , S − l + k ...S − l +1 Y − ...l T − l +1 ...T − k + l (cid:105) = (cid:68) ∆ (2) ( B +1 ...l ) , S − l + k ...S − l +1 ⊗ Y − ...l ⊗ T − l +1 ...T − k + l (cid:69) = Tr (cid:16) [ (cid:101) R + l +1 ,l ... (cid:101) R + k + l, ] B ...l [ R k + l, ...R l +1 ,l ] Y ...l (cid:17) Therefore, the RHS of (2.36) is precisely equal to the LHS of (2.36), as required.Similarly, proving (2.29) for a = X + , b = A − , c = B − boils down to the equality:1 k ! l ! Tr V ⊗ k + l [ R k,k +1 ...R ,k + l ] A ...k [ (cid:101) R − ,k + l ... (cid:101) R − k,k +1 ] B k +1 ...k + l X ...k + l (cid:89) ≤ i TALE OF TWO SHUFFLE ALGEBRAS 17 A = (cid:101) A + ⊗ (cid:101) A − , op , coop such that (cid:101) A + ∼ = (cid:101) A + ⊗ (cid:101) A − , op , coop ∼ = 1 ⊗ (cid:101) A − op , coop are sub-bialgebras of A ,and the commutation of elements coming from the two factors is governed by:(2.39) (cid:104) a , b (cid:105) a b = b a (cid:104) a , b (cid:105) for all a ∈ (cid:101) A + and b ∈ (cid:101) A − , op , coop . Let us now spell out the definingrelations between the generators of the double algebra (2.38). The qua-dratic relations (2.16)–(2.20) hold as stated between the + generators, andhold with the opposite multiplication between the − generators. As for therelations that involve one of the + generators and one of the − generators, we have: Proposition 2.13. We have the following formulas in A : (2.40) S +2 (cid:101) RS − = S − (cid:101) RS +2 , RS +1 T − = T − S +1 R (2.41) T +1 (cid:101) RT − = T − (cid:101) RT +1 , RT +2 S − = S − T +2 R as well as: (2.42) S ∓ · ± X ± ...k = (cid:101) R ± k ... (cid:101) R ± X ± ...k R ...R k · ± S ∓ (2.43) X ± ...k · ± T ∓ = T ∓ · ± R k ...R X ± ...k (cid:101) R ± ... (cid:101) R ± k where · + = · and · − = · op (the opposite multiplication in A ). Finally, we have: (2.44) [ E + ij , E − i (cid:48) j (cid:48) ] = s + j (cid:48) i t + ji (cid:48) − t − j (cid:48) i s − ji (cid:48) for all i, j, i (cid:48) , j (cid:48) ∈ { , ..., n } .Proof. Let us now prove the first formula in (2.40) and leave the second one and(2.41) as exercises for the interested reader. Since:∆( S + ) = (1 ⊗ S + )( S + ⊗ 1) in (cid:101) A + (2.45) ∆( S − ) = ( S − ⊗ ⊗ S − ) in (cid:101) A − , op , coop (2.46)formula (2.39) for a = S +2 and b = S − implies: S +2 (cid:104) S +2 , S +1 (cid:105) S − = S − (cid:104) S +2 , S − (cid:105) S +2 Using (2.30) to evaluate the pairing implies precisely the first formula in (2.40).Let us now prove (2.42) and leave the analogous formula (2.43) as an exercise. Wewill do so in the case ± = +, as ± = − just involves the opposite of all relations.∆( X +1 ...k ) = X +1 ...k ⊗ S + k ...S +1 ⊗ ⊗ X +1 ...k )( T +1 ...T + k ⊗ 1) + ... where the rightmost ellipsis stands for terms which have a non-zero number ofindices on either side of the ⊗ sign, so they pair trivially with S − . Meanwhile,∆( S − ) is given by (2.46). Applying (2.39) for a = X +1 ...k and b = S − yields: (cid:104) S + k ...S +1 , S − (cid:105) X +1 ...k (cid:104) T +1 ...T + k , S − (cid:105) S − = S − X +1 ...k Formulas (2.28), (2.30) and (2.31) transform the formula above precisely into (2.42). , As for (2.44), consider relation (2.23) for k = 1 and X = E + ij :(2.47) ∆( E + ij ) = E + ij ⊗ n (cid:88) x,y =1 s + xi t + jy ⊗ E + xy as well as the (op , coop) version of the above equality that holds in (cid:101) A − , op , coop :(2.48) ∆( E − i (cid:48) j (cid:48) ) = n (cid:88) x (cid:48) ,y (cid:48) =1 E − x (cid:48) y (cid:48) ⊗ t − j (cid:48) y (cid:48) s − x (cid:48) i (cid:48) + 1 ⊗ E − i (cid:48) j (cid:48) Then (2.44) follows by applying (2.39) for a = E + ij and b = E − i (cid:48) j (cid:48) . (cid:3) Quantum toroidal gl n n > sl n be the Kac-Moody Lie algebra of type (cid:98) A n . Thecorresponding Drinfeld-Jimbo quantum group is defined as the associative algebra:(3.1) U q ( ˙ sl n ) = Q ( q ) (cid:68) x ± i , ψ ± s , c (cid:69) i ∈ Z /n Z s ∈{ ,...,n } modulo the fact that c is central, as well as the following relations:(3.2) ψ s ψ s (cid:48) = ψ s (cid:48) ψ s (3.3) ψ s x ± i = q ± ( δ i +1 s − δ is ) x ± i ψ s (3.4) [ x ± i , x ± j ] = 0 if j / ∈ { i − , i + 1 } ( x ± i ) x ± j − ( q + q − ) x ± i x ± j x ± i + x ± j ( x ± i ) = 0 if j ∈ { i − , i + 1 } (3.5) [ x + i , x − j ] = δ ji q − q − (cid:18) ψ i +1 ψ i − ψ i ψ i +1 (cid:19) for all i, j ∈ Z /n Z and s, s (cid:48) ∈ { , ..., n } . We will extend the notation ψ s to all s ∈ Z by setting ψ s + n = cψ s . We also consider the q –deformed Heisenberg algebra:(3.6) U q ( ˙ gl ) = Q ( q ) (cid:68) p ± k , c (cid:69) k ∈ N where c is central, and the p ± k satisfy the commutation relation:(3.7) [ p k , p l ] = kδ k + l · c k − c − k q − q − Then we will consider the algebra:(3.8) U q ( ˙ gl n ) = U q ( ˙ sl n ) ⊗ U q ( ˙ gl ) (cid:46) ( c ⊗ − ⊗ c )which serves as an affine q –version of the Lie algebra isomorphism gl n ∼ = sl n ⊕ gl . We note that the algebra defined below is slightly larger than the usual quantum group, sincethe Cartan part of the latter is generated by the ratios ψ i ψ j , instead of ψ ± , ..., ψ ± n themselves TALE OF TWO SHUFFLE ALGEBRAS 19 U q ( ˙ gl n ) into a bialgebra using the counit ε ( c ) = 1, ε ( ψ s ) =1, ε ( x ± i ) = 0, ε ( p k ) = 0 and the coproduct given by ∆( c ) = c ⊗ c and:(3.9) ∆( ψ s ) = ψ s ⊗ ψ s ∆( x + i ) = ψ i ψ i +1 ⊗ x + i + x + i ⊗ x − i ) = 1 ⊗ x − i + x − i ⊗ ψ i +1 ψ i (3.11)(3.12) ∆( p k ) = 1 c k ⊗ p k + p k ⊗ p − k ) = 1 ⊗ p − k + p − k ⊗ c k Moreover, the sub-bialgebras: U ≥ q ( ˙ gl n ) = Q ( q ) (cid:68) x + i , p k , ψ ± s , c (cid:69) i ∈ Z /n Z ,k ∈ N s ∈{ ,...,n } ⊂ U q ( ˙ gl n )(3.14) U ≤ q ( ˙ gl n ) = Q ( q ) (cid:68) x − i , p − k , ψ ± s , c (cid:69) i ∈ Z /n Z ,k ∈ N s ∈{ ,...,n } ⊂ U q ( ˙ gl n )(3.15)are endowed with a bialgebra pairing:(3.16) U ≥ q ( ˙ gl n ) ⊗ U ≤ q ( ˙ gl n ) (cid:104)· , ·(cid:105) −→ Q ( q )generated by properties (2.28), (2.29) and: (cid:104) ψ s , ψ s (cid:48) (cid:105) = q − δ ss (cid:48) , (cid:104) x + i , x − j (cid:105) = δ ij q − − q , (cid:104) p k , p − l (cid:105) = kδ lk q − k − q k and all other parings between generators are 0. It is well-known that U q ( ˙ gl n ) isthe Drinfeld double corresponding to the datum (3.16) (modulo the identificationof the symbols ψ s in the two factors of (3.16)). The algebra U q ( ˙ gl n ) is Z n –graded:deg c = 0 , deg ψ s = 0 , deg x ± i = ± ς i , deg p ± k = ± k δ where ς i = (0 , ..., , , , ..., (cid:124) (cid:123)(cid:122) (cid:125) i –th position and δ = (1 , ..., Remark 3.3. The elements x ± i of U q ( ˙ gl n ) are called simple generators, while theelements p ± k are called imaginary generators. Up to constant multiples, these areall the primitive elements of U q ( ˙ gl n ) (see Definition 3.7). Definition 3.5. Consider the algebra: (3.17) E = Q ( q ) (cid:68) f ± [ i ; j ) , ψ ± s , c (cid:69) ≤ s ≤ n ( i,j ) ∈ Z n,n ) Z (cid:46) relations (3.90) – (3.91) where c is central, and the quadratic relations (3.91) will be explained later. , The algebra E is a bialgebra with respect to the coproduct ∆( c ) = c ⊗ c and:∆( ψ s ) = ψ s ⊗ ψ s (3.18) ∆( f [ i ; j ) ) = j (cid:88) • = i f [ • ; j ) ψ i ψ • ⊗ f [ i ; • ) (3.19) ∆( f − [ i ; j ) ) = j (cid:88) • = i f − [ i ; • ) ⊗ f − [ • ; j ) ψ • ψ i (3.20)where the notation ψ s is extended to all s ∈ Z by ψ s + n = cψ s . The sub-bialgebras: E ≥ = Q ( q ) (cid:68) f +[ i ; j ) , ψ ± s , c (cid:69) ≤ s ≤ n ( i,j ) ∈ Z n,n ) Z ⊂ E (3.21) E ≤ = Q ( q ) (cid:68) f − [ i ; j ) , ψ ± s , c (cid:69) ≤ s ≤ n ( i,j ) ∈ Z n,n ) Z ⊂ E (3.22)are endowed with a bialgebra pairing:(3.23) E ≥ ⊗ E ≤ (cid:104)· , ·(cid:105) −→ Q ( q )generated by properties (2.28), (2.29) and:(3.24) (cid:104) ψ s , ψ s (cid:48) (cid:105) = q − δ ss (cid:48) , (cid:104) f [ i ; j ) , f − [ i (cid:48) ; j (cid:48) ) (cid:105) = δ [ i ; j )[ i (cid:48) ; j (cid:48) ) q − − q where the delta function is defined as:(3.25) δ ( i,j )( i (cid:48) ,j (cid:48) ) = (cid:40) i, j ) ≡ ( i (cid:48) , j (cid:48) ) mod ( n, n ) Z E is the Drinfeld double corresponding to the datum (3.23).The algebra E is Z n –graded:deg c = 0 , deg ψ s = 0 , deg f ± [ i ; j ) = ± [ i ; j )where [ i ; j ) = ς i + ... + ς j − (we write ς k = ς k mod n ).3.6. The subalgebras: E ⊃ E ± = Q ( q ) (cid:68) f ± [ i ; j ) (cid:69) ( i An element x ∈ E ± d is called primitive if: (3.27) ∆( x ) ∈ (cid:104) ψ ± s (cid:105) s ∈ Z ⊗ x + x ⊗ (cid:104) ψ ± s (cid:105) s ∈ Z We will write E prim ± d ⊂ E ± d for the vector subspace of primitive elements. TALE OF TWO SHUFFLE ALGEBRAS 21 As shown in [21], we have:dim E prim ± d = (cid:40) d is either [ i ; i + 1) or k δ i ∈ { , ..., n } and k ∈ N . Therefore, up to scalar multiples, there is aunique choice of primitive elements:(3.28) x ± i ∈ E ± [ i ; i +1) , p ± k ∈ E ± k δ which will be called simple and imaginary (respectively) primitive generators of E .Comparing this with Remark 3.3 allows us to obtain the following: Theorem 3.8. ( [21] ) Any choice of simple and imaginary primitive elements (3.28) of E gives rise to an isomorphism of Z n –graded bialgebras U q ( ˙ gl n ) ∼ = E . E boils down to controlling the up-to-scalar ambiguity in choosing the primitiveelements. To do this, we consider a formal parameter q and define linear functionals:(3.29) α ± [ i ; j ) : E ± [ i ; j ) −→ Q ( q, q n )for all ( i < j ) ∈ Z ( n,n ) Z , satisfying the following properties:(3.30) α ± [ i ; j ) ( r · r (cid:48) ) = (cid:40) α ± [ • ; j ) ( r ) α ± [ i ; • ) ( r (cid:48) ) if r ∈ E ± [ • ; j ) , r (cid:48) ∈ E ± [ i ; • ) q + = q and q − = ( q n q ) − ):(3.31) α ± [ i ; j ) ( f ± [ i (cid:48) ; j (cid:48) ) ) = δ ( i,j )( i (cid:48) ,j (cid:48) ) (1 − q ) q j − in ± We henceforth fix the elements (3.28) by making the choice of [22], namely:(3.32) α ± [ i ; i +1) ( x ± i ) = ± α ± [ s ; s + nk ) ( p ± k ) = ± ∀ k > i, s ∈ Z /n Z . The fact that the right-hand side of (3.33) does not dependon s is a consequence of the Z /n Z –invariance of the elements p ± k , see [21].3.10. It is easy to note that the bialgebra U q ( ˙ gl n ) ∼ = E possesses an antipode: A ( ψ s ) = ψ − s , A ( x + i ) = − ψ i +1 ψ i x + i , A ( x − i ) = − x − i ψ i ψ i +1 , A ( p ± k ) = − c ± k p ± k In terms of the root generators, we may write:(3.34) A ± ( f ± [ i ; j ) ) = ψ ± j ψ ± i ¯ f ± [ i ; j ) q i − j ) n ∓ The antipode is a bialgebra anti-automorphism A : E → E satisfying certain compatibilityproperties with the product, coproduct and pairing. We choose to write A ( x ) instead of the morecommon S ( x ) so as to not confuse the antipode with the series S ( x ) of Subsection 3.30 , where the elements ¯ f ± [ i ; j ) are inductively defined in terms of f ± [ i ; j ) by the formulas:(3.35) j (cid:88) • = i ¯ f ± [ • ; j ) f ± [ i ; • ) q •− i ) n ∓ = j (cid:88) • = i f ± [ • ; j ) ¯ f ± [ i ; • ) q j −• ) n ∓ = 0Alternatively, the elements ¯ f ± [ i ; j ) are completely determined by their coproduct:∆( ¯ f [ i ; j ) ) = j (cid:88) • = i ψ • ψ j ¯ f [ i ; • ) ⊗ ¯ f [ • ; j ) ∆( ¯ f − [ i ; j ) ) = j (cid:88) • = i ¯ f − [ • ; j ) ⊗ ψ j ψ • ¯ f − [ i ; • ) and by their values under the linear maps (3.96):(3.36) α ± [ i ; j ) ( ¯ f ± [ i (cid:48) ; j (cid:48) ) ) = δ ( i,j )( i (cid:48) ,j (cid:48) ) (1 − q − ) q i − jn ± Since E ∼ = U q ( ˙ gl n ), we will use the notation f ± [ i ; j ) (respectively ¯ f ± [ i ; j ) ) for theelements of either algebra. These elements will be called root generators of eitheralgebra E ∼ = U q ( ˙ gl n ), because [ i ; j ) are positive roots of the affine A n root system.3.11. Affinizations of quantum groups are defined by replacing each generator x ± i as in Subsection 3.1 by an infinite family of generators x ± i,k , ∀ k ∈ Z . To defineaffinizations explicitly, let us consider variables z as being colored by an integer i .Then we may define the following color-dependent rational function:(3.37) ζ (cid:16) zw (cid:17) = zqq kn − wq − zq kn − w if col i − col j = nk i − col j / ∈ {− , } mod nzq kn − wzqq kn − wq − if col i − col j = nk − z, w of colors i and j respectively. Definition 3.12. The quantum toroidal algebra is: U q,q ( ¨ gl n ) = Q ( q, q n ) (cid:68) x ± i,k , ϕ ± i,k (cid:48) , ψ ± s , c, ¯ c (cid:69) k ∈ Z ,k (cid:48) ∈ N i ∈ Z /n Z ,s ∈ Z (cid:46) relations (3.38) – (3.44)Consider the series x ± i ( z ) = (cid:80) k ∈ Z x ± i,k z k and ϕ ± s ( w ) = ψ ± s +1 ψ ± s + (cid:80) ∞ k =1 ϕ ± s,k w ± k , and set:(3.38) c, ¯ c central , ψ s + n = ψ s c, ∀ s ∈ Z (3.39) ψ s commutes with ψ ’s and ϕ ’s, and satisfies (3.3) with x ± i (cid:32) x ± i ( z )(3.40) ϕ ± i ( z ) ϕ ± (cid:48) j ( w ) ζ (cid:16) w ¯ c ± z (cid:17) ζ (cid:16) w ¯ c ±(cid:48) z (cid:17) = ϕ ± (cid:48) j ( w ) ϕ ± i ( z ) ζ (cid:16) z ¯ c ±(cid:48) w (cid:17) ζ (cid:16) z ¯ c ± w (cid:17) (3.41) x ± i ( z ) ϕ ± (cid:48) j ( w ) ζ (cid:18) wz ¯ c δ ∓±(cid:48) (cid:19) ± = ϕ ± (cid:48) j ( w ) x ± i ( z ) ζ (cid:32) z ¯ c δ ∓±(cid:48) w (cid:33) ± (3.42) x ± i ( z ) x ± j ( w ) ζ (cid:16) wz (cid:17) ± = x ± j ( w ) x ± i ( z ) ζ (cid:16) zw (cid:17) ± (3.43) [ x + i ( z ) , x − j ( w )] = δ ji q − q − (cid:104) δ (cid:16) zw ¯ c (cid:17) ϕ + i ( z ) − δ (cid:16) wz ¯ c (cid:17) ϕ − i ( w ) (cid:105) and: x ± i ( z ) x ± i ( z ) x ± j ( w ) − ( q + q − ) x ± i ( z ) x ± j ( w ) x ± i ( z ) + x ± j ( w ) x ± i ( z ) x ± i ( z )+(3.44) + same expression with z and z switched = 0 , if j ∈ { i − , i + 1 } for all choices of ± , ± (cid:48) , i, j , where the variables z and w have color i and j , respec-tively, for the purpose of defining the rational function ζ . Note that we extend theindex i to arbitrary integers, by applying the convention: x ± i + n,k = x ± i,k q − k , ϕ ± i + n,k = ϕ ± i,k q − k We consider the subalgebras U ± q,q ( ¨ gl n ) ⊂ U q,q ( ¨ gl n ) generated by { x ± i,k } i ∈ Z /n Z k ∈ Z . Remark 3.13. In the notation of Subsection 1.2, we have U + q,q ( ¨ gl n ) = U → q,q ( ¨ gl n ) and U − q,q ( ¨ gl n ) = U ← q,q ( ¨ gl n ) , but we will henceforth use the ± notation. U ± q,q ( ¨ gl n ). Con-sider variables z ia of color i , for various i ∈ { , ..., n } and a ∈ N . We call a function R ( z ..., z d , ..., z n ..., z nd n ) color-symmetric if it is symmetric in z i , ..., z id i for all i ∈ { , ..., n } . Depending on the context, the symbol “Sym” will refer to eithercolor-symmetric functions in variables z ia , or to the symmetrization operation:Sym F ( ..., z i , ..., z id i , ... ) = (cid:88) ( σ ,...,σ n ) ∈ S ( d ) × ... × S ( d n ) F ( ..., z i,σ i (1) , ..., z i,σ i ( d i ) , ... )Let d ! = d ! ...d n !. The following construction arose in the context of quantumgroups in [6], by analogy to the work of Feigin-Odesskii on certain elliptic algebras. Definition 3.15. Consider the vector space: (3.45) (cid:77) d =( d ,...,d n ) ∈ N n Q ( q, q n )( ..., z i , ..., z id i , ... ) Sym and endow it with an associative algebra structure, by setting R ∗ R (cid:48) equal to: Sym R ( ..., z i , ..., z id i , ... ) d ! R (cid:48) ( ..., z i,d i +1 , ..., z i,d i + d (cid:48) i , ... ) d (cid:48) ! n (cid:89) i,i (cid:48) =1 1 ≤ a ≤ d i (cid:89) d i (cid:48) e i ≤ i ≤ n ϕ + i ( z ia ¯ c ) ⊗ (cid:105) R + ( z i,a ≤ e i ⊗ z i,a>e i ¯ c ) (cid:81) ≤ i ≤ n ≤ i (cid:48) ≤ n (cid:81) a ≤ e i a (cid:48) >e i (cid:48) ζ ( z i (cid:48) a (cid:48) ¯ c /z ia )(3.50) ∆( R − ) = e ≤ d (cid:88) e ∈ N n R − ( z i,a ≤ e i ¯ c ⊗ z i,a>e i ) (cid:104)(cid:81) a ≤ e i ≤ i ≤ n ⊗ ϕ − i ( z ia ¯ c ) (cid:105)(cid:81) ≤ i ≤ n ≤ i (cid:48) ≤ n (cid:81) a ≤ e i a (cid:48) >e i (cid:48) ζ ( z ia ¯ c /z i (cid:48) a (cid:48) )(3.51)and there exists a pairing between the two halves given by:(3.52) (cid:10) R + , R − (cid:11) = (1 − q − ) | d | d ! (cid:73) R + ( ..., z ia , ... ) R − ( ..., z ia , ... ) (cid:81) ni,j =1 (cid:81) ( i,a ) (cid:54) =( j,b ) a ≤ d i ,b ≤ d j ζ ( z ia /z jb ) ≤ i ≤ n (cid:89) ≤ a ≤ d i dz ia πiz ia for any R + ∈ S d and R − ∈ S − d (we refer the reader to [21] or [22] for details). TALE OF TWO SHUFFLE ALGEBRAS 25 Remark 3.18. To think of (3.50) as a tensor, we expand the right-hand side innon-negative powers of z ia /z i (cid:48) a (cid:48) for a ≤ e i , e i (cid:48) < a (cid:48) , thus obtaining an infinite sumof monomials. In each of these monomials, we put the symbols ϕ + i,d to the very leftof the expression, then all powers of z ia with a ≤ e i , then the ⊗ sign, and finallyall powers of z ia with a > e i . The powers of the central element ¯ c = ¯ c ⊗ areplaced in the first tensor factor. The resulting expression will be a power series,and therefore lies in a completion of S ≥ ⊗S ≥ . The same argument applies to (3.51) . Remark 3.19. In formula (3.52) , the integral is defined in such a way that thevariable z ia traces a contour which surrounds z ib q , z i − ,b and z i +1 ,b q − for all i ∈ { , ..., n } and all a, b (a particular choice of contours which achieves this aimis explained in Proposition 3.8 of [21] ). S ± ∼ = U ± q,q ( ¨ gl n ). More preciselywe construct particular elements of S ± called “PBW generators”, indexed by atotally ordered set, and claim that a linear basis of S ± is given by ordered productsof the PBW generators. In the case of the algebra E ± ∼ = U ± q ( ˙ gl n ), we have alreadyseen in Subsection 3.6 that the PBW generators of E ± are indexed by:( i < j ) ∈ Z ( n, n ) Z It should come as no surprise that the PBW generators of S ± are indexed by: (cid:16) ( i < j ) , k (cid:17) ∈ Z ( n, n ) Z × Z If we write µ = j − ik , we will find it more useful to index the PBW generators by: (cid:16) ( i < j ) , µ (cid:17) ∈ Z ( n, n ) Z × Q such that j − iµ ∈ Z . For any choice of i < j and µ as above, we define: A µ ± [ i ; j ) = Sym (cid:81) j − a = i ( z a q an ) (cid:100) a − i +1 µ (cid:101)−(cid:100) a − iµ (cid:101) (cid:16) − z i q z i +1 (cid:17) ... (cid:16) − z j − q z j − (cid:17) (cid:89) i ≤ a j informula (3.56). If this happens, we make the convention that:[ i ; j ) = − [ j ; i ) if i > j Moreover, formulas (3.56) imply that k := j − iµ is an integer, so we will write:(3.58) f ( k )[ i ; j ) = f µ [ i ; j ) , ¯ f ( k )[ i ; j ) = ¯ f µ [ i ; j ) and set (3.58) equal to 0 if k = j − iµ / ∈ Z . The assignment:deg f ( k )[ i ; j ) = deg ¯ f ( k )[ i ; j ) = ([ i ; j ) , k )makes E µ into a Z n × Z graded algebra, and we write E µ | d for its degree d × Z graded piece. The following is an obvious consequence of Subsections 3.32–3.10. Proposition 3.22. For any µ , the algebra E µ has a coproduct ∆ µ , for which: ∆( f µ [ i ; j ) ) = (cid:88) •∈{ i,...,j } f µ [ • ; j ) ψ i ψ • ⊗ f µ [ i ; • ) ∆( f µ − [ i ; j ) ) = (cid:88) •∈{ i,...,j } f µ − [ i ; • ) ⊗ f µ − [ • ; j ) ψ • ψ i ∆( ¯ f µ [ i ; j ) ) = (cid:88) •∈{ i,...,j } ψ • ψ j ¯ f µ [ i ; • ) ⊗ ¯ f µ [ • ; j ) ∆( ¯ f µ − [ i ; j ) ) = (cid:88) •∈{ i,...,j } ¯ f µ − [ • ; j ) ⊗ ψ j ψ • ¯ f µ − [ i ; • )5 and linear maps: α [ i ; j ) : E µ | [ i ; j ) → Q ( q, q n ) for all ( i, j ) ∈ Z ( n,n ) Z , that satisfy property (3.30) and: α [ i ; j ) ( f ( k )[ i (cid:48) ; j (cid:48) ) ) = δ ( i,j )( i (cid:48) ,j (cid:48) ) (1 − q ) q gcd( j − i,k ) n sign k (3.59) α [ i ; j ) ( ¯ f ( k )[ i (cid:48) ; j (cid:48) ) ) = δ ( i,j )( i (cid:48) ,j (cid:48) ) (1 − q − ) q − gcd( j − i,k ) n sign k (3.60) The notation • ∈ { i, ..., j } means “ • runs between i and j ”. TALE OF TWO SHUFFLE ALGEBRAS 27 If k = 0 , the signs in the right-hand side of the formulas above are defined to be + or − , depending on whether i < j or i > j , respectively. E ± µ ⊂ E µ generated by those elements (3.58)where the sign of k is equal to ± . As in (3.28), we obtain the following elements: { p µ ± [ i ; i + a ) } i ∈ Z /n Z ⊂ E ± µ are simple generators, if n (cid:45) a (3.61) { p µ ± l δ ,r } r ∈{ ,...,g } l ∈ N ,µln ∈ Z ⊂ E ± µ are imaginary generators(3.62) ∀ µ = ab ∈ Q (cid:116)∞ . These elements are all primitive for the coproduct ∆ µ and satisfy: α ± [ i ; j ) (cid:16) p µ ± [ t ; t + a ) (cid:17) = ± δ ( t,t + a )( i,j ) (3.63) α ± [ s ; s + ln ) (cid:16) p µ ± l δ ,r (cid:17) = ± δ rs mod g (3.64)for any i, j, s, t . We may use the notation: p ( ± b ) ± [ i ; i + a ) = p µ ± [ i ; i + a ) and p ( ± µln ) ± l δ ,r = p µ ± l δ ,r to emphasize the fact that deg p ( k ) d = ( d , k ) ∈ Z n × Z . Let us write:(3.65) ¯ i = i − n (cid:22) i − n (cid:23) for all i ∈ Z , and set δ ji = 1 if i ≡ j modulo n , and 0 otherwise. Definition 3.24. Consider the algebras: (3.66) D ± = → (cid:79) µ ∈ Q E ± µ (cid:46) (3.67) – (3.68) whose generators, by the discussion above, are denoted by: (cid:110) p ( ± k ) ± [ i ; j ) , p ( ± k (cid:48) ) ± l δ ,r (cid:111) k,k (cid:48) > ,l ∈ Z \ i,j ) ∈ Z n,n ) Z ,r ∈ Z /g Z Whenever d := det (cid:18) k k (cid:48) j − i nl (cid:19) such that | d | = gcd( k (cid:48) , nl ) , we impose: (3.67) (cid:104) p ( ± k ) ± [ i ; j ) , p ( ± k (cid:48) ) ± l δ ,r (cid:105) = ± p ( ± k ± k (cid:48) ) ± [ i ; j + ln ) (cid:16) δ ri mod g q dn ± − δ rj mod g q − dn ± (cid:17) and whenever det (cid:18) k k (cid:48) j − i j (cid:48) − i (cid:48) (cid:19) = gcd( k + k (cid:48) , j + j (cid:48) − i − i (cid:48) ) , we impose: (3.68) p ( ± k ) ± [ i ; j ) p ( ± k (cid:48) ) ± [ i (cid:48) ; j (cid:48) ) q δ ij (cid:48) − δ ii (cid:48) − p ( ± k (cid:48) ) ± [ i (cid:48) ; j (cid:48) ) p ( ± k ) ± [ i ; j ) q δ jj (cid:48) − δ ji (cid:48) == (cid:88) [ t ; s )=[ i (cid:48) ; j (cid:48) ) f µ ± [ t,j ) ¯ f µ ± [ i ; s ) δ sj (cid:48) q − δ i (cid:48) j (cid:48) q − − q − δ i (cid:48) j (cid:48) q k (cid:48) ( s − i (cid:48) )) n ∓ q ∓ − where we set µ = j + j (cid:48) − i − i (cid:48) k + k (cid:48) . , D = E ∞ , and use the notation p (0) ± [ i ; i +1) and p (0) ± k δ , for itssimple and imaginary generators, respectively, as defined in Subsection 3.23. Definition 3.26. Let us define the double of the algebras (3.66) as: (3.69) D = D + ⊗ D ⊗ D − (cid:46) relations (3.70) – (3.74) where: (3.70) (cid:104) p ( ± ± [ i ; j ) , p ( ± ± l δ , (cid:105) = ± p ( ± ± [ i ; j + ln ) (cid:0) q l ± − q − l ± (cid:1) (3.71) (cid:104) p ( ± ± [ i ; j ) , p (0) ∓ l δ , (cid:105) = ± p ( ± ± [ i ; j − nl ) (cid:0) q l ± c ± l − q − l ± c ∓ l (cid:1) (3.72) p ( ± ± [ i ; j ) p (0) ± [ s ; s +1) q δ is +1 − δ is − p (0) ± [ s ; s +1) p ( ± ± [ i ; j ) q δ js +1 − δ js == ± (cid:16) δ is +1 · q − n ∓ p ( ± ± [ i − j ) − δ js · q n ∓ p ( ± ± [ i ; j +1) (cid:17) (3.73) (cid:104) p ( ± ± [ i ; j ) , p (0) ∓ [ s ; s +1) (cid:105) = ± (cid:32) δ is · q n ∓ p ( ± ± [ i +1; j ) ψ ± i +1 ψ ± i − δ js +1 · q − n ∓ ψ ± j ψ ± j − p ( ± ± [ i ; j − (cid:33) and: (3.74) (cid:104) p (1)[ i ; j ) , p ( − i (cid:48) ; j (cid:48) ) (cid:105) = 1 q − − q (cid:88) (cid:108) i − j (cid:48) n (cid:109) ≤ k ≤ (cid:106) j − i (cid:48) n (cid:107) f (0)[ i (cid:48) + nk ; j ) ψ j (cid:48) ¯ cψ i (cid:48) ¯ f (0)[ i ; j (cid:48) + nk ) − (cid:88) (cid:108) j (cid:48)− in (cid:109) ≤ k ≤ (cid:106) i (cid:48)− jn (cid:107) f (0) − [ j + nk ; i (cid:48) ) ψ j ψ i ¯ c ¯ f (0) − [ j (cid:48) ; i + nk ) Relations (3.71)–(3.74) are “sufficient” to describe all the relations between thethree tensor factors of (3.69), because the algebras D ± are generated by:(3.75) (cid:110) p ( ± ± [ i ; j ) (cid:111) ( i,j ) ∈ Z n,n ) Z (we will prove this in Proposition 3.40). Therefore, relations (3.71)–(3.74) allowsto “straighten” any product of elements from the subalgebras D + , D , D − , i.e. towrite said product as a sum of products of elements from D + , D , D − , in this order. Theorem 3.27. ( [22] ) There is an isomorphism D ∼ = S .Proof. (sketch) The subalgebra: S ⊃ T µ = (cid:68) A µ [ i ; j ) , B µ − [ i ; j ) (cid:69) j − iµ ∈ Z if µ > µ = ∞ (cid:68) A µ − [ i ; j ) , B µ [ i ; j ) (cid:69) j − iµ ∈ Z if µ < TALE OF TWO SHUFFLE ALGEBRAS 29 is isomorphic to E µ of (3.55) for all µ ∈ Q (cid:116) ∞ , by sending: f µ [ i ; j ) (cid:32) ψ j A µ [ i ; j ) ψ i (3.76) f µ − [ i ; j ) (cid:32) ψ i B µ − [ i ; j ) ψ j · q j − i ) (3.77)if µ > µ = ∞ , while: f µ [ i ; j ) (cid:32) B µ [ i ; j ) · (cid:16) − p n q (cid:17) i − j (3.78) f µ − [ i ; j ) (cid:32) A µ − [ i ; j ) · (cid:16) − p n q (cid:17) j − i (3.79)if µ < 0. Similarly, T := S is isomorphic to a tensor product of n Heisenbergalgebras. As in Subsection 3.23, this allows us to construct the images of the simpleand imaginary generators: p ( ± k ) ± [ i ; j ) (cid:32) X ( ± k ) ± [ i ; j ) ∈ S , p ( ± k (cid:48) ) ± l δ ,r (cid:32) X ( ± k (cid:48) ) ± l δ ,r ∈ S In loc. cit. , we showed that the simple and imaginary generators X ...... ∈ S satisfyrelations (3.67)–(3.68) and (3.70) –(3.74) with p ’s replaced by X ’s, hence we obtainan algebra homomorphism:(3.80) Φ : D → S This map is an isomorphism because ordered products of the elements f µ [ i ; j ) (re-spectively their images under the assignments (3.76)–(3.79)) in increasing order of µ were shown in [22] (respectively [21]) to form a linear basis of D (respectively S ). (cid:3) Corollary 3.28. If we combine (3.47) with (3.80) , we obtain an isomorphism: Ψ : D ∼ = U q,q ( ¨ gl n ) If we write: Ψ( D + ) = U ↑ q,q ( ¨ gl n ) , Ψ( D − ) = U ↓ q,q ( ¨ gl n ) and consider the usual triangular decomposition: Ψ( D ) = U q ( ˙ gl n ) = U ≥ q ( ˙ gl n ) ⊗ U ≤ q ( ˙ gl n ) then we obtain the decomposition (1.7) as algebras. D + ⊗ U ≥ q ( ˙ gl n ) and D − ⊗ U ≤ q ( ˙ gl n )into bialgebras, in such a way that D becomes their Drinfeld double. This will bedone by recasting D ± as a new type of shuffle algebra, as described in Subsection 1.4.With this in mind, we will prove the following more explicit version of Theorem 1.5. Theorem 1.5 (explicit): If A + and A − are the shuffle algebras that will intro-duced in Definitions 4.8 and 6.2, respectively, then we have algebra isomorphisms: D + ∼ = A + , D − ∼ = A − , op0 ANDREI NEGUT , Moreover, the extended algebras (cid:101) A ± defined in (6.3) have bialgebra structures anda bialgebra pairing between them, such that the Drinfeld double: A = (cid:101) A + ⊗ (cid:101) A − , op , coop is isomorphic to D (and hence also with S and U q,q ( ¨ gl n ) ) as an algebra. E ∼ = U q ( ˙ gl n )in more detail, and fill the gaps left in the discussion above. Let us consider thefollowing matrix-valued rational function called an R –matrix:(3.81) R ( x ) = (cid:88) ≤ i,j ≤ n E ii ⊗ E jj (cid:18) q − xq − − x (cid:19) δ ji + ( q − q − ) (cid:88) ≤ i (cid:54) = j ≤ n E ij ⊗ E ji x δ i Consider the algebra: (3.82) E := Q ( q ) (cid:68) s [ i ; j ) , t [ i ; j ) , c (cid:69) ≤ i ≤ ni ≤ j ∈ Z (cid:46) relations (3.83) – (3.86) where: (3.83) c is central, and s [ i ; i ) t [ i ; i ) = 1 R (cid:18) xy (cid:19) S ( x ) S ( y ) = S ( y ) S ( x ) R (cid:18) xy (cid:19) (3.84) R (cid:18) xy (cid:19) T ( x ) T ( y ) = T ( y ) T ( x ) R (cid:18) xy (cid:19) (3.85) R (cid:18) xcy (cid:19) S ( x ) T ( y ) = T ( y ) S ( x ) R (cid:18) xyc (cid:19) (3.86) where Z = Z ⊗ Id and Z = Id ⊗ Z for any symbol Z , and: S ( x ) = if d =0 then i ≤ j (cid:88) ≤ i,j ≤ n,d ≥ s [ i ; j + nd ) · E ij x − d (3.87) T ( x ) = if d =0 then i ≤ j (cid:88) ≤ i,j ≤ n,d ≥ t [ i ; j + nd ) · E ji x d (3.88)The series S ( x ), T ( y ) are the transposes of the series denoted T − ( x ), T + ( y ) in [21],which explains the discrepancy between our conventions and those of loc. cit. Relations (3.84) and (3.85) can be made explicit by expanding in either positive or negativepowers of x/y , but (3.86) must be expanded in negative powers of x/y . TALE OF TWO SHUFFLE ALGEBRAS 31 ψ k = s − k ; k ) = t [ k ; k ) ∈ E for all 1 ≤ k ≤ n , and set:(3.89) f [ i ; j ) = s [ i ; j ) ψ i , f − [ i ; j ) = t [ i ; j ) ψ − i ∀ ≤ i ≤ n and i ≤ j ∈ Z . We will extend our notation to all integers by setting: f ± [ i + n ; j + n ) = f ± [ i ; j ) , ψ k + n = cψ k ∀ i ≤ j, k . It is elementary to see that relations (3.84)–(3.86) can be rewritten as:(3.90) ψ k f ± [ i ; j ) = q ± δ jk ∓ δ ik f ± [ i ; j ) ψ k and:(3.91) (cid:88) ± [ i ; j ) ± (cid:48) [ i (cid:48) ; j (cid:48) )= d coefficient · f ± [ i ; j ) f ± (cid:48) [ i (cid:48) ; j (cid:48) ) = 0for all ± , ± (cid:48) ∈ { + , −} and d ∈ Z n . We will not need to spell out the coefficientsin (3.91) explicitly, but they can easily be obtained by expanding (3.84)–(3.86) aspower series in x/y and equating matrix coefficients of every E ij ⊗ E i (cid:48) j (cid:48) .The bialgebra (and Drinfeld double) structure on E = E ≥ ⊗ E ≤ from (3.18)–(3.20)can be presented in terms of the matrix-valued power series (3.87)–(3.88) as:∆( S ( x )) = (1 ⊗ S ( xc )) · ( S ( x ) ⊗ T ( x )) = (1 ⊗ T ( x )) · ( T ( xc ) ⊗ · denotes matrix multiplication (i.e. the formula E ij · E i (cid:48) j (cid:48) = δ i (cid:48) j E ij (cid:48) ), and c = c ⊗ c = 1 ⊗ c . It is straightforward to check that these coproducts respectrelations (3.18)–(3.20), i.e. extend to well-defined coproducts on the algebra E .The pairing (3.24) takes the form:(3.94) (cid:68) S ( x ) , T ( y ) (cid:69) = R (cid:18) xy (cid:19) − ⇔ (cid:68) S ( x ) , T ( y ) − (cid:69) = R (cid:18) xy (cid:19) It is elementary to show that (3.94) generates a bialgebra pairing, i.e. it intertwinesthe product with the coproduct on E , and that E is the Drinfeld double of its halveswith respect to this pairing. Moreover, the linear maps (3.29) which were used tonormalize primitive elements of E can be easily seen to come from the assignment:(3.95) E ± α ± −→ End( C n )[[ x ∓ ]] , (cid:40) α + ( r ) = (cid:104) r, T ( x ) − (cid:105) if r ∈ E + α − ( r ) = (cid:104) S ( x ) − , r (cid:105) if r ∈ E − Indeed, we define:(3.96) α ± [ i ; j ) : E ± [ i ; j ) −→ Q ( q, q n )as the coefficients of the maps α ± , appropriately renormalized as follows: α + ( r ) = (cid:88) ( i ≤ j ) ∈ Z n,n ) Z α [ i ; j ) ( r ) · E ji x (cid:98) j − n (cid:99) − (cid:98) i − n (cid:99) q i − jn + (3.97) α − ( r ) = (cid:88) ( i ≤ j ) ∈ Z n,n ) Z α − [ i ; j ) ( r ) · E ij x (cid:98) i − n (cid:99) − (cid:98) j − n (cid:99) q i − jn − (3.98) , Then it is elementary to observe that the bialgebra property of the pairing,together with definition (3.95), imply the multiplicativity property (3.30).3.33. We will henceforth write S + ( x ) = S ( x ) and T − ( x ) = T ( x ), so we have: S + ( x ) = if d =0 then i ≤ j (cid:88) ≤ i,j ≤ n,d ≥ f [ i ; j + nd ) ψ − i (cid:124) (cid:123)(cid:122) (cid:125) s +[ i ; j ) · E ij x − d T − ( x ) = if d =0 then i ≤ j (cid:88) ≤ i,j ≤ n,d ≥ f − [ i ; j + nd ) ψ i (cid:124) (cid:123)(cid:122) (cid:125) t − [ i ; j ) · E ji x d as elements of E ⊗ End( C n )[[ x ± ]]. We define series S − ( x ) and T + ( x ) by:(3.99) S − ( x ) T − ( xq ) = 1(3.100) D − S + ( xq n q ) † DT + ( x ) † = 1where D = diag( q , ..., q n ). The coefficients of these series will be denoted by: T + ( x ) = if d =0 then i ≤ j (cid:88) ≤ i,j ≤ n,d ≥ ψ j ¯ f [ i ; j + nd ) q j − i ) n c − d (cid:124) (cid:123)(cid:122) (cid:125) t +[ i ; j ) · E ij x − d S − ( x ) = if d =0 then i ≤ j (cid:88) ≤ i,j ≤ n,d ≥ ψ − j ¯ f − [ i ; j + nd ) q i − j ) n c d (cid:124) (cid:123)(cid:122) (cid:125) s − [ i ; j ) · E ji x d It is straightforward to show that formulas (3.99) and (3.100) imply (3.35). Thefollowing identities are easy to prove, as consequences of (3.84), (3.85), (3.86):(3.101) T +1 ( x ) R (cid:18) yxq (cid:19) S +2 ( y ) = S +2 ( y ) R (cid:18) yxq (cid:19) T +1 ( x )(3.102) S − ( x ) R (cid:18) yxq (cid:19) T − ( y ) = T − ( y ) R (cid:18) yxq (cid:19) S − ( x )(3.103) R (cid:18) xcy (cid:19) S +1 ( x ) T − ( y ) = T − ( y ) S +1 ( x ) R (cid:18) xyc (cid:19) (3.104) R (cid:18) xcy (cid:19) T +2 ( y ) S − ( x ) = S − ( x ) T +2 ( y ) R (cid:18) xyc (cid:19) (3.105) S +1 ( x ) R (cid:18) xycq (cid:19) S − ( y ) = S − ( y ) R (cid:18) xcyq (cid:19) S +1 ( x )(3.106) T +2 ( y ) R (cid:18) xycq (cid:19) T − ( x ) = T − ( x ) R (cid:18) xcyq (cid:19) T +2 ( y ) TALE OF TWO SHUFFLE ALGEBRAS 33 D + , D , D − (specifically relations (3.70)–(3.72) and(3.71)–(3.73)) in terms of the series S ± ( x ) and T ± ( x ). Proposition 3.35. Under the substitution: (3.107) p (1)[ i,j + nd ) (cid:32) E ji z d · q in for all ≤ i, j ≤ n and d ∈ Z , the following relations hold in D : X +1 ( z ) · S +2 ( w ) = S +2 ( w ) · R (cid:0) zw (cid:1) f (cid:0) zw (cid:1) X +1 ( z ) R (cid:18) wzq (cid:19) (3.108) T +2 ( w ) · X +1 ( z ) = R (cid:18) zwq (cid:19) X +1 ( z ) R (cid:0) wz (cid:1) f (cid:0) wz (cid:1) · T +2 ( w )(3.109) S − ( w ) · X +1 ( z ) = R (cid:18) zwcq (cid:19) X +1 ( z ) R (cid:16) wcz (cid:17) · S − ( w )(3.110) X +1 ( z ) · T − ( w ) = T − ( w ) · R (cid:16) zwc (cid:17) X +1 ( z ) R (cid:18) wczq (cid:19) (3.111) for any X + ( z ) ∈ End( C n )[ z ± ] . To understand the meaning of relations (3.108)–(3.111), let us spell out the first ofthese. Letting S + ( w ) = (cid:80) k ≥ u,v s +[ u ; v + nk ) E uv w k and X ( z ) = E ij z d , formula (3.108) reads:(3.112) k ≥ (cid:88) ≤ u,v ≤ n E ij z d ⊗ E uv w k · s +[ u ; v + nk ) = k,a,b ≥ (cid:88) ≤ u,v, • , ∗ ,x,y ≤ n s +[ u ; • + n ( k − a − b )) (cid:18) ⊗ E u • w k − a − b (cid:19) (cid:18) r •∗ xi,a · E xi ⊗ E •∗ z a w a (cid:19) (cid:18) E ij z d ⊗ (cid:19) (cid:18) r ∗ vjy,b · E yj ⊗ E ∗ v z b w b (cid:19) where r and r (cid:48) are the coefficients of the power series expansions: R (cid:16) zw (cid:17) · f (cid:16) zw (cid:17) − = k ≥ (cid:88) i,j,u,v r uvij,k · E ij ⊗ E uv z k w k R (cid:18) wzq (cid:19) = k ≥ (cid:88) i,j,u,v r uvij,k · E ij ⊗ E uv z k w k Equating the coefficients of ... ⊗ E uv w k in the two sides of (3.112) yields the identity: E ij z d · s +[ u ; v + nk ) = a,b ≥ (cid:88) ≤• , ∗ ,x,y ≤ n r •∗ xi,a r ∗ vjy,b s +[ u ; • + n ( k − a − b )) · E xy z d − a − b which is a relation in the algebra D , once we perform the substitution (3.107). Proof. We will only prove relation (3.108), and leave the analogous formulas(3.109)–(3.111) as exercises to the interested reader. Let us rewrite (3.108) as:(3.113) S +2 ( w ) − X +1 ( z ) S +2 ( w ) = R (cid:16) wz (cid:17) − X +1 ( z ) R (cid:18) wzq (cid:19) , If we let A : E → E denote the antipode, then we have A − ( S ( w )) = S ( wc ) − as aconsequence of (3.92). With this in mind, (3.113) reads:(3.114) X +1 ( z ) ♠ S +2 ( w ) = X +1 ( z ) ♣ S +2 ( w )where for any e ∈ U ≥ q ( ˙ gl n ), we write: X +1 ( z ) ♠ e = A − ( e ) X +1 ( z ) e (3.115) X +1 ( z ) ♣ e = (cid:68) A − ( e ) , T ( z ) (cid:69) X +1 ( z ) (cid:68) e , T ( zq ) − (cid:69) (3.116)Indeed, when e = S ( w ), the right-hand sides of (3.115) and (3.116) match the LHSand RHS of (3.113), respectively, due to (3.94). It is easy to see that the operations ♠ and ♣ are additive in e , and moreover: X +1 ( z ) ♠ ( ee (cid:48) ) = ( X +1 ( z ) ♠ e ) ♠ e (cid:48) X +1 ( z ) ♣ ( ee (cid:48) ) = ( X +1 ( z ) ♣ e ) ♣ e (cid:48) Since the series coefficients of S + ( w ) generate E + , (3.114) implies that:(3.117) X +1 ( z ) ♠ e = X +1 ( z ) ♣ e ∀ e ∈ U ≥ q ( ˙ gl n )When e = p (0)[ l δ , (whose coproduct is ∆( e ) = e ⊗ c − l ⊗ e ), relation (3.117) reads: X +1 ( z ) p (0)[ l δ , − p (0)[ l δ , X +1 ( z ) = (cid:68) p (0)[ l δ , , T ( zq ) − (cid:69) X +1 ( z ) − X +1 ( z ) (cid:68) p (0)[ l δ , , T ( z ) (cid:69) Formulas (3.64) and (3.95) imply that the pairings in the right-hand side of theformula above are equal to q l and q − l , respectively, so we have: (cid:104) X +1 ( z ) , p (0)[ l δ , (cid:105) = z l X +1 ( z )( q l − q − l )Plugging in X +1 ( z ) = E ij z d and using the correspondence (3.107) implies precisely(3.70) for ± = +. Similarly, plugging in e = p (0)[ s ; s +1) into (3.117) implies (3.72). (cid:3) By analogy with Proposition 3.35, we have: Proposition 3.36. Under the substitution: (3.118) p ( − i,j + nd ) (cid:32) E ji z d · q − jn the following relations hold in D (recall that · op denotes the opposite product): X − ( z ) · op S − ( w ) = S − ( w ) · op R (cid:0) zw (cid:1) f (cid:0) zw (cid:1) X − ( z ) D R (cid:18) wq q n z (cid:19) D − (3.119) T − ( w ) · op X − ( z ) = D R (cid:18) zq q n w (cid:19) D − X − ( z ) R (cid:0) wz (cid:1) f (cid:0) wz (cid:1) · op T − ( w )(3.120) S +2 ( w ) · op X − ( z ) = D R (cid:18) zq q n wc (cid:19) D − X − ( z ) R (cid:16) wcz (cid:17) · op S +2 ( w )(3.121) X − ( z ) · op T +2 ( w ) = T +2 ( w ) · op R (cid:16) zwc (cid:17) X − ( z ) D R (cid:18) wcq q n z (cid:19) D − (3.122) for any X − ( z ) ∈ End( C n )[ z ± ] , where D = diag( q , ..., q n ) . TALE OF TWO SHUFFLE ALGEBRAS 35 Finally, let us perform both substitutions (3.107) and (3.118) simultaneously: p (+1)[ i,j + nd ) (cid:32) (cid:0) E ji z d (cid:1) + · q in p ( − i,j + nd ) (cid:32) (cid:0) E ji z d (cid:1) − · q − jn (the notation X ± is just meant to differentiate among p ( ± i,j + nd ) ). Then we have:(3.123) (cid:34)(cid:18) E ij z d (cid:19) + , (cid:18) E i (cid:48) j (cid:48) z d (cid:48) (cid:19) − (cid:35) == ( q − (cid:88) k ∈ Z (cid:16) s +[ j (cid:48) ; i + nk ) t +[ j ; i (cid:48) + n ( − d − d (cid:48) − k )) c − d (cid:48) ¯ c − t − [ i ; j (cid:48) + nk ) s − [ i (cid:48) ; j + n ( d + d (cid:48) − k )) c − d ¯ c − (cid:17) as an immediate consequence of formula (3.74) (we set s ± [ i ; j ) = t ± [ i ; j ) = 0 if i > j ).3.37. We will now prove a useful Lemma about the structure of the algebra E + ofSubsection 3.6. Let us write LHS d for the quantity in the left-hand side of (3.91),when the signs are ± = ± (cid:48) = +. Then we have: E + = Q ( q ) (cid:68) f [ i ; j ) (cid:69) ( i Assume B + is a N n –graded Q ( q ) –algebra, such that: B ≥ = (cid:68) B + , ψ ± s , c ± (cid:69) s ∈ Z (cid:16) ψ s x − q −(cid:104) deg x, ς s (cid:105) xψ s , ψ s + n − cψ s , c central (cid:17) ∀ x ∈B + ,s ∈ Z is a bialgebra. Assume there exist elements (cid:54) = f (cid:48) [ i ; j ) ∈ B [ i ; j ) and linear maps: α (cid:48) [ i ; j ) : B [ i ; j ) → Q ( q ) ∀ i ∈ { , ..., n } and j > i such that the analogues of (3.19) , (3.20) , (3.30) , (3.31) hold. If: (3.124) (cid:110) x primitive and α (cid:48) [ i ; j ) ( x ) = 0 , ∀ i, j (cid:111) ⇒ x = 0 then the map E + Υ → B + , Υ( f [ i ; j ) ) = f (cid:48) [ i ; j ) is an injective algebra homomorphism. In the formula above, (cid:104)· , ·(cid:105) is the bilinear form on Z n given by (cid:104) ς i , ς j (cid:105) = δ ji − δ j − i , Proof. Let us consider the left-hand side of (3.91) with f (cid:48) [ i ; j ) instead of the f [ i ; j ) :LHS (cid:48) d = (cid:88) [ i ; j )+[ i (cid:48) ; j (cid:48) )= d coefficient · f (cid:48) [ i ; j ) f (cid:48) [ i (cid:48) ; j (cid:48) ) ∈ B d To show that Υ is an algebra homomorphism, we would need to show that LHS (cid:48) d =0, which we will prove by induction on d ∈ N n . The base case d = 0 is trivial, sowe will only prove the induction step. We have:∆(LHS (cid:48) d ) ∈ (cid:104) ψ ± s (cid:105) s ∈ Z ⊗ LHS (cid:48) d + ... + LHS (cid:48) d ⊗ (cid:104) ψ ± s (cid:105) s ∈ Z where the middle terms denoted by the ellipsis are equal to Υ ⊗ Υ applied to themiddle terms of ∆(LHS d ). Since the latter are 0 (as LHS d = 0 in E + ), we concludethat LHS (cid:48) d is primitive. Moreover, the analogues of (3.30) and (3.31) imply that: α (cid:48) [ i ; j ) (LHS (cid:48) d ) = α [ i ; j ) (LHS d ) = 0 ∀ d , i < j Therefore, assumption (3.124) implies that LHS (cid:48) d = 0 for all d , thus establishing thefact that Υ is a well-defined algebra homomorphism. To show that Υ is injective,assume that its kernel is non-empty. Since Υ preserves degrees, we may choose0 (cid:54) = x ∈ E + of minimal degree d ∈ N n such that Υ( x ) = 0. Since Υ preserves thecoproduct and is injective in degrees < d (by the minimality of d ), we concludethat x is primitive. However, since Υ intertwines the linear maps α [ i ; j ) with α (cid:48) [ i ; j ) ,we conclude that x is also annihilated by the linear maps α [ i ; j ) , hence x = 0. (cid:3) Since an injective linear map of finite-dimensional vector spaces Φ : V (cid:44) → V is anisomorphism if dim V ≤ dim V (as well as the similar statement in the gradedcase, if V and V have finite-dimensional graded pieces which are preserved by Φ)we obtain the following: Corollary 3.39. If the assumption of Lemma 3.38 holds, and moreover, if dim B d ≤ the RHS of (3.26) for all d ∈ N n , then Υ : E + ∼ = B + is an isomorphism. Proposition 3.40. The Q ( q, q n ) –algebra D ± is generated by the elements: (3.125) (cid:110) p ( ± ± [ i ; j ) (cid:111) ( i,j ) ∈ Z n,n ) Z Proof. Without loss of generality, let ± = +. We will prove that p ( k )[ i ; j ) (resp. p ( k (cid:48) ) l δ ,r )lies in the subalgebra generated by the elements (3.125) for all choices of indices i, j, l, r , by induction on k (respectively k (cid:48) ). To this end, let us choose a latticetriangle of minimal size with the vector ( j − i, k ) (respectively ( nl, k (cid:48) )) as an edge: TALE OF TWO SHUFFLE ALGEBRAS 37 (cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:0)(cid:0)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16) (0 , 0) ( nl, k (cid:48) )( a, b ) respectively (cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:0)(cid:0)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:8)(cid:8)(cid:8)(cid:8) (0 , 0) ( j − i, k )( a, b ) In the case of the picture on the left, namely that of the element p ( k )[ i ; j ) , we have:(3.126) det (cid:18) b k − ba j − i − a (cid:19) = 1If a ≡ j − i modulo n , then relation (3.67) gives us: (cid:104) p ( b )[ i ; i + a ) , p ( k − b ) j − i − an δ ,i (cid:105) = p ( k )[ i ; j ) · non-zero constantwhile a (cid:54)≡ j − i modulo n , then relation (3.68) gives us: p ( b )[ i ; i + a ) p ( k − b )[ i + a ; j ) q − − p ( k − b )[ i + a ; j ) p ( b )[ i ; i + a ) q δ ij − δ ii + a = f ( k )[ i ; j ) q − − q = p ( k )[ i ; j ) · non-zero constantIn either of the two formulas above, the induction hypothesis implies that theleft-hand side lies in the algebra generated by the elements (3.125). Therefore, sodoes the right-hand side and the induction step is complete.The case of the picture on the right, namely that of the element p ( k (cid:48) ) l δ ,r , is provedanalogously. We will therefore only sketch the main idea, and leave the details tothe interested reader. For any s ≡ r mod g , (3.68) implies:a q − commutator of p ( b )[ s ; s + a ) and p ( k (cid:48) − b )[ s + a ; s + nl ) = (cid:88) [ i ; j )+[ i ; j )= l δ ¯ f µ [ i ; j ) f µ [ i ; j ) coefficientwhere µ = k (cid:48) nl . By the induction hypothesis, the left-hand side of the expressionabove lies in the subalgebra generated by the elements (3.125), hence so does theright-hand side, which we henceforth denote RHS. Clearly, RHS ∈ B + µ , and it cantherefore be expressed as a sum of products of the simple and imaginary generators:RHS = α · p µl δ ,r + sum of products of more than one generator of B + µ It was shown in the last paragraph of Section 4 of [22] that the coefficient α aboveis non-zero. By the induction hypothesis, all products of more than one simple orimaginary generator lie in the subalgebra generated by (3.125). Since the quantityRHS also lies in this subalgebra, we conclude that p µl δ ,r also does. (cid:3) The shuffle algebra with spectral parameters V ⊗ k ) to be rational functions. We will recycle all the , notations from Section 2, so let V be an n –dimensional vector space and consider:(4.1) R ( x ) ∈ End Q ( q ) ( V ⊗ V )( x )given by (3.81). For a parameter q , we define:(4.2) (cid:101) R ( x ) = R (cid:18) xq (cid:19) ∈ End Q ( q,q n ) ( V ⊗ V )( x )It is well-known that R ( x ) satisfies the Yang-Baxter equation with parameter:(4.3) R (cid:18) z z (cid:19) R (cid:18) z z (cid:19) R (cid:18) z z (cid:19) = R (cid:18) z z (cid:19) R (cid:18) z z (cid:19) R (cid:18) z z (cid:19) and it is easy to show that (cid:101) R ( z ) satisfies the following analogue of (2.2)–(2.3):(4.4) (cid:101) R (cid:18) z z (cid:19) (cid:101) R (cid:18) z z (cid:19) R (cid:18) z z (cid:19) = R (cid:18) z z (cid:19) (cid:101) R (cid:18) z z (cid:19) (cid:101) R (cid:18) z z (cid:19) (4.5) R (cid:18) z z (cid:19) (cid:101) R (cid:18) z z (cid:19) (cid:101) R (cid:18) z z (cid:19) = (cid:101) R (cid:18) z z (cid:19) (cid:101) R (cid:18) z z (cid:19) R (cid:18) z z (cid:19) Finally, we note that the R –matrix (4.1) is (almost) unitary, in the sense that:(4.6) R ( x ) R (cid:18) x (cid:19) = f ( x ) · Id V ⊗ V where:(4.7) f ( x ) = (1 − xq )(1 − xq − )(1 − x ) It is easy to see that “half” of the rational function f ( x ) could have been absorbedin the definition of R ( x ), but we will prefer our current conventions.4.2. We will represent elements of End( V ⊗ k )( z , ..., z k ) as braids on k strands.The only difference between the present setup and that of Section 2 is that eachstrand carries not only a label i ∈ { , ..., k } but also a variable z i . With this inmind, we make the convention that the endomorphism corresponding to a positivecrossing of strands labeled i and j , endowed with variables z i and z j respectively,is: R ij (cid:18) z i z j (cid:19) Because of (4.2), we can represent both R and (cid:101) R as crossings of braids of the samekind (i.e. we do not need the dichotomy of straight strands versus squiggly strands,of Subsection 2.2) if we remember to change the variable on one of our strands.Because of this, we will always write the variable next to each strand. For example,the braids: Figure 11. Braids decorated with variables TALE OF TWO SHUFFLE ALGEBRAS 39 represent the following compositions ∈ End( V ⊗ )( z , z ): R (cid:18) z z (cid:19) A ( z ) (cid:101) R (cid:18) z z (cid:19) B ( z ) and A ( z ) (cid:101) R (cid:18) z z (cid:19) B ( z ) R (cid:18) z z (cid:19) respectively. The variable does not change along a strand, except at a box.4.3. Note that R ( x ) has a pole at x = 1, hence:(4.8) Res x = q − (cid:101) R ( x ) = ( q − − q ) · (12) ∈ End Q ( q,q ) ( V ⊗ V )where (12) denotes the permutation operator of the two factors. Pictorially, theendomorphism (4.8) will be represented by two black dots indicating a color change(recall that the color encodes the index ∈ { , ..., k } of a strand) along two strands: Figure 12. Black dots can slide past arbitrary strandsThe equality of braids depicted in Figure 12 means that one can move the blackdots as far left or as far right as we wish, no matter how many other strands wepass over or under. Explicitly, the equality depicted above reads:( q − − q )(12) · R i (cid:18) z i y (cid:19) R − j (cid:18) y z j (cid:19) = R i (cid:18) z i y (cid:19) R − j (cid:18) y z j (cid:19) · ( q − − q )(12)Finally, we note that due to formula (4.6), we can always change a crossing in abraid, at the cost of multiplying with the function (4.7): Figure 13. Changing a crossing , Proposition 4.5. Let A = A ...k ( z , ..., z k ) , B = B ...l ( z , ..., z l ) . The assignment: (4.9) A ∗ B = a <...b j yields an associative algebra structure on the vector space: ∞ (cid:77) k =0 End Q ( q,q n ) ( V ⊗ k )( z , ..., z k ) with unit ∈ End Q ( q,q n ) ( V ⊗ ) . We call (4.9) the “shuffle product”. Proposition 4.6. The shuffle product above preserves the vector space: A +big ⊂ ∞ (cid:77) k =0 End Q ( q,q n ) ( V ⊗ k )( z , ..., z k ) consisting of tensors X = X ...k ( z , ..., z k ) which simultaneously satisfy: • X = x ( z , ..., z k ) (cid:81) ≤ i (cid:54) = j ≤ k ( z i − z j q ) for some x ∈ End Q ( q,q n ) ( V ⊗ k )[ z ± , ..., z ± k ] . • X is symmetric, in the sense that: (4.10) X = R σ · ( σXσ − ) · R − σ ∀ σ ∈ S ( k ) , where R σ = R σ ( z , ..., z k ) is any braid lift of the permutation σ , and: σXσ − = X σ (1) ...σ ( k ) ( z σ (1) , ..., z σ ( k ) ) Proof. The fact that the shuffle product (4.9) preserves the vector space of sym-metric tensors is proved word-for-word like Proposition 2.6. Therefore, it remainsto prove that if A and B only have simple poles at z i = z j q , then A ∗ B has thesame property. Since R ab ( z ) has a simple pole at z = 1, a priori A ∗ B could havesimple poles at z i = z j , so it remains to show that the residues at these poles vanish.Without loss of generality, we will prove the vanishing of the residue at z = z k .We will show that any symmetric tensor X = X ...k ( z , ..., z k ) with at most asimple pole at z = z k is actually regular there. Since only the indices/variables 1and k will play an important role in the following, we will use ellipses ... for theindices/variables 2 , ..., k − 1. Let us consider (4.10) in the particular case σ = (1 k ):(4.11) − Y ...k − ,k ( z , z , ..., z k − , z k ) · R σ = R σ · Y k ...k − , ( z k , z , ..., z k − , z ) TALE OF TWO SHUFFLE ALGEBRAS 41 where Y ( z , ..., z k ) = X ( z , ..., z k ) · ( z − z k ) is regular at z = z k . We may choose: R σ = R (cid:18) z z (cid:19) ...R ,k − (cid:18) z z k − (cid:19) R k (cid:18) z z k (cid:19) R − k,k − (cid:18) z k z k − (cid:19) ...R − k (cid:18) z k z (cid:19) Since the residue of R k (cid:16) z z k (cid:17) at z = z k is ( q − q − ) · (1 k ), the residue of (4.11) is: − Y ...k ( x, ..., x ) · R (cid:18) xz (cid:19) ...R ,k − (cid:18) xz k − (cid:19) · (1 k ) · R − k,k − (cid:18) xz k − (cid:19) ...R − k (cid:18) xz (cid:19) == R (cid:18) xz (cid:19) ...R ,k − (cid:18) xz k − (cid:19) · (1 k ) · R − k,k − (cid:18) xz k − (cid:19) ...R − k (cid:18) xz (cid:19) · Y k... ( x, ..., x )(we wrote x = z = z k and canceled an overall scalar factor of q − q − ). We maymove the permutation (1 k ) to the very right of the equations above, obtaining: − Y ...k ( x, ..., x ) · R (cid:18) xz (cid:19) ...R ,k − (cid:18) xz k − (cid:19) R − ,k − (cid:18) xz k − (cid:19) ...R − (cid:18) xz (cid:19) · (1 k ) == R (cid:18) xz (cid:19) ...R ,k − (cid:18) xz k − (cid:19) R − ,k − (cid:18) xz k − (cid:19) ...R − (cid:18) xz (cid:19) · Y ...k ( x, ..., x ) · (1 k )After canceling all the R factors and the permutation operators (1 k ), we are leftwith Y ...k ( x, ..., x ) = 0, which implies that X was regular at z = z k to begin with. (cid:3) X ( z , ..., z k ) with at most simple poles, we let:(4.12) Res { z = y,z = yq ,...,z i = yq i − } X be the rational function in y, z i +1 , ..., z k obtained by successively taking the residueat z = z q , then at z = z q ,..., then at z i = z q i − and finally relabeling thevariable z (cid:32) y . More generally, for any collection of natural numbers:1 = c < c < ... < c u < c u +1 = k + 1we will write: Res { z cs = y s ,z cs +1 = y s q ,...,z cs +1 − = y s q cs +1 − cs − } ∀ s ∈{ ,...,u } X for the rational function in y , .., y u obtained by applying the iterated residue (4.12)construction for the groups of variables indexed by { c , c +1 , ..., c − } , ..., { c u , c u +1 , ..., c u +1 − } . Moreover, we set: k (cid:89) i =1 x i = (cid:89) ≤ i ≤ k x i = x x ...x k (cid:89) i = k x i = (cid:89) k ≥ i ≥ x i = x k x k − ...x for any collection of potentially non-commuting symbols x , ..., x k . Definition 4.8. Let A + ⊂ A +big be the vector subspace of elements X such that forany composition k = λ + ... + λ u we have (let λ s = c s +1 − c s for all s ): (4.13) Res { z cs = y s ,z cs +1 = y s q ,...,z cs +1 − = y s q λs − } ∀ s ∈{ ,...,u } X == ( q − − q ) k − u unordered pairs ( s,d ) (cid:54) =( t,e ) (cid:89) with ≤ s,t ≤ u, ≤ d<λ s , ≤ e<λ t f (cid:18) y s q d y t q e (cid:19) , (cid:89) u ≥ s ≥ (cid:89) s ≤ t ≤ u ( t,e ) (cid:54) =( s, (cid:89) ≤ e<λ t R c s ,c t + e (cid:18) y s y t q e (cid:19) · X ( λ ,...,λ u ) c ...c u ( y , ..., y u ) · (cid:89) ≤ s ≤ u (cid:89) u ≥ t>s (cid:89) λ t >e ≥ R c t + e,c s (cid:18) y t q e y s q λ s (cid:19) (cid:34) u (cid:89) s =1 (cid:18) c s ... c s +1 − c s +1 − c s + 1 ... c s +1 − c s (cid:19)(cid:35) for some X ( λ ,...,λ u ) ∈ End( V ⊗ u )( y , ..., y u ) . Pictorially, the RHS of (4.13) may be represented as follows: Figure 14. The RHS of (4.13) for u = 2, λ = 4, λ = 3Note the symbol “blue over blue” to the right of Figure 14. Given two colors γ and γ , placing γ over γ is a prescription that indicates that the braid inquestion be multiplied by the product of f ( y/y (cid:48) ), where y (respectively y (cid:48) ) goesover all variables on strands whose left endpoint has color γ (respectively γ ), andthe leftmost endpoint with variable y is above the leftmost endpoint with variable y (cid:48) . Remark 4.9. We will call (4.13) the wheel conditions in the current matrix-valued setting, because the E ⊗ ... ⊗ E coefficient of any X satisfying (4.13) is a symmetric rational function in z , ..., z k that satisfies the wheel conditions of [9] . Remark 4.10. Note that when k = 1 , the wheel condition (4.13) is vacuous, butit is already non-trivial for k = 2 (as opposed from the n = 1 case of loc. cit.) Proposition 4.11. The vector subspace A + of Definition 4.8 is preserved by theshuffle product (and will henceforth be called the “shuffle algebra”).Proof. Assume that two matrix-valued rational functions A and B in k and l vari-ables, respectively, satisfy the wheel condition (4.13). To prove that their shuffleproduct A ∗ B also satisfies the wheel condition, we must take the iterated residueof the right-hand side of (4.9) at: z c s = y s , z c s +1 = y s q , ..., z c s +1 − = y s q λ s − 1) TALE OF TWO SHUFFLE ALGEBRAS 43 for any composition k + l = λ + ... + λ u . We will show that, at such a specialization,each summand in the RHS of (4.9) has the form predicated in the RHS of (4.13),so we henceforth fix a shuffle a < ... < a k , b < ... < b l . Because (cid:101) R ( z ) has a simplepole at z = q − , the only way such a shuffle can have a non-zero residue is if: { a , ..., a k } = u (cid:71) s =1 { c s , c s + 1 , ..., r s − , r s − }{ b , ..., b l } = u (cid:71) s =1 { r s , r s + 1 , ..., c s +1 − , c s +1 − } for some choice of r s ∈ { c s , ..., c s +1 − } for all s ∈ { , ..., u } . We will indicate thischoice by using the following colors for the strands of our braids:red for c s , blue for c s + 1 , ..., r s − r s , green for r s + 1 , ..., c s +1 − u = 2, but the modifications that lead to the general case arestraightforward; although we only depict a single blue and green strand in each ofthe u groups, the reader may obtain the general case by replacing each of themwith any number of parallel blue and green strands, respectively): Figure 15. The black dots in the middle of the braid appear because the variables on the braidsin question are set equal to each other in the iterated residue. By sliding the blackdots as far to the right as possible (which is allowed, due to Figure 12), we obtain: Figure 16. , One readily notices that certain pairs of braids are twisted twice around each other,and these twists can be canceled up to a factor of f ( y/y (cid:48) ) (due to the identity inFigure 13), where y and y (cid:48) are the variables on the braids in question. Keepingin mind that the variables on the red strands are modified to the right of the redboxes, this yields the following braid: Figure 17. Note that the black dots on the right side of the braid above yield the same per-mutation as the black dots on the right side of the braid in Figure 14, due to thefollowing identity: (cid:18) ... k − k ... k (cid:19) = (12)(23) ... ( k − , k ) = (1 k )(1 , k − ... (12)in the symmetric group S ( k ). Therefore, the braid in Figure 17 is precisely of theform predicated in the right-hand side of (4.13), which concludes our proof. (cid:3) Proposition 4.12. For any X ∈ A + and any composition k = λ + ... + λ u , thetensor Y = X ( λ ,...,λ u ) that appears in (4.13) has at most simple poles at: (4.14) y s q d − y t q − and y s q d − y t q λ t for all ≤ s < t ≤ u and any ≤ d < λ s . Moreover, if λ s = λ t then: (4.15) Y ...,s,...t,... ( ..., y s , ..., y t , ... ) = R ( st ) · Y ...,t,...s,... ( ..., y t , ..., y s , ... ) · R − st ) for any braid lift R ( st ) = R ( st ) ( y , ..., y u ) of the transposition ( st ) .Proof. Let us first prove the statement about the poles of Y . In the course of thisproof, all poles will be counted with multiplicities, in the sense that whenever werefer to a “set of poles”, the reader should assume this means “multiset of poles”.Because of the first bullet of Proposition 4.6, which determines the allowable polesof X ∈ A + , the left-hand side of (4.13) has a simple pole at:(4.16) y s q d − y t q e ± ∀ ≤ s < t ≤ u, ≤ d < λ s , ≤ e < λ t On the other hand, the right-hand side of (4.13) has a double pole at:(4.17) y s q d − y t q e ∀ ≤ s < t ≤ u, ≤ d < λ s , ≤ e < λ t because of the f factors, and a simple pole at:(4.18) (cid:40) y s − y t q e , and y s q λ s − y t q e ∀ ≤ s < t ≤ u, ≤ e < λ t TALE OF TWO SHUFFLE ALGEBRAS 45 because of the simple pole of R ( z ) at z = 1. Eliminating the multiset of poles in(4.17) and (4.18) from the multiset of poles in (4.16) yields the allowable polesof Y ( y , ..., y u ), and it is elementary to see that they are precisely of the form (4.14).As for (4.15), we will prove it pictorially. To keep the pictures legible, we will onlyshow the case s = 1, t = u = 2, but the interested reader may easily generalize theargument. Because of property (4.10), the tensor X ( y , y q , ..., y , y q , ... ) (whichis represented by a braid akin to Figure 14) is also equal to the following braid: Figure 18. (we ignore the scalar-valed rational functions f in the diagrams above, as theycommute with all the braids involved). The braid called R σ interchanges the twocollections of λ s = λ t braids corresponding to the variables y s q ∗ and y t q ∗ . Al-though we could choose the crossings of R σ arbitrarily, the choice we make aboveis that the two red strands cross above all other ones, then the two blue strandsnext to the red strands cross above all remaining ones, then the two blue strandsnext to the previous blue strands cross etc. In virtue of Figure 12, we may movethe black dots to the very right of the picture above, obtaining the braid below: Figure 19. Then we pull the red strands as far up as possible, and notice that the blue strandsare all unlinked, thus yielding the braid in Figure 20 below. , Figure 20. The red strands in Figure 20 correspond to the endomorphism: R ( st ) · ( st ) Y ( st ) · R − st ) ∈ End( V ⊗ )( y s , y t )which we may equate with Y due to the braid equivalences described above. (cid:3) A + has a “vertical” and a “horizontal” grading: N (cid:51) vdeg f ( z , ..., z k ) E i j ⊗ ... ⊗ E i k j k = k (4.19) Z n (cid:51) hdeg f ( z , ..., z k ) E i j ⊗ ... ⊗ E i k j k = (hom deg f ) δ + k (cid:88) a =1 deg E i a j a (4.20)where δ = (1 , ..., 1) and the grading on End( V ) is defined by:(4.21) deg E ij = − [ i ; j )We will find it convenient to extend the notation E ij to all i, j ∈ Z , according to:(4.22) E ij = E ¯ i ¯ j z (cid:98) i − n (cid:99) − (cid:98) j − n (cid:99) ∈ End( V )[ z ± ]where ¯ i denotes the residue class of i in the set { , ..., n } . Then the grading (4.20)makes sense for arbitrary integer indices E i a j a , and formula (4.21) also makes sensefor all integers i, j . We will denote the graded pieces of the shuffle algebra by: A + = ∞ (cid:77) k =0 A k , A k = (cid:77) d ∈ Z n A d ,k and refer to ( d , k ) as the degree of homogeneous elements. Finally, we write: | d | = d + ... + d n for any d = ( d , ..., d n ) ∈ Z and refer to the number:(4.24) µ = | (hom deg f ) δ + (cid:80) ka =1 deg E i a j a | k ∈ Q More generally, we will extend the notation above to a k –fold tensor, by the rule:(4.23) E i j ⊗ ... ⊗ E i k j k = E ¯ i ¯ j ⊗ ... ⊗ E ¯ i k ¯ j k z (cid:106) i − n (cid:107) − (cid:106) j − n (cid:107) ...z (cid:106) ik − n (cid:107) − (cid:106) jk − n (cid:107) k as elements of End( V ⊗ k )[ z ± , ..., z ± k ] TALE OF TWO SHUFFLE ALGEBRAS 47 as the slope of the matrix-valued rational function f ( z , ..., z k ) E i j ⊗ ... ⊗ E i k j k .We will consider the partial ordering on Z n given by:(4.25) ( d , ..., d n ) ≤ ( d (cid:48) , ..., d (cid:48) n ) if (cid:40) d n < d (cid:48) n or d n = d (cid:48) n and (cid:80) n − i =1 d i ≤ (cid:80) n − i =1 d (cid:48) i The extended shuffle algebra with spectral parameter R –matrix with spectral parameter (3.81). Definition 5.2. Consider the extended shuffle algebra: (cid:101) A + = (cid:68) A + , s [ i ; j ) (cid:69) i ≤ j ≤ i ≤ n (cid:46) relations (5.1) and (5.2) In order to concisely state the relations, it makes sense to package the new genera-tors s [ i ; j ) into the generating function: S ( x ) = d =0 only if i ≤ j (cid:88) ≤ i,j ≤ n, d ≥ s [ i ; j + nd ) ⊗ E ij x d ∈ (cid:101) A + ⊗ End( V )[[ x − ]] We impose the following analogues of relations (2.16) and (2.19) : (5.1) R (cid:18) xy (cid:19) S ( x ) S ( y ) = S ( y ) S ( x ) R (cid:18) xy (cid:19) (5.2) X · S ( y ) = S ( y ) · R k (cid:16) z k y (cid:17) ...R (cid:16) z y (cid:17) f (cid:16) z k y (cid:17) ...f (cid:16) z y (cid:17) X (cid:101) R (cid:18) z y (cid:19) ... (cid:101) R k (cid:18) z k y (cid:19) for any X = X ...k ( z , ..., z k ) ∈ A + ⊂ (cid:101) A + . Note that the grading on A + extends to one on (cid:101) A + , by setting:deg s [ i ; j ) = ([ i ; j ) , ∈ Z n × N t [ i ; j ) ∈ (cid:101) A + defined by (3.100), where:(5.3) T ( x ) = d =0 only if i ≤ j (cid:88) ≤ i,j ≤ n, d ≥ t [ i ; j + nd ) ⊗ E ij x d Then it is a straightforward computation (which we leave as an exercise to theinterested reader) to see that (5.1), (5.2) imply analogues of (2.17), (2.18), (2.20):(5.4) T ( x ) T ( y ) R (cid:18) xy (cid:19) = R (cid:18) xy (cid:19) T ( y ) T ( x )(5.5) T ( x ) (cid:101) R (cid:18) xy (cid:19) S ( y ) = S ( y ) (cid:101) R (cid:18) xy (cid:19) T ( x ) , (5.6) T ( y ) · X = (cid:101) R k (cid:18) yz k (cid:19) ... (cid:101) R (cid:18) yz (cid:19) X R (cid:16) yz (cid:17) ...R k (cid:16) yz k (cid:17) f (cid:16) yz (cid:17) ...f (cid:16) yz k (cid:17) · T ( y )Therefore, we conclude that the series S ( x ) and T ( x ) satisfy the same relations asin Definition 2.8 (modified in order to account for the variables z i ), even thoughthe s ’s and the t ’s are not independent of each other anymore. We will write: s − i ; i ) = ψ i = t [ i ; i ) and note that formulas (5.1) imply that:(5.7) ψ i ψ j = ψ j ψ i (5.8) ψ s X = q −(cid:104) hdeg X, ς s (cid:105) Xψ s ∀ X ∈ (cid:101) A + , where (cid:104)· , ·(cid:105) is the bilinear form on Z n given by (cid:104) ς i , ς j (cid:105) = δ ji − δ j − i .5.4. Consider the following topological coproduct on the algebra (cid:101) A + , which is thenatural analogue of the coproduct studied in Proposition 2.9:(5.9) ∆( S ( x )) = (1 ⊗ S ( x )) · ( S ( x ) ⊗ T ( x )) = ( T ( x ) ⊗ · (1 ⊗ T ( x )) by (5.3) and:(5.10) ∆( X ...k ) = k (cid:88) i =0 ( S k ( z k ) ...S i +1 ( z i +1 ) ⊗ ·· X ...i ( z , ..., z i ) ⊗ X i +1 ...k ( z i +1 , ..., z k ) (cid:81) ≤ u ≤ i Because S ( x ) is a power series in x , the coproduct defined abovetakes values in a completion of (cid:101) A + ⊗ (cid:101) A + . Specifically, to make sense of the secondline of (5.10) , we must expand the rational function: X ...k ( z , ..., z k ) (cid:81) ≤ u ≤ i 0) ( | d + d | , k + k ) ( | d | , k ) F IGURE 21. The hinge of a tensorThe intersection of the arrows, namely ( | d | , k ), is called the hinge of X ⊗ X . Definition 5.7. Let µ ∈ Q . We let A + ≤ µ ⊂ A + be the set of those X such that: (5.11) ∆( X ) = ∆ µ ( X ) + ( anything ) ⊗ ( slope < µ ) where ∆ µ ( X ) consists only of summands X ⊗ X with slope X = µ , as in (4.24) . In terms of the pictorial definitions of hinges above, X ∈ A + ≤ µ if and onlyif all summands in ∆( X ) have hinge at slope | d | /k ≤ µ as measured from the origin. It is easy to see that A + ≤ µ is a vector space. Let us define its graded pieces:(5.12) A ≤ µ | k = A + ≤ µ ∩ A k , A ≤ µ | d ,k = A + ≤ µ ∩ A d ,k and note that A ≤ µ | d ,k (cid:54) = 0 only if | d | ≤ kµ . Proposition 5.8. For any µ ∈ Q , the subspace A + ≤ µ is a subalgebra of A + , and: ∆ µ ( X ∗ Y ) = ∆ µ ( X ) ∗ ∆ µ ( Y ) for all X, Y ∈ A + ≤ µ .Proof. Note that degree is multiplicative, i.e. (assume the LHS is non-zero):deg (cid:16) f ( z , ..., z k ) E i j ⊗ ... ⊗ E i k j k (cid:17)(cid:16) f (cid:48) ( z , ..., z k ) E i (cid:48) j (cid:48) ⊗ ... ⊗ E i (cid:48) k j (cid:48) k (cid:17) == deg f ( z , ..., z k ) E i j ⊗ ... ⊗ E i k j k + deg f (cid:48) ( z , ..., z k ) E i (cid:48) j (cid:48) ⊗ ... ⊗ E i (cid:48) k j (cid:48) k Therefore, if ∆( X ) = X ⊗ X and ∆( Y ) = Y ⊗ Y with slope X , slope Y ≤ µ ,then slope X Y ≤ µ . Since ∆( XY ) = X Y ⊗ X Y , this implies the conclusion. (cid:3) {A ≤ µ | d ,k } µ ∈ Q yielda filtration of A d ,k by finite-dimensional vector spaces. , Lemma 5.10. The dimension of A ≤ µ | d ,k as a vector space over Q ( q, q n ) is at mostthe number of unordered collections: (5.13) ( i , j , λ ) , ..., ( i u , j u , λ u ) where: • λ s ∈ N with (cid:80) us =1 λ s = k • ( i s , j s ) ∈ Z ( n,n ) Z with (cid:80) us =1 [ i s ; j s ) = d • j s − i s ≤ µλ s for all s ∈ { , ..., u } In (5.72), we will show that the dimension of A ≤ µ | d ,k is equal to the number ofunordered collections (5.13). The argument below follows that of [9], [20], [21]. Proof. To any partition λ = ( λ ≤ ... ≤ λ u ) of k ∈ N , we associate the linear map: A ≤ µ | d ,k ϕ λ −→ End( V ⊗ u )( y , ..., y u ) X (cid:32) X ( λ ,...,λ u ) of (4.13)Consider the dominance ordering λ (cid:48) > λ on partitions, and define: A λ ≤ µ | d ,k = (cid:92) λ (cid:48) >λ Ker ϕ λ (cid:48) Since A ( k ) ≤ µ | d ,k = A ≤ µ | d ,k , then the desired bound on dim A ≤ µ | d ,k would follow from:(5.14) dim ϕ λ (cid:16) A λ ≤ µ | d ,k (cid:17) ≤ (cid:26) unordered ( i , j ) , ..., ( i u , j u ) ∈ Z ( n, n ) Z , such that u (cid:88) s =1 [ i s ; j s ) = d and j s − i s ≤ µλ s for all s ∈ { , ..., u } (cid:41) for any λ = ( λ ≤ ... ≤ λ u ). By Proposition 4.12, any Y ∈ Im ϕ λ is of the form: Y ( y , ..., y u ) ∈ End( V ⊗ u )[ y ± , ..., y ± u ] (cid:81) ≤ s Similarly, one can show that the residue of Y vanishes at y s q d − y t q λ t for any s < t and 0 ≤ d < λ s and this precisely implies that Y is a Laurent polynomial:(5.16) Y ( y , ..., y u ) = h ,...,h u ∈ Z (cid:88) ≤ α ,β ,...,α u ,β u ≤ n coefficient · y h ...y h u u E α β ⊗ ... ⊗ E α u β u Since the matrices R and (cid:101) R have total degree 0, the horizontal degree of Y is equalto that of X , namely d , so we conclude that the only summands with non-zerocoefficient in (5.16) satisfy:( h + ... + h u ) δ + deg E α β + ... + deg E α u β u = d Finally, the slope condition on X implies an analogous slope condition on Y : ineach variable y s , we have: nh s + α s − β s ≤ µλ s Therefore, the number of coefficients that one gets to choose in (5.16) is at mostthe number of collections ( j s = α s + nh s , i s = β s ) satisfying the conditions in theright-hand side of (5.14). The reason why we need to take unordered collections isthe symmetry property of Y proved in Proposition 4.12. (cid:3) X = X ...k ( z , ..., z k ) ∈ A + the matrix-valued power series X ( k ) ( y ) ∈ End( V )[ y ± ]. Proposition 5.12. For any A ∈ A d ,k and B ∈ A e ,l , we have: (5.17) ( A ∗ B ) ( k + l ) ( y ) = A ( k ) ( y ) B ( l ) ( y ) q ke n where e n is the last component of the vector e = ( e , ..., e n ) ∈ Z n .Proof. The proof is precisely the u = 1 case of the proof of Proposition 4.11, sincethe equality of braids therein indicates the fact that:(5.18) ( A ∗ B ) ( k + l ) ( y ) = A ( k ) ( y ) B ( l ) ( yq k )The fact that B ∈ A e ,l implies the homogeneity property: B ( z ξ, ..., z k ξ ) = ξ e n B ( z , ..., z k )Since R –matrices are invariant under rescaling variables, the rational function B ( l ) ( y ) of (4.13) also satisfies B ( l ) ( yξ ) = ξ e n B ( l ) ( y ). Then (5.18) implies (5.17). (cid:3) For all ( i, j ) ∈ Z ( n,n ) Z , define the linear maps:(5.19) ∞ (cid:77) k =0 A [ i ; j ) ,k α [ i ; j ) −→ Q ( q, q n ) X ...k ( z , ..., z k ) α [ i ; j ) −→ coefficient of E ji in X ( k ) ( y )(1 − q ) k q k ( i − j )+( j − i )+ k − k ¯ in (recall that E ij = E ¯ i ¯ j y (cid:98) j − n (cid:99) − (cid:98) i − n (cid:99) ). As a consequence of Proposition 5.12, we have: , Corollary 5.13. For any k, l ∈ N and ( i, j ) ∈ Z ( n,n ) Z , we have: (5.20) α [ i ; j ) ( A ∗ B ) = α [ • ; j ) ( A ) α [ i ; • ) ( B ) · q k ( •− i ) − l ( j −• ) n whenever deg A = ([ • ; j ) , k ) and deg B = ([ i ; • ) , l ) for some • between i and j . Ifsuch an • does not exist, then the RHS of (5.20) is set equal to 0, by convention.Proof. The corollary is an immediate consequence of (5.17) and (5.19). The onlything we need to check is that the power of q is the same in the left as in theright-hand sides of (5.20), which happens due to the elementary identity: q ( k + l )( i − j ) − k + l )¯ in = q k ( •− j ) − k • n q l ( i −• ) − l ¯ in q − k ( (cid:98) •− n (cid:99) − (cid:98) i − n (cid:99) ) q k ( •− i ) − l ( j −• ) n since if e = [ i ; • ), then e n = (cid:4) •− n (cid:5) − (cid:4) i − n (cid:5) . (cid:3) B µ | d = A ≤ µ | d , | d | µ . Particular importance will be given to the subalgebra:(5.21) B + µ = | d |∈ µ N (cid:77) d ∈ Z n B µ | d As a consequence of Proposition 5.8, the leading order term ∆ µ of (5.11) restrictsto a coproduct on the enhanced subalgebra: B ≥ µ = (cid:68) B + µ , ψ ± s (cid:69) s ∈ Z (cid:46) relation (5.8) Lemma 5.15. If X ∈ B + µ is primitive with respect to the coproduct ∆ µ , and: (5.22) α [ i ; j ) ( X ) = 0 for all ( i, j ) ∈ Z ( n,n ) Z such that deg X ∈ [ i ; j ) × N , then X = 0 .Proof. The assumption X ∈ B + µ implies that deg X = ( d , k ) with:(5.23) | d | = µk As we observed in the proof of Lemma 5.10, it suffices to show that ϕ λ ( X ) = 0 forall partitions λ , which we will do in reverse dominance ordering of the partition λ .The base case is when λ = ( k ), which is satisfied because ϕ ( k ) ( X ) = 0 is preciselythe content of the assumption (5.22). For a general partition λ , we may invoke theinduction hypothesis to conclude that ϕ λ (cid:48) ( X ) = 0 for all partitions λ (cid:48) > λ , and inthis case ϕ λ ( X ) takes the form of (5.16). However, the fact that X is a primitiveelement requires every summand appearing in the RHS of (5.16) to satisfy: nh s + α s − β s < µλ s for all 1 ≤ s ≤ u (we have u = l ( λ ) > 1, since we are dealing with the case λ (cid:54) = ( k )).However, relation (5.23) forces the following identity: u (cid:88) s =1 ( nh s + α s − β s ) = µk = µ u (cid:88) s =1 λ s This yields a contradiction, hence ϕ λ ( X ) = 0, thus completing the induction step. (cid:3) TALE OF TWO SHUFFLE ALGEBRAS 53 B + µ , which together withLemma 5.10 will yield a PBW basis of the shuffle algebra, leading to the proof ofTheorem 1.5. Consider the following notion of symmetrization, analogous to (2.8)and (4.10):(5.24) Sym X = (cid:88) σ ∈ S ( k ) R σ · X σ (1) ...σ ( k ) ( z σ (1) , ..., z σ ( k ) ) · R − σ where R σ is the product of R ij (cid:16) z i z j (cid:17) associated to any braid lift of σ . For instance:(5.25) R ω k ( z , ..., z k ) = k − (cid:89) i =1 k (cid:89) j = i +1 R ij (cid:18) z i z j (cid:19) lifts the longest element of S ( k ). Consider the matrix-valued rational functions: Q ( x ) = q − (cid:88) ≤ i,j ≤ n ( xq ) δ i For any ( i, j ) ∈ Z ( n,n ) Z and µ ∈ Q such that j − iµ ∈ N , set: (5.30) F µ [ i ; j ) = Sym R ω k ( z , ..., z k ) k (cid:89) a =1 (cid:20) (cid:101) R a (cid:18) z z a (cid:19) ... (cid:101) R a − ,a (cid:18) z a − z a (cid:19) Q a − ,a (cid:18) z a − z a (cid:19) E ( a ) s a − s a q san (cid:21) (5.31) ¯ F µ [ i ; j ) = ( − q q n ) − k · Sym R ω k ( z , ..., z k ) k (cid:89) a =1 (cid:20) (cid:101) R a (cid:18) z z a (cid:19) ... (cid:101) R a − ,a (cid:18) z a − z a (cid:19) ¯ Q a − ,a (cid:18) z a − z a (cid:19) E ( a ) s (cid:48) a − s (cid:48) a q s (cid:48) an (cid:21) (recall (4.22) ) where s a = j − (cid:100) µa (cid:101) , s (cid:48) a = j − (cid:98) µa (cid:99) . Then F µ [ i ; j ) , ¯ F µ [ i ; j ) ∈ B + µ , and: ∆ µ (cid:16) F µ [ i ; j ) (cid:17) = (cid:88) •∈{ i,...,j } F µ [ • ; j ) ψ i ψ • ⊗ F µ [ i ; • ) (5.32) ∆ µ (cid:16) ¯ F µ [ i ; j ) (cid:17) = (cid:88) •∈{ i,...,j } ψ • ψ j ¯ F µ [ i ; • ) ⊗ ¯ F µ [ • ; j ) (5.33) where we set F µ [ i ; j ) = ¯ F µ [ i ; j ) = 0 if j − iµ / ∈ N . , We will often write F ( k )[ i ; j ) = F µ [ i ; j ) if j − i = µk (and the analogous notation for ¯ F )in order to emphasize the fact that this shuffle element has vertical degree k . Proof. Note that if Q , ¯ Q were replaced by (cid:101) R in formulas (5.30), (5.31), then theright-hand sides of the aforementioned formulas would precisely equal:(5.34) E s s q s n ∗ ... ∗ E s k − s k q skn and E s (cid:48) s (cid:48) q s (cid:48) n ∗ ... ∗ E s (cid:48) k − s (cid:48) k q s (cid:48) kn The fact that the shuffle elements (5.34) satisfy the wheel conditions is simply aconsequence of iterating Proposition 4.11 a number of k − F µ [ i ; j ) , ¯ F µ [ i ; j ) are concerned, the fact that they satisfy the wheel conditionsis proved similarly with the fact that (5.34) satisfies the wheel conditions: thisis because Proposition 4.11 uses the fact that Res x = q − (cid:101) R ( x ) is a multiple of thepermutation matrix, and we have already seen in (5.28) that the residues of Q ( x ),¯ Q ( x ), (cid:101) R ( x ) are the same up to scalar. We leave the details to the interested reader.Let us prove that the shuffle elements (5.30) and (5.31) lie in B + µ . We will onlyprove the former case, since the latter case is analogous. We have:(5.35) F := F µ [ i ; j ) = Sym R ω k ( z , ..., z k ) X ...k ( z , ..., z k )where X is the expression on the second line of (5.30). To this end, for any l ∈{ , ..., k } we need to look at the first l tensor factors of Sym R ω k X and isolatethe terms of minimal | hdeg | . If Y is a k –tensor, we will henceforth use the phrase“initial degree of Y ” instead of “total | hdeg | of the first l factors of Y ”. Because:lim x → R ab ( x ) = n (cid:88) i,j =1 q δ ji E ( a ) ii ⊗ E ( b ) jj + (cid:88) i>j ( q − q − ) E ( a ) ij ⊗ E ( b ) ji (5.36) lim x →∞ R ab ( x ) = n (cid:88) i,j =1 q − δ ji E ( a ) ii ⊗ E ( b ) jj − (cid:88) i For any ≤ a (cid:54) = b ≤ k , multiplying a k –tensor Y by: R ab (cid:18) z a z b (cid:19) or (cid:101) R ab (cid:18) z a z b (cid:19) or Q ab (cid:18) z a z b (cid:19) or ¯ Q ab (cid:18) z a z b (cid:19) (either on the left or on the right) cannot decrease the minimal initial degree of Y . Therefore, it suffices to compute the minimal initial degree of:(5.38) X σ (1) ...σ ( k ) ( z σ (1) , ..., z σ ( k ) ) = k (cid:89) a =1 (cid:20) (cid:101) R σ (1) σ ( a ) (cid:18) z σ (1) z σ ( a ) (cid:19) ... (cid:101) R σ ( a − ,σ ( a ) (cid:18) z σ ( a − z σ ( a ) (cid:19) Q σ ( a − ,σ ( a ) (cid:18) z σ ( a − z σ ( a ) (cid:19) E ( σ ( a )) s a − s a q san (cid:21) for any permutation σ of { , ..., k } . Claim 5.18 implies that the minimal initialdegree (henceforth denoted “m.i.d.”) comes from the various E s a − s a factors:(5.39) m.i.d. of (5.38) = (cid:88) a ∈ A ( s a − − s a ) + TALE OF TWO SHUFFLE ALGEBRAS 55 where A = { σ − (1) , ..., σ − ( l ) } and the number a ∈ A such that a − / ∈ A . This number must be subtracted from the maximal initial degree because Q ( ∞ ) has 0 on the diagonal, by definition. It is elementary to show that:(5.40) RHS of (5.39) = (cid:88) a ∈ A ( (cid:100) µa (cid:101) − (cid:100) µ ( a − (cid:101) ) + ≥ t (cid:88) s =1 ( (cid:100) µβ s (cid:101) − (cid:98) µα s (cid:99) )if A splits up into consecutive blocks of integers:(5.41) A = { α + 1 , ..., β , α + 1 , ..., β , ..., α t + 1 , ..., β t } with 0 ≤ α , while β s < α s +1 for all s , and β t ≤ k . Since:(5.42) RHS of (5.40) ≥ µ t (cid:88) s =1 ( β s − α s ) = µ · A = µl we conclude that F ∈ A + ≤ µ . Because | hdeg F | = j − i and vdeg F = k , then F ∈ B + µ .Moreover, the terms of minimal initial degree in F correspond to those situationswhere we have equality in all the inequalities above, and these require µl ∈ Z and: A = { , ..., l } Let us now compute the summands which achieve the minimal initial degree in:(5.43) R ω k X = R ω l ( z , ..., z l ) (cid:20) R ,l +1 (cid:18) z z l +1 (cid:19) ...R l,k (cid:18) z l z k (cid:19)(cid:21) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) R ω k − l ( z l +1 , ..., z k ) E (1 ...l ) s | ... | s l (cid:20) (cid:101) R ,l +1 (cid:18) z z l +1 (cid:19) ...Q l,l +1 (cid:18) z l z l +1 (cid:19) ... (cid:101) R l,k (cid:18) z l z k (cid:19)(cid:21) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) E ( l +1 ...k ) s l | ... | s k where the notation E ( u...v ) c u − | c u | ... | c v − | c v is shorthand for:(5.44) v (cid:89) a = u (cid:20) (cid:101) R ua (cid:18) z u z a (cid:19) ... (cid:101) R a − ,a (cid:18) z a − z a (cid:19) Q a − ,a (cid:18) z a − z a (cid:19) E ( a ) c a − c a q can (cid:21) If the terms with the squiggly red underline were not present in (5.43), then wewould conclude that the terms of minimal initial degree would be precisely:(5.45) m.i.d. R ω k X = R ω l E (1 ...l ) s | ... | s l ⊗ R ω k − l E ( l +1 ...k ) s l | ... | s k and upon symmetrization, this would almost imply (5.32). However, we must dealwith the contribution of the terms with the squiggly red underline. As we have seenin the discussion above (specifically (5.36) and (5.37)), these factors only contributea diagonal matrix to the terms of minimal initial degree. Specifically, if: E (1 ...l ) s | ... | s l = n (cid:88) x a ,y a =1 coefficient · E x y ⊗ ... ⊗ E x l y l ⊗ ⊗ k − l E ( l +1 ...k ) s l | ... | s k = n (cid:88) x a ,y a =1 coefficient · ⊗ l ⊗ E x l +1 y l +1 ⊗ ... ⊗ E x k y k , (above, y l = s l = x l +1 in all summands with non-zero coefficient) then the terms ofminimal initial degree in R ω k X yield the following value for the coproduct (5.11):∆ µ ( R ω k X ) = n (cid:88) x a ,y a =1 ψ − x l +1 ...ψ − x k q (cid:80) ≤ a ≤ ll
The quantity (5.44) is a sum of tensors E x u y u ⊗ ... ⊗ E x v y v where: { x u ≡ r mod n } − { y u ≡ r mod n } = { c u ≡ r mod n } − { d u ≡ r mod n } for any r ∈ Z /n Z . we may rewrite (5.46) as:∆ µ ( R ω k X ) = R ω l E (1 ...l ) s | ... | s l ψ s l +1 ...ψ s k ψ s l ...ψ s k − ⊗ R ω k − l E ( l +1 ...k ) s l | ... | s k Upon symmetrization with respect to ( l, k − l )–shuffles (i.e. those permutations σ ∈ S ( k ) which preserve the set { , ..., l } ), this leads precisely to (5.32). (cid:3) F µ [ i ; j ) , ¯ F µ [ i ; j ) ∈ B + µ are completelydetermined by the coproduct relations (5.32) and (5.33), together with their valueunder the linear functionals (5.19). Let us therefore compute the latter: Proposition 5.21. For any ( i, j ) ∈ Z ( n,n ) Z and µ ∈ Q such that j − iµ ∈ N , we have: α [ u ; v ) (cid:16) F µ [ i ; j ) (cid:17) = δ ( i,j )( u,v ) (1 − q ) q gcd( k,j − i ) n (5.47) α [ u ; v ) (cid:16) ¯ F µ [ i ; j ) (cid:17) = δ ( i,j )( u,v ) (1 − q − ) q − gcd( k,j − i ) n (5.48) for any ( u, v ) ∈ Z ( n,n ) Z such that [ u ; v ) = [ i ; j ) .Proof. Recall that F = F µ [ i ; j ) is given by the symmetrization (5.35), namely: F = (cid:88) σ ∈ S ( k ) R σ · σ (cid:16) R ω k · second line of (5.30) (cid:17) σ − · R − σ TALE OF TWO SHUFFLE ALGEBRAS 57 where R σ is an arbitrary braid which lifts the permutation σ . Of the k ! summandsin the right-hand side, only the one corresponding to the identity permutation isinvolved in the iterated residue of F at z k = z k − q ,..., z = z q , hence we obtain:(5.49) Res { z = y,z = yq ,...,z k = yq k − } F = R ω k ( y, yq , ..., yq k − ) k (cid:89) a =1 (cid:20) (cid:101) R a (cid:0) q − a (cid:1) ... (cid:101) R a − ,a (cid:0) q − (cid:1) · Res x = q − Q a − ,a ( x ) · E ( a ) s a − s a q san (cid:12)(cid:12)(cid:12) z a (cid:55)→ yq a − (cid:21) (as E ( a ) ij = E ( a )¯ i ¯ j z (cid:98) i − n (cid:99) − (cid:98) j − n (cid:99) a , we must specialize z a = yq a − in (5.49)). Note that: R ω k ( z , ..., z k ) = k − (cid:89) a =1 k (cid:89) b = a +1 R ab (cid:18) z a z b (cid:19) = R (cid:18) z z (cid:19) ...R k (cid:18) z z k (cid:19) (cid:89) b = k (cid:89) a = b − R ab (cid:18) z a z b (cid:19) due to (4.3). Since (cid:101) R is given by (4.2) and Res x = q − Q ( x ) = q − · (12), we have:LHS of (5.49) = q − k R (cid:0) q − (cid:1) ...R k (cid:0) q − k (cid:1) (cid:89) b = k (cid:89) a = b − R ab (cid:0) q a − b (cid:1) k (cid:89) b =1 (cid:34) b − (cid:89) a =1 R ba (cid:0) q b − a − (cid:1) · ( b − , b ) · E ( b ) s b − s b q sbn (cid:12)(cid:12)(cid:12) z b (cid:55)→ yq b − (cid:35) If we move the permutations ( b − , b ) all the way to the right, then (5.49) equals:LHS of (5.49) = q − k R (cid:0) q − (cid:1) ...R k (cid:0) q − k (cid:1) (cid:89) b = k (cid:89) a = b − R ab (cid:0) q a − b (cid:1) k (cid:89) b =3 (cid:34) b − (cid:89) a =1 R b,a +1 (cid:0) q b − a − (cid:1)(cid:35) · k (cid:89) b =1 E (1) s b − s b q sbn (cid:12)(cid:12)(cid:12) z b (cid:55)→ yq b − · (cid:18) ... k ... (cid:19) (4.6) == q − k (cid:89) ≤ a
Theorem 5.24. We have an algebra isomorphism: (5.57) D + Υ + ∼ = A + , p ( k ) d (cid:55)→ P ( k ) d where D + is the explicit algebra of Definition 3.24. B + µ satisfy the analogues of relations (3.67) and (3.68). Proposition 5.26. Assume gcd( j − i, k ) = 1 , and consider the lattice triangle T : (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:64)(cid:64) (cid:67)(cid:67)(cid:67)(cid:67)(cid:67)(cid:67)(cid:67)(cid:67) (0 , j − i, k ) µT F IGURE . Two types of lattice trianglesor (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:0)(cid:0) (0 , j − i, k ) µT uniquely determined as the triangle of maximal area situated completely to the rightof the vector ( j − i, k ) , which does not contain any lattice points inside. Let µ denotethe slope of one of the edges of T , as indicated in the pictures above. Then: ∆ (cid:16) P ( k )[ i ; j ) (cid:17) = P ( k )[ i ; j ) ⊗ ψ i ψ j ⊗ P ( k )[ i ; j ) + (cid:16) tensors with hinge strictly right of T (cid:17) +(5.58) + (cid:88) i ≤ u We have the identity: (5.65) E (1 ...k ) c | c | ... | c k − | c k = ¯ E (1 ...k ) c | c +1 | ... | c k − +1 | c k · ( − q q n ) − k where ¯ E ( u...v ) c u − | ... | c v = (cid:81) va = u (cid:104) (cid:101) R ua (cid:16) z u z a (cid:17) ... (cid:101) R a − ,a (cid:16) z a − z a (cid:17) ¯ Q a − ,a (cid:16) z a − z a (cid:17) E ( a ) c a − c a q can (cid:105) . Let us first show how the Claim allows us to complete the proof of the Proposition.Because of (5.65), formula (5.64) is equivalent to:(5.66) ¯ F µ [ i ; u ) = ( − q q n ) α − k · Sym R ω k − α ¯ E (1 ...k − α ) s α +1 | s α +1 +1 | ... | s k − +1 | s k Then (5.62), (5.63), (5.66) follow from (5.30), (5.31) and the formulas below: j − (cid:24) ( j − i ) tk (cid:25) = j − (cid:100) µt (cid:101) ∀ t ∈ { , ..., β } j − (cid:24) ( j − i ) tk (cid:25) + δ αt = v − (cid:24) ( v − u )( t − β ) α − β (cid:25) ∀ t ∈ { β + 1 , ..., α } j − (cid:24) ( j − i ) tk (cid:25) + 1 = u − (cid:98) µ ( t − α ) (cid:99) ∀ t ∈ { α + 1 , ..., k } which are all straightforward consequences of our assumption on the triangle T .This completes the proof of formula (5.58) for the picture on the left in Figure22 (as we said, the case of the picture on the right is analogous, and left to theinterested reader). As for Claim 5.27, we start with the following identity:(5.67) ( E j,t ⊗ Q (cid:18) z z (cid:19) (1 ⊗ E t,i ) q tn == ( E j,t +1 ⊗ 1) ¯ Q (cid:18) z z (cid:19) (1 ⊗ E t +1 ,i ) q t +1 n · ( − q q n ) − for all i, j, t ∈ Z . Indeed, by plugging in (5.26)–(5.27), formula (5.67) reads: q − n (cid:88) u =1 z (cid:98) j − n (cid:99) − (cid:98) t − n (cid:99) z (cid:98) t − n (cid:99) − (cid:98) i − n (cid:99) (cid:16) z q z (cid:17) δ t
We have already seen that, for fixed µ , the assignment: f µ [ i ; j ) of (3.57) (cid:32) F µ [ i ; j ) of (5.30)yields an algebra homomorphism E + µ → A + . To extend this to a homomorphism:(5.68) Υ + : D + −→ A + we need to prove that formulas (3.67) and (3.68) hold with p , f replaced by P , F .In order to show that (3.67) holds in this setup, let us first show that the linearmaps α [ u ; v ) take the same value on both sides of the equation. By (5.20), we have: α [ u ; v ) (LHS of (3.67)) = α [ u ; v ) (cid:16) P ( k )[ i ; j ) P ( k (cid:48) ) l δ ,r (cid:17) − α [ u ; v ) (cid:16) P ( k (cid:48) ) l δ ,r P ( k )[ i ; j ) (cid:17) = TALE OF TWO SHUFFLE ALGEBRAS 63 = α [ u + nl ; v ) (cid:16) P ( k )[ i ; j ) (cid:17) α [ u ; u + nl ) (cid:16) P ( k (cid:48) ) l δ ,r (cid:17) q dn − α [ v − nl ; v ) (cid:16) P ( k (cid:48) ) l δ ,r (cid:17) α [ u ; v − nl ) (cid:16) P ( k )[ i ; j ) (cid:17) q − dn == δ ( i,j )( u + nl,v ) δ ru mod g q dn − δ ( i,j )( u,v − nl ) δ rv mod g q − dn = α [ u ; v ) (RHS of (3.67)) ∀ u, v . Therefore, Lemma 5.15 implies that the equality (3.67) would follow from:LHS of (3.67) ∈ B + j + ln − ik + k (cid:48) We may depict the degree vectors of the elements P ( k )[ i ; j ) , P ( k (cid:48) ) l δ ,r , P ( k + k (cid:48) )[ i ; j + nl ) as: (cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:0)(cid:0)(cid:0)(cid:0)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:0)(cid:0)(cid:0)(cid:0) (0 , j − i, k ) ( j − i + nl, k + k (cid:48) )( nl, k (cid:48) ) We need to show that all the hinges of summands of ∆(LHS of (3.67)) are to theright the vector ( j − i + nl, k + k (cid:48) ). Since coproduct is multiplicative, the hingesof ∆( XY ) are all among the sums of hinges of ∆( X ) and ∆( Y ), as vectors in Z .By definition, the hinges of ∆( P ( k )[ i ; j ) ) and ∆( P ( k (cid:48) ) l δ ,r ) lie to the right of the vectors( j − i, k ) and ( nl, k (cid:48) ), respectively. The sum of any two such hinges lies to the rightof the parallelogram in the picture, except for the sum of the two hinges below:∆( P ( k )[ i ; j ) ) = ... + P ( k )[ i ; j ) ⊗ ... has a hinge at ( j − i, k )∆( P ( k (cid:48) ) l δ ,r ) = ... + 1 ⊗ P ( k (cid:48) ) l δ ,r + ... has a hinge at (0 , P ( k )[ i ; j ) P ( k (cid:48) ) l δ ,r ) and ∆( P ( k (cid:48) ) l δ ,r P ( k )[ i ; j ) ) both have a hinge at the point ( j − i, k ),but the corresponding summand in both coproducts is: P ( k )[ i ; j ) ⊗ P ( k (cid:48) ) l δ ,r We conclude that this summand vanishes in ∆(LHS of (3.67)), which thereforehas all the hinges to the right of ( j − i + nl, k + k (cid:48) ). This completes the proof of (3.67).Let us now prove (3.68) by induction on k + k (cid:48) (the base case k + k (cid:48) = 1 is trivial).Recall that µ = j + j (cid:48) − i − i (cid:48) k + k (cid:48) , and let us represent the degrees of P ( k )[ i ; j ) and P ( k (cid:48) )[ i (cid:48) ; j (cid:48) ) as: (cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:65)(cid:65)(cid:65)(cid:65)(cid:1)(cid:1)(cid:1)(cid:1) (cid:65)(cid:65)(cid:65)(cid:65)(cid:1)(cid:1)(cid:1)(cid:1) (0 , j − i, k )( j + j (cid:48) − i − i (cid:48) , k + k (cid:48) )( j (cid:48) − i (cid:48) , k (cid:48) )4 ANDREI NEGUT , We have the following formulas, courtesy of Proposition 5.26:∆ (cid:16) P ( k )[ i ; j ) (cid:17) = P ( k )[ i ; j ) ⊗ ψ i ψ j ⊗ P ( k )[ i ; j ) + (cid:88) i ≤ u The elements (5.71) are all linearly independent in A + . Let us first show that the Claim completes the proof of the Theorem. The linearindependence of the elements (5.71) implies that:(5.72) dim A ≤ µ | d ,k ≥ (cid:110) unordered collections (5.13) (cid:111) for all µ , k , d (indeed, formula (3.26) implies that the number of products (5.71)with µ bounded above is precisely equal to the number in the RHS of (5.72)).Combining this with Lemma 5.10, we conclude that the products (5.71) actuallyform a linear basis of A + , and therefore A + is generated by the elements (5.54)and (5.55). This implies that Υ + is an isomorphism, since we showed in [22] that(5.71) also form a linear basis of D + .Let us now prove Claim 5.28. We assume that the basis vectors x ( i ) µ of B + µ areordered in non-decreasing order of | hdeg | , i.e.:(5.73) i ≥ i (cid:48) ⇒ | hdeg x ( i ) µ | ≥ | hdeg x ( i (cid:48) ) µ | Now suppose we have a non-trivial linear relation among the various products(5.71). We may rewrite this hypothetical relation as:(5.74) x ( i ) µ j (cid:89) ν>µ x ( j ) ν = (cid:88) coefficient · x ( i (cid:48) ) µ j (cid:48) (cid:89) ν>µ x ( j (cid:48) ) ν where all terms in the RHS have i (cid:48) < i . Since the coproduct is multiplicative, thenall the hinges of ∆( XY ) are sums of hinges of ∆( X ) and hinges of ∆( Y ), as vectorsin Z . Therefore, ∆(LHS of (5.74)) has a single summand with hinge at the latticepoint:(5.75) (cid:16) hdeg x ( i ) µ , vdeg x ( i ) µ (cid:17) and the corresponding summand is precisely:(5.76) ∆(LHS of (5.74)) = ... + ψx ( i ) µ ⊗ j (cid:89) ν>µ x ( j ) ν + ... where ψ stands for a certain (unimportant) product of ψ ± a ’s. Meanwhile, thecoproduct of the RHS of (5.74) can only have a hinge at a lattice point (5.75) if , equality is achieved in (5.73). The corresponding summand in the coproduct is:(5.77) ∆(RHS of (5.74)) = ... + (cid:88) coefficient · ψx ( i (cid:48) ) µ ⊗ j (cid:48) (cid:89) ν>µ x ( j (cid:48) ) ν + ... Since x ( i ) µ cannot be expressed as a linear combination of x ( i (cid:48) ) µ with i (cid:48) < i , the right-hand sides of expressions (5.76) and (5.77) cannot be equal. This contradictionimplies that there can be no relation (5.74), which proves Claim 5.28. (cid:3) Corollary 5.29. The algebra A + is generated by the vdeg = 1 elements: (5.78) (cid:110) E ij (cid:111) ( i,j ) ∈ Z n,n ) Z The corollary is an immediate consequence of Proposition 3.40 and Theorem 5.24.6. The double shuffle algebra with spectral parameters In the previous Section, we constructed the extended shuffle algebra correspondingto the R –matrix with spectral parameters (3.81). We will now take two suchextended shuffle algebras and construct their double, as was done in Subsections2.10 and 2.12 for R –matrices without spectral parameters. This will conclude theproof of Theorem 1.5.6.1. Let q + = q and q − = q − n q − . If (cid:101) R + ( x ) = (cid:101) R ( x ) is given by (4.2), then: (cid:101) R − ( x ) = (cid:34) (cid:101) R † (cid:18) x (cid:19) − (cid:35) † ∈ End( V ⊗ V )( x )is given by: (cid:101) R − ( x ) = (cid:88) ≤ i,j ≤ n E ii ⊗ E jj (cid:18) q − − xqq − − xq − (cid:19) δ ji − ( q − q − ) (cid:88) ≤ i (cid:54) = j ≤ n E ij ⊗ E ji q j − i ) ( xq − ) δ i The shuffle algebra A − is defined just like in Definition 4.8,using q − instead of q , and the multiplication (4.9) uses (cid:101) R − instead of (cid:101) R . Because of (6.1), the map:(6.2) A + Φ −→ A − , X ...k ( z , ..., z k ) (cid:55)→ D ...D k X ...k ( z , ..., z k ) (cid:12)(cid:12)(cid:12) q + (cid:55)→ q − TALE OF TWO SHUFFLE ALGEBRAS 67 is a Q ( q )–linear algebra isomorphism. Therefore, it is straightforward to see thatthe images of the elements (5.30)–(5.31) under Φ are given by: F − µ [ i ; j ) = Sym R ω k k (cid:89) a =1 (cid:20) (cid:101) R − a (cid:18) z z a (cid:19) ... (cid:101) R − a − ,a (cid:18) z a − z a (cid:19) Q − a − ,a (cid:18) z a − z a (cid:19) E ( a ) s a − s a q san − (cid:21) ¯ F − µ [ i ; j ) = ( − q n + ) k Sym R ω k k (cid:89) a =1 (cid:20) (cid:101) R − a (cid:18) z z a (cid:19) ... (cid:101) R − a − ,a (cid:18) z a − z a (cid:19) ¯ Q − a − ,a (cid:18) z a − z a (cid:19) E ( a ) s (cid:48) a − s (cid:48) a q s (cid:48) an − (cid:21) where Q − , ¯ Q − are defined just like Q + := Q and ¯ Q + := ¯ Q of (5.26) and (5.27),but substituting q + = q by q − . By applying the isomorphism Φ, one concludesthat the analogues of Theorem 5.24 and Corollary 5.29 hold for A − instead of A + .6.3. In Definition 5.2, we defined the extended shuffle algebra by introducing newgenerators. We will now add two more central elements c and ¯ c , and define instead:(6.3) (cid:101) A ± = (cid:68) A ± , s ± [ i ; j ) , c, ¯ c (cid:69) i ≤ j ≤ i ≤ n c, ¯ c central and relations (6.4) and (6.5)where:(6.4) R (cid:18) xy (cid:19) S ± ( x ) S ± ( y ) = S ± ( y ) S ± ( x ) R (cid:18) xy (cid:19) (6.5) X ± · S ± ( y ) = S ± ( y ) · R k (cid:16) z k y (cid:17) ...R (cid:16) z y (cid:17) f (cid:16) z k y (cid:17) ...f (cid:16) z y (cid:17) X ± (cid:101) R ± (cid:18) z y (cid:19) ... (cid:101) R ± k (cid:18) z k y (cid:19) for any X ± = X ± ...k ( z , ..., z k ) ∈ A ± ⊂ (cid:101) A ± , where: S + ( x ) = d =0 only if i ≤ j (cid:88) ≤ i,j ≤ n, d ≥ s +[ i ; j + nd ) ⊗ E ij x d S − ( x ) = d =0 only if i ≤ j (cid:88) ≤ i,j ≤ n, d ≥ s − [ i ; j + nd ) ⊗ E ji x d Define the series T ± ( x ) by (3.99) and (3.100), which imply the following:(6.6) T ± ( y ) · X ± = (cid:101) R ± k (cid:18) yz k (cid:19) ... (cid:101) R ± (cid:18) yz (cid:19) X ± R (cid:16) yz (cid:17) ...R k (cid:16) yz k (cid:17) f (cid:16) yz (cid:17) ...f (cid:16) yz k (cid:17) · T ± ( y )6.4. The algebras (cid:101) A ± are graded by Z n × Z , with:deg X ± ...k ( z , ..., z k ) = ( d , ± k ) , ∀ X ± ∈ A ± deg s ± [ i ; j ) = ( ± [ i ; j ) , , ∀ i ≤ j where d ∈ Z n is defined in (4.20). We write deg X = (hdeg X, vdeg X ) to specifythe components of the degree vector in Z n and Z , respectively. The reason why weintroduced central elements c and ¯ c to the algebras (6.3) is to twist the coproduct.Specifically, let ∆ old be the coproduct of (5.9)–(5.10), and define:(6.7) ∆ : (cid:101) A ± −→ (cid:101) A ± (cid:98) ⊗ (cid:101) A ± , by the formulas ∆( c ) = c ⊗ c , ∆(¯ c ) = ¯ c ⊗ ¯ c , as well as:(6.8) if ∆ old ( X ) = X ⊗ X then ∆( X ) = X c − (hdeg X ) n ¯ c (vdeg X ) ⊗ X Since deg X is multiplicative in X , the fact that ∆ old is co-associative and analgebra homomorphism implies the analogous statements for the coproduct ∆.6.5. We must now prove an analogue of Proposition 2.11, but the main difficulty indoing so is (2.32): given X, Y ∈ End( V ⊗ k )( z , ..., z k ), we can still define the trace of XY , but the answer will be a rational function in z , ..., z k . To obtain a number, onemust integrate out the variables z , ..., z k , and the choice of contours will be crucial.Let us consider the following expressions, for any σ ∈ S ( k ): R σ = ≤ i There is a pairing (of vector spaces): (6.10) A + ⊗ A − (cid:104)· , ·(cid:105) −→ Q ( q, q n ) given by: (6.11) (cid:68) I +1 ∗ ... ∗ I + k , X − ...k ( z , ..., z k ) (cid:69) = ( q − k (cid:90) | z |(cid:28) ... (cid:28)| z k | Tr R ω k k (cid:89) a =1 (cid:34) I ( a ) a ( z a ) k (cid:89) b = a +1 (cid:101) R + ab (cid:18) z a z b (cid:19)(cid:35) X ...k ( z , ..., z k ) (cid:81) ≤ i Formula (6.11) is well-defined as a linear functional in the second argument,while (6.12) is well-defined as a linear functional in the first argument. Therefore,to show that (6.10) is well-defined as a linear functional in both arguments, weonly need to show that (6.11) and (6.12) produce the same result when X ± is ofthe form (6.9) (this statement implicitly uses Corollary 5.29, which states that anyelement in A ± is a linear combination of the elements (6.9)). To this end, we have:1( q − k (cid:68) I +1 ∗ ... ∗ I + k , J − ∗ ... ∗ J − k (cid:69) according to (6.11) = (cid:90) | z |(cid:28) ... (cid:28)| z k | (cid:88) σ ∈ S ( k ) (6.13) Tr (cid:89) ≤ i There exists a bialgebra pairing: (6.21) (cid:101) A + ⊗ (cid:101) A − (cid:104)· , ·(cid:105) −→ Q ( q, q n ) generated by (6.10) and: (cid:10) S +2 ( y ) , S − ( x ) (cid:11) = (cid:101) R + (cid:18) xy (cid:19) (cid:10) T +2 ( y ) , T − ( x ) (cid:11) = (cid:101) R − (cid:18) xy (cid:19) (6.22) (cid:10) S +2 ( y ) , T − ( x ) (cid:11) = R (cid:18) xy (cid:19) f − (cid:18) xy (cid:19) (cid:10) T +2 ( y ) , S − ( x ) (cid:11) = R (cid:18) xy (cid:19) (6.23) (all rational functions above are expanded in | x | (cid:28) | y | ).Proof. The proof follows that of Proposition 2.11 very closely, so we will only sketchthe main ideas and leave the details to the interested reader. Take any: a, b ∈ { X + , S + ( x ) , T + ( x ) , for X ∈ A + } c ∈ { X − , S − ( x ) , T − ( x ) , for X ∈ A − } and define (cid:104) ab, c (cid:105) to be the RHS of (2.28). Then if (cid:80) i a i b i = 0 holds in (cid:101) A + , wemust show that the pairing: (cid:42)(cid:88) i a i b i , c (cid:43) TALE OF TWO SHUFFLE ALGEBRAS 73 thus defined is 0. If at least one of a, b, c is either S ± ( x ) or T ± ( x ), thenthe statement in question is proved just like in Proposition 2.11, if one iscareful to expand x around ∞ ± (since (6.11)–(6.12) also involve integrals, weneed to stipulate that x should be closer to ∞ ± than any of the variables z , .., z k ).The remaining case is when a, b, c are all in A ± , and we must prove that:(6.24) (cid:104) A + ∗ B + , Y − (cid:105) = (cid:104) B + ⊗ A + , ∆( Y − ) (cid:105) for all A ...k ( z , ..., z k ) , B ...l ( z , ..., z l ) ∈ A + and Y ...k + l ( z , ..., z k + l ) ∈ A − . ByCorollary 5.29, it suffices to consider A = I ∗ ... ∗ I k and B = I k +1 ∗ ... ∗ I k + l forvarious I a ∈ End( V )[ z ± ]. In this case, we may rewrite (6.11) as:(6.25) (cid:68) I +1 ∗ ... ∗ I + k , X − ...k ( z , ..., z k ) (cid:69) = ( q − k (cid:90) | z |(cid:29) ... (cid:29)| z k | Tr (cid:89) b = k (cid:34) I ( b ) k +1 − b ( z b ) (cid:89) a = b − (cid:101) R + ba (cid:18) z b z a (cid:19)(cid:35) R ω k S ...k ( z , ..., z k ) (cid:81) ≤ i We have the following commutation relations in the algebra A : (6.28) S ∓ ( w ) · ± X ± == (cid:101) R ± k (cid:18) wcz k (cid:19) ... (cid:101) R ± (cid:18) wcz (cid:19) X ± R (cid:18) wcz (cid:19) ...R k (cid:18) wcz k (cid:19) · ± S ∓ ( w )(6.29) X ± · ± T ∓ ( w ) == T ∓ ( w ) · R k (cid:16) z k wc (cid:17) ...R (cid:16) z wc (cid:17) X ± (cid:101) R ± (cid:16) z wc (cid:17) ... (cid:101) R ± k (cid:16) z k wc (cid:17) if X ± = X ± ...k ( z , ..., z k ) ∈ A ± , where we recall that · + = · and · − = · op . Finally: (6.30) (cid:104) E + ij , E − i (cid:48) j (cid:48) (cid:105) == ( q − c (cid:98) i − n (cid:99) − (cid:98) j − n (cid:99) (cid:88) k ∈ Z (cid:16) s +[ j (cid:48) ; i + nk ) t +[ j ; i (cid:48) − nk ) ¯ c − t − [ i ; j (cid:48) + nk ) s − [ i (cid:48) ; j − nk ) ¯ c − (cid:17) (we set s ± [ i ; j ) = t ± [ i ; j ) = 0 if i > j ).Proof. Formulas (6.28) and (6.29) are proved just like (2.42) and (2.43) (the pres-ence of the variable c is due to the twist in the coproduct (6.7)), and so we leavethem as exercises to the interested reader. As far as (6.30) is concerned, we note thedefinition (6.8) of the coproduct implies the following analogue of formula (2.47):(6.31) ∆ (cid:18) E ij z d (cid:19) = E ij z d ⊗ a,b ≥ (cid:88) ≤ x,y ≤ n s +[ x ; i + na ) t +[ j ; y + nb ) c d + a + b ¯ c ⊗ E xy z d + a + b in (cid:101) A + , as well as the following analogue of (2.48):(6.32) ∆ (cid:18) E i (cid:48) j (cid:48) z d (cid:48) (cid:19) = a,b ≥ (cid:88) ≤ x,y ≤ n E xy z d (cid:48) − a − b ⊗ t − [ y ; j (cid:48) + nb ) s − [ i (cid:48) ; x + na ) c d (cid:48) − a − b ¯ c − + 1 ⊗ E i (cid:48) j (cid:48) z d (cid:48) in (cid:101) A − , op , coop . Then (6.30) is simply an application of (2.39). (cid:3) Proposition 6.10. We have an isomorphism of vector spaces A = A + ⊗ A ⊗ A − .Proof. We mush show that, for all a ∈ A − ⊗ A and b ∈ A ⊗ A + , we have:(6.33) ab = (cid:88) i x + i x i x − i for certain x + i ∈ A + , x i ∈ A , x − i ∈ A − with | vdeg x + i | ≤ | vdeg a | and | vdeg x − i | ≤| vdeg b | for all i . We will prove this fact by induction on d = min( | vdeg a | , | vdeg b | ).The base case d = 0 follows from (6.5)–(6.6) and (6.28)–(6.29), while the case | vdeg a | = | vdeg b | = 1 follows from (6.30). Without loss of generality, we mayuse Corollary 5.29 more precisely, its analogue for A − instead of A + ) to reduce tothe case a = a a where | vdeg a | , | vdeg a | ∈ { , ..., | vdeg a | − } . Therefore, theinduction hypothesis implies that: a b = (cid:88) i x + i x i x − i ⇒ ab = (cid:88) i a x + i x i x − i TALE OF TWO SHUFFLE ALGEBRAS 75 We may apply the induction hypothesis to write a x + i as in the RHS of (6.33),obtaining: ab = (cid:88) i,j y + j y j y − j x i x − i and finally, formulas (6.28)–(6.29) allow us to move y − j to the very right of theexpression, thus completing the induction step. (cid:3) Proof. of Theorem 1.5 (in the formulation of Subsection 3.29): Theorem 5.24 (andits analogue when A + is replaced by A − ) give rise to algebra isomorphisms:Υ ± : D ± → A ± Moreover, D ∼ = A because they have the same generators (the series S ± , T ± )satisfying the same relations. Therefore, we obtain an isomorphism of vector spaces: D = D + ⊗ D ⊗ D − Υ → A + ⊗ A ⊗ A − = A To show that Υ is an algebra isomorphism, one needs to show that:Υ( ab ) = Υ( a )Υ( b )for all a ∈ A − ⊗ A and b ∈ A ⊗ A + , and we will do so by induction on: | vdeg a | + | vdeg b | (the base case vdeg a = vdeg b = 0 follows from the fact that D ∼ = A ): • when vdeg a = 0, vdeg b = 1, compare (3.108)–(3.109) with (6.5)–(6.6)and (3.110)–(3.111) with (6.28)–(6.29) • when vdeg a = − 1, vdeg b = 0, compare (3.119)–(3.120) with (6.5)–(6.6)and (3.121)–(3.122) with (6.28)–(6.29) • when vdeg a = − 1, vdeg b = 1, compare (3.123) with (6.30)As for the induction step, it is proved akin to that of Proposition 6.10, by usingCorollary 5.29 to conclude that either a or b can be written as the product of twoelements whose | vdeg | is strictly smaller, and thus reducing to the cases treatedin the bullets above. (cid:3) References [1] Beck J., Braid group action and quantum affine algebras , Commun. Math. 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