Quantum loop groups and shuffle algebras via Lyndon words
QQUANTUM LOOP GROUPS AND SHUFFLEALGEBRAS VIA LYNDON WORDS
ANDREI NEGUT , AND ALEXANDER TSYMBALIUK
Abstract.
We study PBW bases of the untwisted quantum loop group U q ( L g ) (in the Drinfeld new presentation) using the combinatorics of loopwords, by generalizing the treatment of [26, 27, 40] in the finite type case.As an application, we prove that Enriquez’ homomorphism [10] from thepositive half of the quantum loop group to the trigonometric degeneration ofFeigin-Odesskii’s elliptic algebra [13] associated to g is an isomorphism. Introduction g be the Kac-Moody Lie algebra corresponding to a root system of finitetype. Associated with a decomposition of the set of roots ∆ = ∆ + (cid:116) ∆ − , thereexists a triangular decomposition:(1.1) g = n + ⊕ h ⊕ n − where:(1.2) n + = (cid:77) α ∈ ∆ + Q · e α and analogously for n − . The elements e α will be called root vectors. Formula (1.1)induces a triangular decomposition of the universal enveloping algebra:(1.3) U ( g ) = U ( n + ) ⊗ U ( h ) ⊗ U ( n − )Then the PBW theorem asserts that a linear basis of U ( n + ) is given by the products:(1.4) U ( n + ) = k ∈ N (cid:77) γ ≥···≥ γ k ∈ ∆ + Q · e γ . . . e γ k and analogously for U ( n − ), for any total order of the set of positive roots ∆ + . Theroot vectors (1.2) can be normalized so that we have:(1.5) [ e α , e β ] = e α e β − e β e α ∈ Z ∗ · e α + β whenever α, β and α + β are positive roots. Thus we see that formula (1.5) pro-vides an algorithm for constructing, up to scalar multiple, all the root vectors (1.2)inductively starting from e i = e α i , where { α i } i ∈ I ⊂ ∆ + are the simple roots of g .The upshot is that all the root vectors e α , and with them the PBW basis (1.4), canbe read off from the combinatorics of the root system. Given subalgebras { A k } Nk =1 of an algebra A , the decomposition A = A ⊗ · · · ⊗ A N will meanthat the multiplication in A induces a vector space isomorphism m : A ⊗ · · · ⊗ A N ∼ −→ A . a r X i v : . [ m a t h . R T ] F e b ANDREI NEGUT , AND ALEXANDER TSYMBALIUK U q ( g ) is a q -deformation of the universal enveloping al-gebra U ( g ), and we will focus on emulating the features of the previous Subsection.For one thing, there exists a triangular decomposition analogous to (1.3):(1.6) U q ( g ) = U q ( n + ) ⊗ U q ( h ) ⊗ U q ( n − )and there exists a PBW basis analogous to (1.4):(1.7) U q ( n + ) = k ∈ N (cid:77) γ ≥···≥ γ k ∈ ∆ + Q ( q ) · e γ . . . e γ k The q -deformed root vectors e α ∈ U q ( n + ) are defined via Lusztig’s braid groupaction, which requires one to choose a reduced decomposition of the longest elementin the Weyl group of type g . It is well-known ([35]) that this choice precisely ensuresthat the order ≥ on ∆ + is convex, in the sense of Definition 2.18. Moreover, the q -deformed root vectors satisfy the following q -analogue of relation (1.5), where α, β and α + β are any positive roots that satisfy α < α + β < β as well as theminimality property (5.9):(1.8) [ e α , e β ] q = e α e β − q ( α,β ) e β e α ∈ Z [ q, q − ] ∗ · e α + β where ( · , · ) denotes the scalar product corresponding to the root system of type g .As in the Lie algebra case, we conclude that the q -deformed root vectors can bedefined (up to scalar multiple) as iterated q -commutators of e i = e α i (with i ∈ I ),using the combinatorics of the root system and the chosen convex order on ∆ + .1.3. There is a well-known incarnation of U q ( n + ) due to Green [15], Rosso [39],and Schauenburg [41] in terms of quantum shuffles:(1.9) U q ( n + ) Φ (cid:44) −→ F = k ∈ N (cid:77) i ,...,i k ∈ I Q ( q ) · [ i . . . i k ]where the right-hand side is endowed with the quantum shuffle product (Defini-tion 4.10). As shown by Lalonde-Ram in [27], there is a one-to-one correspondencebetween positive roots and so-called standard Lyndon (or Shirshov) words in thealphabet I (these notions will be recalled in Subsections 2.3 - 2.11):(1.10) (cid:96) : ∆ + ∼ −→ (cid:110) standard Lyndon words (cid:111) The lexicographic order on standard Lyndon words, given by a fixed total order ofthe indexing set I of simple roots, induces a total order on the positive roots:(1.11) α < β ⇔ (cid:96) ( α ) < (cid:96) ( β ) lexicographicallyIt was shown in [26, 40] that this total order is convex, and hence can be applied toobtain root vectors e α ∈ U q ( n + ) for any positive root α , as in (1.8). Moreover, [26]shows that the root vector e α is uniquely characterized (up to a scalar multiple) bythe property that Φ( e α ) is an element of Im Φ whose leading order term [ i . . . i k ](in the lexicographic order) is precisely (cid:96) ( α ). UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 3 U q ( L g ) = U q ( L n + ) ⊗ U q ( L h ) ⊗ U q ( L n − )where U q ( L n + ) is a q -deformation of the universal enveloping algebra of n + [ t, t − ].The latter Lie algebra has the property that all its root spaces are one-dimensional,so we are able to adapt many of the results mentioned in the previous Subsection: Theorem 1.5.
There exists an injective algebra homomorphism: U q ( L n + ) Φ L (cid:44) −→ F L = k ∈ N (cid:77) i ,...,i k ∈ Id ,...,d k ∈ Z Q ( q ) · (cid:104) i ( d )1 . . . i ( d k ) k (cid:105) where the right-hand side is made into an algebra in Definition 4.26. Fix a totalorder of I , which induces the following total order on the set { i ( d ) } d ∈ Z i ∈ I : (1.12) i ( d ) < j ( e ) if d > e or d = e and i < j This induces the lexicographic order on the words [ i ( d )1 . . . i ( d k ) k ] with respect towhich we may define the notion of standard Lyndon loop words by analogy with [27] (see Subsections 2.21 - 2.26 for details). Then, there exists a 1-to-1 correspondence: (1.13) (cid:96) : ∆ + × Z ∼ −→ (cid:110) standard Lyndon loop words (cid:111) The lexicographic order on the right-hand side induces a convex order on the left-hand side, with respect to which one can define elements: (1.14) e (cid:96) ( α,d ) ∈ U q ( L n + ) for all ( α, d ) ∈ ∆ + × Z . We have the following analogue of the PBW theorem: (1.15) U q ( L n + ) = k ∈ N (cid:77) (cid:96) ≥···≥ (cid:96) k standard Lyndon loop words Q ( q ) · e (cid:96) . . . e (cid:96) k There are also analogues of the constructions above with + ↔ − and e ↔ f .Remark . We will show in Proposition 4.39 that elements of Φ L ( U q ( L n + )) haveleading order terms precisely given by the concatenations: (cid:96) . . . (cid:96) k as (cid:96) ≥ · · · ≥ (cid:96) k go over all standard Lyndon loop words. In Conjecture 4.42, wehypothesize (following [26] in finite type) that for any ( α, d ) ∈ ∆ + × Z , the element:Φ L (cid:0) e ( α,d ) (cid:1) ∈ F L ANDREI NEGUT , AND ALEXANDER TSYMBALIUK is the unique (up to scalar multiple) element of Im Φ L of its degree whose leadingorder term is (cid:96) ( α, d ). As a consequence of this conjecture, one would conclude thatthe PBW basis vector featured in the RHS of (1.15) has the property that:Φ L ( e (cid:96) . . . e (cid:96) k ) ∈ Q ( q ) ∗ · [ (cid:96) . . . (cid:96) k ] + smaller wordsAs we already mentioned, the total order on ∆ + × Z given by:(1.16) ( α, d ) < ( β, e ) ⇔ (cid:96) ( α, − d ) < (cid:96) ( β, − e ) lexicographicallyis convex; this fact will be proved in Proposition 2.33. As such, this order comesfrom a certain reduced word in the affine Weyl group associated to g (= the Coxetergroup associated to (cid:98) g ), in accordance with Theorem 3.14. Therefore, the rootvectors (1.14) are obtained by the classical construction of [2, 4, 30], once we passit through the “affine to loop” isomorphism of Theorem 5.19.We note that our notion of standard Lyndon loop words, as well as the order (1.16)on ∆ + × Z , are not the same as the similarly named notions of [19]. In general,our order between ( α, d ) and ( β, e ) is not determined by the order between α and β , as was the case in loc. cit. U q ( L n + ) Υ −→ A + ⊂ (cid:77) k =( k i ) i ∈ I ∈ N I Q ( q )( . . . , z i , . . . , z ik i , . . . ) Sym where the direct sum is made into an algebra using the multiplication (6.2) (werefer the reader to Definition 6.2 for the precise definition of the inclusion ⊂ abovein terms of pole and wheel conditions). In the present paper, we prove that: Theorem 1.8.
The map Υ is an isomorphism. In type A n , this result follows immediately from the type (cid:98) A n case proved in [34](see also [44] for the rational, super, and two-parameter generalizations), but themethods of loc. cit. are difficult to generalize to our current setup. Instead, we usethe framework of the preceding Subsection to prove Theorem 1.8. Explicitly, inSubsection 6.20, we construct an algebra homomorphism: A + ι (cid:44) −→ F L such that Φ L = ι ◦ ΥWe show that applying Υ to the ordered products of the root vectors (1.14) givesa basis of A + as a vector space, which will be shown to imply Theorem 1.8.The homomorphism ι can be construed as connecting the two (a priori) differentinstances of shuffle algebras that appear in the study of quantum loop groups. UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 5 • Theorems on convex PBW bases of affine quantum groups [2, 4, 25, 29] inspiredby the constructions of [24, 28, 38] for quantum groups of finite type. • Shuffle algebra incarnations of quantum groups [15, 39, 41], which we generalize tothe case of quantum loop groups, thus obtaining the algebra F L that features inTheorem 1.5. Our definition of this algebra is a reformulation of the constructionof [17], and in fact our presentation is to loc. cit. as Green’s presentation [15] isto Rosso’s presentation [39] of shuffle algebras in the finite type case. • Feigin-Odesskii shuffle algebras [13] and their trigonometric degenerations [10],which have recently had numerous applications to mathematical physics; we referthe interested reader to [12] for a recent survey.1.10. The structure of the present paper is the following: • In Section 2, we study the Lie algebras g and L g , recall the notion of standardLyndon words for the former, and extend this notion to the latter. • In Section 3, we show that the lexicographic order on ∆ + × Z induced by (1.13)corresponds to a certain reduced decomposition of a translation in the extendedaffine Weyl group of g . • In Section 4, we study the quantum groups U q ( g ) and U q ( L g ), and their PBWbases defined with respect to standard Lyndon words. We construct the objectsfeaturing in Theorem 1.5. • In Section 5, we wrap up the proof of Theorem 1.5 by tying it in with the well-known construction of PBW bases of affine quantum groups ([2, 4]). • In Section 6, we recall the trigonometric degeneration of the Feigin-Odesskiishuffle algebra, and prove Theorem 1.8 using the results of Theorem 1.5. • In the Appendix, we give explicit combinatorial data pertaining to standardLyndon loop words for all (untwisted) classical types, corresponding to an orderof the simple roots of our choice. For any other order of the simple roots, as wellas for the exceptional types, computer code performing these tasks in reasonabletime is available on demand from the authors.1.11. We would like to thank Pavel Etingof and Boris Feigin for their help andnumerous stimulating discussions over the years. A.N. would like to gratefully ac-knowledge NSF grants DMS-1760264 and DMS-1845034, as well as support from theAlfred P. Sloan Foundation and the MIT Research Support Committee. A.T. wouldlike to gratefully acknowledge NSF grant DMS-2037602. A close relative of the algebra F L also appeared (in a different context) in [42]. ANDREI NEGUT , AND ALEXANDER TSYMBALIUK Lie algebras and Lyndon words
It is a classical result that the free Lie algebra on a set of generators { e i } i ∈ I has abasis indexed by Lyndon words (see Definition 2.4) in the alphabet I . If we imposea certain collection of relations among the e i ’s, then [27] showed that a basis of theresulting Lie algebra is given by standard Lyndon words (see Definition 2.12), anddetermined the latter in the particular case of the maximal nilpotent subalgebra ofa simple Lie algebra. In the present Section, we will extend the treatment of loc. cit. to the situation of loops into simple Lie algebras.2.1. Let us consider a root system of finite type:∆ + (cid:116) ∆ − ⊂ Q (where Q denotes the root lattice) associated to the symmetric pairing:( · , · ) : Q ⊗ Q −→ Z Let { α i } i ∈ I ⊂ ∆ + denote a choice of simple roots. The Cartan matrix ( a ij ) i,j ∈ I and the symmetrized Cartan matrix ( d ij ) i,j ∈ I of this root system are:(2.1) a ij = 2( α i , α j )( α i , α i ) and d ij = ( α i , α j ) Definition 2.2.
To the root system above, one associates the Lie algebra: g = Q (cid:68) e i , f i , h i (cid:69) i ∈ I (cid:46) relations (2.2) , (2.3) , (2.4) where we impose the following relations for all i, j ∈ I : (2.2) [ e i , [ e i , . . . , [ e i , e j ] , . . . ]] (cid:124) (cid:123)(cid:122) (cid:125) − a ij Lie brackets = 0 , if i (cid:54) = j (2.3) [ h j , e i ] = d ji e i , [ h i , h j ] = 0 as well as the opposite relations with e ’s replaced by f ’s, and finally the relation: (2.4) [ e i , f j ] = δ ji h i We will consider the triangular decomposition (1.1), where n + , h , n − are the Liesubalgebras of g generated by the e i , h i , f i , respectively. We will write: Q ± ⊂ Q for the monoids generated by ± α i . The Lie algebra g is graded by Q , if we let:deg e i = α i , deg h i = 0 , deg f i = − α i The subalgebras n ± are graded by Q ± accordingly. UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 7 i . . . i k ]for various i , . . . , i k ∈ I . Let us fix a total order on the set I of simple roots, whichinduces the following total lexicographic order on the set of all words:[ i . . . i k ] < [ j . . . j l ] if i = j , . . . , i a = j a , i a +1 < j a +1 for some a ≥ i = j , . . . , i k = j k and k < l Definition 2.4.
A word (cid:96) = [ i . . . i k ] is called Lyndon (such words were alsostudied independently by Shirshov) if it is smaller than all of its cyclic permutations: [ i . . . i a − i a . . . i k ] < [ i a . . . i k i . . . i a − ] for all a ∈ { , . . . , k } . The following is an elementary exercise, that we leave to the interested reader.
Claim . If (cid:96) < (cid:96) are Lyndon, then (cid:96) (cid:96) is also Lyndon, and so (cid:96) (cid:96) < (cid:96) (cid:96) .Given a word w = [ i . . . i k ], the subwords: w a | = [ i . . . i a ] and w | a = [ i k − a +1 . . . i k ]with 0 ≤ a ≤ k will be called a prefix and a suffix of w , respectively. Such a prefix ora suffix is called proper if a / ∈ { , k } . It is straightforward to show that a word w isLyndon iff it is smaller than all of its proper suffixes, i.e. w < w | a for all 0 < a < k . Proposition 2.6. (see [27, § for a survey) Any Lyndon word (cid:96) has a factorization: (2.6) (cid:96) = (cid:96) (cid:96) defined by the property that (cid:96) is the longest proper suffix of (cid:96) which is also a Lyndonword. Under these circumstances, (cid:96) is also a Lyndon word. Proposition 2.7.
Any word w has a canonical factorization as a concatenation: (2.7) w = (cid:96) . . . (cid:96) k where (cid:96) ≥ · · · ≥ (cid:96) k are all Lyndon words. w = [ i . . . i k ], we define:(2.8) w e = e i . . . e i k ∈ U ( n + )On the other hand, Propositions 2.6 and 2.7 yield the following construction. Definition 2.9.
For any word w , define e w ∈ U ( n + ) inductively by e [ i ] = e i and: (2.9) e (cid:96) = [ e (cid:96) , e (cid:96) ] ∈ n + if (cid:96) is a Lyndon word with factorization (2.6) , and: (2.10) e w = e (cid:96) . . . e (cid:96) k ∈ U ( n + ) if w is an arbitrary word with canonical factorization (cid:96) . . . (cid:96) k , as in (2.7) . ANDREI NEGUT , AND ALEXANDER TSYMBALIUK
Remark . Because [ e α , e β ] ∈ Q ∗ · e α + β for all positive roots α, β such that α + β is also a root ([20, Proposition 8.4(d)]), then choosing a different factorization (2.6)for various Lyndon words will in practice produce bracketings (2.9) which are non-zero multiples of each other. Thus various choices will simply lead to PBW bases(1.4) which are renormalizations of each other.It is well-known that the elements (2.8) and (2.10) both give rise to bases of U ( n + ),and indeed are connected by the following triangularity property:(2.11) e w = (cid:88) v ≥ w c vw · v e for various integer coefficients c vw such that c ww = 1.2.11. If n + were a free Lie algebra, then it would have a basis given by the ele-ments (2.9), as (cid:96) goes over all Lyndon words (and similarly, U ( n + ) would have abasis given by the elements (2.10) as w goes over all words). But since we have tocontend with the relations (2.2) between the generators e i ∈ n + , we must restrictthe set of Lyndon words which appear. The following definition is due to [27]. Definition 2.12. (a) A word w is called standard if w e cannot be expressed as alinear combination of v e for various v > w , with w e as in (2.8) .(b) A Lyndon word (cid:96) is called standard Lyndon if e (cid:96) cannot be expressed as a linearcombination of e m for various Lyndon words m > (cid:96) , with e (cid:96) as in (2.9) . The following Proposition is non-trivial, and it justifies the above terminology.
Proposition 2.13. ( [27] ) A Lyndon word is standard iff it is standard Lyndon. According to [27, § n + has a basis consisting of the e (cid:96) ’s, as (cid:96) goes over allstandard Lyndon words. Since the Lie algebra n + is Q + -graded by deg e i = α i , itis natural to extend this grading to words as follows:(2.12) deg[ i . . . i k ] = α i + · · · + α i k Because of the decomposition (1.2) of n + , and the fact that the basis vectors e α ∈ n + all live in distinct degrees α ∈ Q + , we conclude that there exists a bijection:(2.13) (cid:96) : ∆ + ∼ −→ (cid:110) standard Lyndon words (cid:111) such that deg (cid:96) ( α ) = α , for all α ∈ ∆ + . The interested reader may find someexamples of the bijection (2.13) for the classical finite types in the Appendix.2.14. The following description of the bijection (2.13) was proved in [26, Proposi-tion 25], and allows one to inductively construct the bijection (cid:96) :(2.14) (cid:96) ( α ) = max γ + γ = α, γ k ∈ ∆ + (cid:96) ( γ ) <(cid:96) ( γ ) (cid:110) concatenation (cid:96) ( γ ) (cid:96) ( γ ) (cid:111) We also have the following simple property of standard words.
Proposition 2.15. ( [27, § ) Any subword of a standard word is standard. UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 9
Combining Propositions 2.7, 2.13, 2.15, we conclude that any standard word canbe uniquely written in the form (2.7), where (cid:96) ≥ · · · ≥ (cid:96) k are all standard Lyndonwords. The converse also holds (by a dimension count argument, see [27, § Proposition 2.16. ( [27] ) A word w is standard if and only if it can be written(uniquely) as w = (cid:96) . . . (cid:96) k , where (cid:96) ≥ · · · ≥ (cid:96) k are standard Lyndon words. Thus we obtain the following reformulation of (1.4):(2.15) U ( n + ) = k ∈ N (cid:77) (cid:96) ≥···≥ (cid:96) k standard Lyndon words Q · e (cid:96) . . . e (cid:96) k By the triangularity property (2.11), we could also get a basis of U ( n + ) by replacing e w = e (cid:96) . . . e (cid:96) k in (2.15) by w e , for any standard word w .2.17. The bijection (2.13) yields a total order on the set of positive roots ∆ + , in-duced by the lexicographic order of standard Lyndon words, see (1.11). As observedin [26, 40], this order is convex, in the following sense. Definition 2.18.
A total order on the set of positive roots ∆ + is called convex if: (2.16) α < α + β < β for all α < β ∈ ∆ + such that α + β is also a root. It is well-known ([35]) that convex orders of the positive roots are in 1-to-1 corre-spondence with reduced decompositions of the longest element of the Weyl groupassociated to our root system. We will consider this issue, and its affine version, inmore detail in Section 3.
Proposition 2.19. ( [26, Proposition 26] ) The order (1.11) on ∆ + is convex. We will prove the loop version of the Proposition above in Proposition 2.33.2.20. We will now extend the description above to the Lie algebra of loops into g : L g = g [ t, t − ] = g ⊗ Q Q [ t, t − ]where the Lie bracket is simply given by:(2.17) [ x ⊗ t m , y ⊗ t n ] = [ x, y ] ⊗ t m + n for all x, y ∈ g and m, n ∈ Z . The triangular decomposition (1.1) extends to asimilar decomposition at the loop level, and our goal is to describe L n + along thelines of Subsections 2.11 - 2.14. To this end, we think of L n + as being generatedby: e ( d ) i = e i ⊗ t d ∀ i ∈ I, d ∈ Z . Associate to e ( d ) i the letter i ( d ) , and call d the exponent of i ( d ) . Wefix a total order on I , which induces the total order (1.12) on the letters { i ( d ) } d ∈ Z i ∈ I .Any word in these letters will be called a loop word:(2.18) (cid:104) i ( d )1 . . . i ( d k ) k (cid:105) , AND ALEXANDER TSYMBALIUK
We have the total lexicographic order on loop words (2.18) induced by (1.12). Allthe results of Subsection 2.3 continue to hold in the present setup, so we have anotion of Lyndon loop words. Since L n + is Q + × Z -graded by:deg e ( d ) i = ( α i , d )it makes sense to extend this grading to loop words as follows:(2.19) deg (cid:104) i ( d )1 . . . i ( d k ) k (cid:105) = ( α i + · · · + α i k , d + · · · + d k )The obvious generalization of (1.2) is:(2.20) L n + = (cid:77) α ∈ ∆ + (cid:77) d ∈ Z Q · e ( d ) α with e ( d ) α = e α ⊗ t d . If deg x = ( α, d ) ∈ Q + × Z , then we will use the notation:hdeg x = α and vdeg x = d and call these two notions the horizontal and the vertical degree, respectively. Whileobviously infinite-dimensional, L n + still has one-dimensional Q + × Z -graded pieces,which is essential for the treatment of [27] to carry through.2.21. We now wish to extend Definition 2.12 in order to obtain a notion of standard(Lyndon) loop words, but here we must be careful, because the alphabet { i ( d ) } d ∈ Z i ∈ I is infinite. To deal with this issue, we consider the increasing filtration: L n + = ∞ (cid:91) s =0 L ( s ) n + defined with respect to the finite-dimensional Lie subalgebras:(2.21) L n + ⊃ L ( s ) n + = (cid:77) α ∈ ∆ + s | α | (cid:77) d = − s | α | Q · e ( d ) α where | α | denotes the height of a root, i.e. | α | = (cid:88) i ∈ I k i if α = (cid:80) i ∈ I k i α i .As a Lie algebra, L ( s ) n + is generated by { e ( d ) i | i ∈ I, − s ≤ d ≤ s } . Therefore,we may apply Definition 2.12 to yield a notion of standard (Lyndon) loop wordswith respect to the finite-dimensional Lie algebras L ( s ) n + , where the correspondingwords will only be made up of the symbols i ( d ) with i ∈ I, d ∈ {− s, . . . , s } . Proposition 2.22.
There exists a bijection: (2.22) (cid:96) : (cid:110) ( α, d ) ∈ ∆ + × Z , | d | ≤ s | α | (cid:111) ∼ −→ (cid:110) standard Lyndon words for L ( s ) n + (cid:111) explicitly determined by (cid:96) ( α i , d ) = (cid:2) i ( d ) (cid:3) and the following property: (2.23) (cid:96) ( α, d ) = max ( γ ,d )+( γ ,d )=( α,d ) γ k ∈ ∆ + , | d k |≤ s | γ k | (cid:96) ( γ ,d ) <(cid:96) ( γ ,d ) (cid:110) concatenation (cid:96) ( γ , d ) (cid:96) ( γ , d ) (cid:111) UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 11
In view of Proposition 2.16, this also gives a parametrization of standard words for L ( s ) n + . We note that both the property (2.23), as well as the main idea of thesubsequent proof, are direct adaptations of the analogous results in [26] (cf. (2.14)). Proof of Proposition 2.22.
Because the root spaces of L ( s ) n + are one-dimensional,as in (2.21), then for any Lyndon word (cid:96) of degree ( α, d ) ∈ Q + × Z with | d | ≤ s | α | ,we have:(2.24) e (cid:96) ∈ Q · e ( d ) α The right-hand side is 0 if α / ∈ ∆ + . By Definition 2.12(b), a word (cid:96) is standard Lyn-don if and only if it is the maximal Lyndon word of its given degree, with the prop-erty that e (cid:96) (cid:54) = 0. Together with the fact [27, § { e (cid:96) | (cid:96) − standard Lyndon } is a basis of L ( s ) n + , this establishes the existence of a bijection (2.22).Let us now prove that this bijection takes the form (2.23). Consider any γ , γ ∈ ∆ + such that γ + γ ∈ ∆ + , and any integers d , d such that | d k | ≤ s | γ k | for all k ∈ { , } . Let us write (cid:96) k = (cid:96) ( γ k , d k ) for all k ∈ { , } and (cid:96) = (cid:96) ( γ + γ , d + d );we may assume without loss of generality that (cid:96) < (cid:96) . We have:(2.25) e (cid:96) k = (cid:88) v ≥ (cid:96) k c v(cid:96) k · v e ∀ k ∈ { , } , due to property (2.11) (which holds in L ( s ) n + as it did in n + ). Thus:(2.26) e (cid:96) e (cid:96) = (cid:88) v ≥ (cid:96) (cid:96) x v · v e for various coefficients x v . As a consequence of Claim 2.5, we have an analogue offormula (2.26) when the indices 1 and 2 are swapped in the left-hand side. Hencewe obtain the following formula for the commutator:(2.27) [ e (cid:96) , e (cid:96) ] = (cid:88) v ≥ (cid:96) (cid:96) y v · v e for various coefficients y v . Furthermore, we may restrict the sum above to standard v ’s, since by the very definition of this notion, any v e can be inductively written asa linear combination of u e ’s for standard u ≥ v (this uses the fact that there existfinitely many words of any given degree, as we use a finite alphabet { i ( d ) } − s ≤ d ≤ si ∈ I ).By this very same reason, we may restrict the right-hand side of (2.11) to standard v ’s, and conclude that { e w | w − standard } yield a basis which is upper triangular interms of the basis { w e | w − standard } . With this in mind, (2.27) implies:(2.28) [ e (cid:96) , e (cid:96) ] = (cid:88) v ≥ (cid:96) (cid:96) v − standard z v · e v for various coefficients z v .However, [ e γ , e γ ] ∈ Q ∗ · e γ + γ implies [ e ( d ) γ , e ( d ) γ ] ∈ Q ∗ · e ( d + d ) γ + γ , so that:(2.29) [ e (cid:96) , e (cid:96) ] ∈ Q ∗ · e (cid:96) Here we are using the fact that if v ≥ (cid:96) and v ≥ (cid:96) , then v v ≥ (cid:96) (cid:96) ; this fact is not truefor arbitrary words v and v , because we could have v = (cid:96) u for some word u < (cid:96) . However,such counterexamples are not allowed because the words v k which appear in (2.25) have the samenumber of letters as (cid:96) k , for degree reasons. , AND ALEXANDER TSYMBALIUK As { e v | v − standard } is a basis of U ( L ( s ) n + ) ([27, § (cid:96) ≥ (cid:96) (cid:96) . This proves the inequality ≥ in (2.23). As for theopposite inequality ≤ , it follows from the fact that (cid:96) ( α, d ) admits a factorization(2.6) (cid:96) ( α, d ) = (cid:96) (cid:96) (with (cid:96) < (cid:96) ( α, d ) < (cid:96) ), and Propositions 2.13, 2.15 imply that (cid:96) k = (cid:96) ( γ k , d k ) for some decomposition ( α, d ) = ( γ , d ) + ( γ , d ). (cid:3) Since standard Lyndon loop words give rise to bases of the finite-dimensional Liealgebra L ( s ) n + , then the analogue of property (2.15) gives us:(2.30) U ( L ( s ) n + ) = k ∈ N (cid:77) (cid:96) ≥···≥ (cid:96) k standard Lyndon loopwords with all exponents in {− s,...,s } Q · e (cid:96) . . . e (cid:96) k By the triangularity property (2.11), we could also get a basis of U ( L ( s ) n + ) byreplacing e w = e (cid:96) . . . e (cid:96) k in (2.30) by w e , for any standard loop word w with allexponents in {− s, . . . , s } .2.23. Property (2.23) will allow us to deduce some facts about the bijection (2.22). Proposition 2.24.
For any positive root α ∈ ∆ + and integer d ∈ Z , we have: (2.31) (cid:96) ( α, d ) < (cid:96) ( α, d − where (cid:96) is the function of (2.22) , which a priori depends on a natural number s (sowe implicitly need d − , d ∈ {− s | α | , . . . , s | α |} in order for (2.31) to make sense).Proof. Let us prove (2.31) by induction on | α | , the base case | α | = 1 being trivial.According to (2.23), there exist decompositions α = γ + γ , d = d + d such that: (cid:96) ( α, d ) = (cid:96) ( γ , d ) (cid:96) ( γ , d )with (cid:96) ( γ , d ) < (cid:96) ( γ , d ). Note that γ (cid:54) = γ as γ + γ is a root. Because weassume d > − s | α | , then at least one of the following two options holds: • d > − s | γ | , in which case the induction hypothesis implies (cid:96) ( γ , d − >(cid:96) ( γ , d ). Then we either have (cid:96) ( γ , d − < (cid:96) ( γ , d ), in which case: (cid:96) ( α, d − ≥ (cid:96) ( γ , d − (cid:96) ( γ , d ) > (cid:96) ( γ , d ) (cid:96) ( γ , d ) = (cid:96) ( α, d )or (cid:96) ( γ , d − > (cid:96) ( γ , d ), in which case: (cid:96) ( α, d − ≥ (cid:96) ( γ , d ) (cid:96) ( γ , d − > (cid:96) ( γ , d ) (cid:96) ( γ , d ) > (cid:96) ( γ , d ) (cid:96) ( γ , d ) = (cid:96) ( α, d ) • d > − s | γ | , in which case the induction hypothesis implies (cid:96) ( γ , d − >(cid:96) ( γ , d ). Then we either have (cid:96) ( γ , d − > (cid:96) ( γ , d ), in which case: (cid:96) ( α, d − ≥ (cid:96) ( γ , d ) (cid:96) ( γ , d − > (cid:96) ( γ , d ) (cid:96) ( γ , d ) = (cid:96) ( α, d )or (cid:96) ( γ , d − < (cid:96) ( γ , d ), in which case: (cid:96) ( α, d − ≥ (cid:96) ( γ , d − (cid:96) ( γ , d ) > (cid:96) ( γ , d ) (cid:96) ( γ , d ) > (cid:96) ( γ , d ) (cid:96) ( γ , d ) = (cid:96) ( α, d )In all chains of two or three inequalities above, the first inequality is due to (2.23),while the third inequality uses Claim 2.5. (cid:3) Next, we estimate the exponents of letters in the standard Lyndon words for L ( s ) n + . UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 13
Proposition 2.25.
For all α ∈ ∆ + and d ∈ {− sk, . . . , sk } with k = | α | , we have: (2.32) (cid:96) ( α, d ) = (cid:104) i ( d )1 . . . i ( d k ) k (cid:105) for various d , . . . , d k ∈ (cid:26)(cid:22) dk (cid:23) , (cid:24) dk (cid:25)(cid:27) Proof.
We will prove (2.32) by induction on k , the base case k = 1 being trivial.If dk = t ∈ Z , then we must show that all exponents of (cid:96) ( α, d ) are equal to t . Indeed,pick a decomposition α = γ + γ into positive roots, and assume without loss ofgenerality that (cid:96) ( γ , t | γ | ) < (cid:96) ( γ , t | γ | ) (otherwise, swap their order). Then: (cid:96) ( α, d ) ≥ (cid:96) ( γ , t | γ | ) (cid:96) ( γ , t | γ | )by (2.23). By the induction hypothesis, the word on the right has all exponentsequal to t , which implies that the first letter of (cid:96) ( α, d ) has exponent ≤ t . Butbecause the first letter of a Lyndon word is its smallest one, this implies that allletters of (cid:96) ( α, d ) have exponent ≤ t . Because vdeg (cid:96) ( α, d ) = d = tk is also thesum of the exponents of (cid:96) ( α, d ), this implies that all letters of (cid:96) ( α, d ) must haveexponent equal to t , as we needed to prove.If tk < d < ( t + 1) k for some t ∈ Z , then we must show that all exponents of (cid:96) ( α, d ) are equal to either t or t + 1. By a slight modification of the argument inthe preceding paragraph, we conclude that the first letter of (cid:96) ( α, d ) has exponent= t + 1, which implies that all letters of (cid:96) ( α, d ) have exponent ≤ t + 1. Then assumefor the purpose of contradiction that there is some letter of (cid:96) ( α, d ) with exponent ≤ t −
1. Consider the factorization (2.6):(2.33) (cid:96) ( α, d ) = (cid:96) ( γ , d ) (cid:96) ( γ , d )for some decomposition α = γ + γ , d = d + d with | d k | ≤ s | γ k | for k ∈ { , } .Since the first letter of (cid:96) ( γ , d ) has exponent t + 1, the induction hypothesis doesnot allow (cid:96) ( γ , d ) to have any letters with exponents ≤ t −
1. Therefore, the letterswith exponents ≤ t − (cid:96) ( γ , d ), and so the induction hypothesis yields: d > t | γ | and d < t | γ | However, if (cid:96) ( γ , d − < (cid:96) ( γ , d +1) then the word (cid:96) ( γ , d − (cid:96) ( γ , d +1) wouldbe greater than (cid:96) ( γ , d ) (cid:96) ( γ , d ) = (cid:96) ( α, d ), by Proposition 2.24, thus contradictingthe maximality of (cid:96) ( α, d ) provided by (2.23). The only other possibility is that (cid:96) ( γ , d − > (cid:96) ( γ , d + 1), at which point the same property (2.23) implies that: (cid:96) ( α, d ) ≥ (cid:96) ( γ , d + 1) (cid:96) ( γ , d − (cid:96) ( γ , d + 1) have exponents ≤ t , which contradicts the fact that the first letter of (cid:96) ( α, d ) has exponent t + 1. (cid:3) s . Proposition 2.27.
Any loop word w with exponents in {− s, . . . , s } is standard(Lyndon) with respect to L ( s ) n + iff it is standard (Lyndon) with respect to L ( s +1) n + . , AND ALEXANDER TSYMBALIUK
Proof.
Due to Proposition 2.16, it suffices to consider the case of standard Lyndonloop words. In other words, we must show that if α is a positive root and d is aninteger such that | d | ≤ s | α | , then the Lyndon words: (cid:96) = (cid:96) ( α, d ) of (2.22) with respect to L ( s ) n + (cid:96) (cid:48) = (cid:96) ( α, d ) of (2.22) with respect to L ( s +1) n + are equal. We may do so by induction on | α | , the base case | α | = 1 being trivial.Due to property (2.23), both (cid:96) and (cid:96) (cid:48) are defined as the maximum over variousconcatenations, but the set of concatenations defining (cid:96) (cid:48) is a priori larger. In otherwords, the only situation in which (cid:96) (cid:54) = (cid:96) (cid:48) would be if: (cid:96) (cid:48) = (cid:96) ( γ , d ) (cid:96) ( γ , d ) > (cid:96) with (cid:96) ( γ , d ) or (cid:96) ( γ , d ) having an exponent ± ( s + 1). However, this can nothappen due to (2.32) applied to (cid:96) (cid:48) , since it would force | d | > s | α | . (cid:3) Proposition 2.27 implies that the notion “standard Lyndon loop word” does notdepend on the particular L ( s ) n + with respect to which it is defined. We concludethat there exists a bijection:(2.34) (cid:96) : ∆ + × Z ∼ −→ (cid:110) standard Lyndon loop words (cid:111) satisfying properties (2.23) and (2.32) (with s = ∞ ).2.28. Because of the Lie algebra isomorphism: L n + ∼ −→ L n + given by e ( d ) α (cid:55)→ e ( d + | α | ) α the procedure:(2.35) (cid:104) i ( d )1 . . . i ( d k ) k (cid:105) (cid:32) (cid:104) i ( d +1)1 . . . i ( d k +1) k (cid:105) preserves the property of a loop word being standard. It obviously also preservesthe property of a loop word being Lyndon, hence also of being standard Lyndon,due to Proposition 2.13. This implies the following result. Proposition 2.29.
For any ( α, d ) ∈ ∆ + × Z , (cid:96) ( α, d + | α | ) is obtained from (cid:96) ( α, d ) by adding 1 to all the exponents of its letters, i.e. by the procedure (2.35) . Therefore, to describe the bijection (2.34), it suffices to specify a finite amount ofdata, i.e. the standard Lyndon loop words corresponding to ( α, d ) for all α ∈ ∆ + and d ∈ { , . . . , | α |} . This will be done in the Appendix for all classical types,corresponding to a specific order of the simple roots. Proposition 2.30.
The restriction of (2.34) to ∆ + × { } matches (2.13) . The result above is simply the s = 0 case of Proposition 2.27. Since U ( L n + ) is thedirect limit as s → ∞ of the U ( L ( s ) n + ), then (2.30) implies:(2.36) U ( L n + ) = k ∈ N (cid:77) (cid:96) ≥···≥ (cid:96) k standard Lyndon loop words Q · e (cid:96) . . . e (cid:96) k UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 15
By Proposition 2.16, we then have:(2.37) U ( L n + ) = (cid:77) w standard loop words Q · e w The following result will be used in Section 4.
Corollary 2.31.
For any loop word w , there exist finitely many standard loopwords ≤ w in any fixed degree ( α, d ) ∈ Q + × Z .Proof. Any standard loop word v admits a canonical factorization v = (cid:96) . . . (cid:96) k where (cid:96) ≥ · · · ≥ (cid:96) k are all standard Lyndon loop words. If v ≤ w , then we notethat all the (cid:96) r ’s are bounded from above by w , due to (cid:96) r ≤ (cid:96) ≤ v . Combining thiswith (2.32), we see that the exponents which appear among the letters of the (cid:96) r ’s arebounded from below. Therefore, there are only finitely many choices of (cid:96) , . . . , (cid:96) k with a fixed number of letters, whose exponents sum up to precisely d . (cid:3) + × Z , by trans-porting the total lexicographic order on loop words. We will now show that thisorder is convex, a notion which is the direct generalization of Definition 2.18. Proposition 2.33.
For all ( α, d ) , ( β, e ) , ( α + β, d + e ) ∈ ∆ + × Z , we have: (2.38) (cid:96) ( α, d ) < (cid:96) ( α + β, d + e ) < (cid:96) ( β, e ) if (cid:96) ( α, d ) < (cid:96) ( β, e ) .Proof. We will prove the required statement by induction on | α + β | , the base casebeing vacuous. By (2.23), we have: (cid:96) ( α + β, d + e ) ≥ (cid:96) ( α, d ) (cid:96) ( β, e ) > (cid:96) ( α, d )Therefore, it remains to show that (cid:96) ( α + β, d + e ) < (cid:96) ( β, e ). Let us assume for thepurpose of contradiction that the opposite inequality holds:(2.39) (cid:96) ( α + β, d + e ) > (cid:96) ( β, e ) > (cid:96) ( α, d )By (2.23), we have:(2.40) (cid:96) ( α + β, d + e ) = (cid:96) ( α (cid:48) , d (cid:48) ) (cid:96) ( β (cid:48) , e (cid:48) )where (cid:96) ( α (cid:48) , d (cid:48) ) < (cid:96) ( β (cid:48) , e (cid:48) ), for certain positive roots α (cid:48) , β (cid:48) satisfying α + β = α (cid:48) + β (cid:48) and integers d (cid:48) , e (cid:48) satisfying d + e = d (cid:48) + e (cid:48) . Comparing the formulas above, wehave two options: Case 1 : (cid:96) ( α (cid:48) , d (cid:48) ) > (cid:96) ( β, e )Case 2 : (cid:96) ( α (cid:48) , d (cid:48) ) < (cid:96) ( β, e )(note that the equality ( α (cid:48) , d (cid:48) ) = ( β, e ) would imply ( α, d ) = ( β (cid:48) , e (cid:48) ), which wouldcontradict the various inequalities above). In Case 1, we would have:(2.41) (cid:96) ( β (cid:48) , e (cid:48) ) > (cid:96) ( α (cid:48) , d (cid:48) ) > (cid:96) ( β, e ) > (cid:96) ( α, d )We will use (2.41) to obtain a contradiction, but first we make an elementary claim: , AND ALEXANDER TSYMBALIUK
Claim . Given positive roots α, β, α (cid:48) , β (cid:48) such that α + β = α (cid:48) + β (cid:48) , then: α (cid:48) = α + γ and β (cid:48) = β − γ or: α (cid:48) = β + γ and β (cid:48) = α − γ for some γ ∈ ∆ ± (cid:116) { } .The Claim is proved as follows. Suppose first that ( α, α (cid:48) ) >
0. Then, the reflection s α (cid:48) ( α ) = α − kα (cid:48) is also a root, for some positive integer k >
0. This impliesthat α − α (cid:48) is either a root or 0, hence α − α (cid:48) = γ for some γ ∈ ∆ ± (cid:116) { } , thusproving the claim. The analogous argument applies if ( α, β (cid:48) ) >
0, ( β, α (cid:48) ) >
0, or( β, β (cid:48) ) >
0. However, one of the aforementioned 4 inequalities must hold, or else0 ≥ ( α + β, α (cid:48) + β (cid:48) ) = ( α + β, α + β ), a contradiction.Using Claim 2.34, we conclude that there exist γ ∈ ∆ ± (cid:116) { } and x ∈ Z such that:(2.42) ( α (cid:48) , d (cid:48) ) = ( α + γ, d + x ) and ( β (cid:48) , e (cid:48) ) = ( β − γ, e − x )or:(2.43) ( α (cid:48) , d (cid:48) ) = ( β + γ, e + x ) and ( β (cid:48) , e (cid:48) ) = ( α − γ, d − x )(one just needs to pick the integer x such that the equalities above hold). First of all,we cannot have γ = 0, as Proposition 2.24 and the chain of inequalities (2.41) wouldsimultaneously require x > x <
0. If γ (cid:54) = 0, then the induction hypothesis of(2.38) contradicts the chain of inequalities in (2.41), as per the following: • If (2.42) holds and γ ∈ ∆ + , the contradiction arises from the fact that (cid:96) ( γ, x )would have to be simultaneously bigger than (cid:96) ( α (cid:48) , d (cid:48) ) and smaller than (cid:96) ( β, e ). • If (2.42) holds and γ ∈ ∆ − , the contradiction arises from the fact that (cid:96) ( − γ, − x )would have to be simultaneously bigger than (cid:96) ( β (cid:48) , e (cid:48) ) and smaller than (cid:96) ( α, d ). • If (2.43) holds and γ ∈ ∆ + , the contradiction arises from the fact that (cid:96) ( γ, x )would have to be simultaneously bigger than (cid:96) ( α (cid:48) , d (cid:48) ) and smaller than (cid:96) ( α, d ). • If (2.43) holds and γ ∈ ∆ − , the contradiction arises from the fact that (cid:96) ( − γ, − x )would have to be simultaneously bigger than (cid:96) ( β (cid:48) , e (cid:48) ) and smaller than (cid:96) ( β, e ).In Case 2, the only situation when (2.39) and (2.40) are compatible would be if:(2.44) (cid:96) ( β, e ) = (cid:96) ( α (cid:48) , d (cid:48) ) w for some loop word w , which would need to satisfy: (cid:96) ( β (cid:48) , e (cid:48) ) > w > (cid:96) ( β, e )(the first inequality is a consequence of (2.39) and (2.40), while the second inequalityis a consequence of the fact that (cid:96) ( β, e ) is Lyndon). However, being a suffix of astandard word, w is also standard and hence admits a canonical factorization: w = (cid:96) ( γ , f ) . . . (cid:96) ( γ k , f k )for various ( γ r , f r ) ∈ ∆ + × Z which satisfy (cid:96) ( γ r , f r ) ≤ (cid:96) ( γ , f ) ≤ w < (cid:96) ( β (cid:48) , e (cid:48) ) forall 1 ≤ r ≤ k . However, (2.44) implies:( β, e ) = ( α (cid:48) , d (cid:48) ) + k (cid:88) r =1 ( γ r , f r ) ⇒ ( β (cid:48) , e (cid:48) ) = ( α, d ) + k (cid:88) r =1 ( γ r , f r ) UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 17
Because α, γ , . . . , γ k , β (cid:48) are all positive roots, we claim that there exist positiveroots (cid:15) , . . . , (cid:15) k and a permutation σ ∈ S ( k ) such that:(2.45) (cid:15) r = α + γ σ (1) + · · · + γ σ ( r ) ∀ r ∈ { , . . . , k } Since (cid:96) ( α, d ) and all the (cid:96) ( γ r , f r ) are < (cid:96) ( β (cid:48) , e (cid:48) ), then the induction hypothesis of(2.38) implies (inductively in r ) that: (cid:96) ( (cid:15) r , d + f σ (1) + · · · + f σ ( r ) ) < (cid:96) ( β (cid:48) , e (cid:48) )However, ( (cid:15) k , d + f + · · · + f k ) = ( β (cid:48) , e (cid:48) ), which provides the required contradiction.It remains to prove (2.45), which we will do by induction on k , the base case k = 1being trivial. If ( α, γ r ) < r , then the reflection s α ( γ r ) = γ r + sα is also a root, for some positive integer s >
0. This implies that α + γ r is aroot, hence we can apply the induction hypothesis for the collection of positiveroots ( α + γ r , γ , . . . , γ r − , γ r +1 , . . . , γ k , β (cid:48) ). The analogous argument applies if( β (cid:48) , γ r ) > r , in which case we can apply the induction hypothesis for thecollection of positive roots ( α, γ , . . . , γ r − , γ r +1 , . . . , γ k , β (cid:48) − γ r ). Hence the onlysituation when we could not prove the claim via the argument above would be if:( α, γ r ) ≥ ≥ ( β (cid:48) , γ r ) ∀ r ⇒ ( α, β (cid:48) − α ) ≥ ≥ ( β (cid:48) , β (cid:48) − α )But this would imply ( β (cid:48) − α, β (cid:48) − α ) ≤
0, which is impossible since β (cid:48) − α (cid:54) = 0. (cid:3) Remark . We note that such “lexicographic order on Lyndon words are convex”results are well-known in representation theory, see e.g. [1] for slightly different (butmore systematic and general) setting from ours.
Corollary 2.36.
Consider any k, k (cid:48) ≥ and any: ( γ , d ) , . . . , ( γ k , d k ) , ( γ (cid:48) , d (cid:48) ) , . . . , ( γ (cid:48) k (cid:48) , d (cid:48) k (cid:48) ) ∈ ∆ + × Z such that: (2.46) ( γ , d ) + · · · + ( γ k , d k ) = ( γ (cid:48) , d (cid:48) ) + · · · + ( γ (cid:48) k (cid:48) , d (cid:48) k (cid:48) ) Then we have: (2.47) min (cid:110) (cid:96) ( γ , d ) , . . . , (cid:96) ( γ k , d k ) (cid:111) ≤ max (cid:110) (cid:96) ( γ (cid:48) , d (cid:48) ) , . . . , (cid:96) ( γ (cid:48) k (cid:48) , d (cid:48) k (cid:48) ) (cid:111) Proof.
Proposition 2.33 is simply the ( k, k (cid:48) ) ∈ { (1 , , (2 , } case of the Corollary.Let us prove the Corollary by induction on min( k, k (cid:48) ), and to break ties, by k + k (cid:48) .This means that we must start with the case min( k, k (cid:48) ) = 1, and we will show howto deal with the k (cid:48) = 1 case (as the k = 1 case is an analogous exercise that weleave to the interested reader). The assumption implies that γ + · · · + γ k ∈ ∆ + , inwhich case (2.45) shows that we can relabel indices such that γ + γ ∈ ∆ + . Thenthe induction hypothesis shows that:min (cid:110) (cid:96) ( γ + γ , d + d ) , (cid:96) ( γ , d ) , . . . , (cid:96) ( γ k , d k ) (cid:111) ≤ (cid:96) ( γ + · · · + γ k , d + · · · + d k )Then Proposition 2.33 for ( γ , d ) and ( γ , d ) implies that the left-hand side is ≥ the minimum of all the (cid:96) ( γ s , d s )’s, as we needed to prove.Let us now assume that k, k (cid:48) >
1. Since: γ + · · · + γ k = γ (cid:48) + · · · + γ (cid:48) k (cid:48) , AND ALEXANDER TSYMBALIUK there exist s, s (cid:48) such that ( γ s , γ (cid:48) s (cid:48) ) >
0. Let us relabel indices such that s = s (cid:48) = 1.As we saw in the proof of Claim 2.34, this implies that:( γ (cid:48) , d (cid:48) ) = ( γ , d ) + ( (cid:15), x )for some (cid:15) ∈ ∆ ± (cid:116) { } and some x ∈ Z . Then (2.46) implies:( γ , d ) + · · · + ( γ k , d k ) = ( γ (cid:48) , d (cid:48) ) + · · · + ( γ (cid:48) k (cid:48) , d (cid:48) k (cid:48) ) + ( (cid:15), x )If (cid:15) ∈ ∆ + , then the induction hypothesis gives us:min (cid:110) (cid:96) ( γ , d ) , (cid:96) ( (cid:15), x ) (cid:111) ≤ (cid:96) ( γ (cid:48) , d (cid:48) )min (cid:110) (cid:96) ( γ , d ) , . . . , (cid:96) ( γ k , d k ) (cid:111) ≤ max (cid:110) (cid:96) ( (cid:15), x ) , (cid:96) ( γ (cid:48) , d (cid:48) ) , . . . , (cid:96) ( γ (cid:48) k (cid:48) , d (cid:48) k (cid:48) ) (cid:111) which implies (2.47). If (cid:15) ∈ ∆ − , then the induction hypothesis gives us: (cid:96) ( γ , d ) ≤ max (cid:110) (cid:96) ( − (cid:15), − x ) , (cid:96) ( γ (cid:48) , d (cid:48) ) (cid:111) min (cid:110) (cid:96) ( − (cid:15), − x ) , (cid:96) ( γ , d ) , . . . , (cid:96) ( γ k , d k ) (cid:111) ≤ max (cid:110) (cid:96) ( γ (cid:48) , d (cid:48) ) , . . . , (cid:96) ( γ (cid:48) k (cid:48) , d (cid:48) k (cid:48) ) (cid:111) which also implies (2.47). Finally, if (cid:15) = 0 and x ≤
0, then Proposition 2.24 impliesthat (cid:96) ( γ , d ) ≤ (cid:96) ( γ (cid:48) , d (cid:48) ), which easily yields (2.47). If (cid:15) = 0 and x >
0, then:min (cid:110) (cid:96) ( γ , d ) , . . . , (cid:96) ( γ k , d k ) (cid:111) ≤ min (cid:110) (cid:96) ( γ , d − x ) , (cid:96) ( γ , d ) , . . . , (cid:96) ( γ k , d k ) (cid:111) ≤≤ max (cid:110) (cid:96) ( γ (cid:48) , d (cid:48) ) , . . . , (cid:96) ( γ (cid:48) k (cid:48) , d (cid:48) k (cid:48) ) (cid:111) where the first inequality is due to (2.31) and the second inequality holds becauseof the induction hypothesis. The chain of inequalities above implies (2.47). (cid:3) Proposition 2.37. If (cid:96) < (cid:96) are standard Lyndon loop words such that (cid:96) (cid:96) isalso a standard Lyndon loop word, then we cannot have: (cid:96) < (cid:96) (cid:48) < (cid:96) (cid:48) < (cid:96) for standard Lyndon loop words (cid:96) (cid:48) , (cid:96) (cid:48) such that deg (cid:96) + deg (cid:96) = deg (cid:96) (cid:48) + deg (cid:96) (cid:48) .Proof. Assume such (cid:96) (cid:48) , (cid:96) (cid:48) existed. Then by (2.23), we would have:(2.48) (cid:96) (cid:48) (cid:96) (cid:48) ≤ (cid:96) (cid:96) The only way this is compatible with (cid:96) < (cid:96) (cid:48) is if: (cid:96) (cid:48) = (cid:96) w for some loop word w , which must be standard due to Proposition 2.15 (or moreprecisely, its straightforward loop generalization). However, (2.48) then implies:(2.49) w(cid:96) (cid:48) ≤ (cid:96) If we consider the canonical factorization (2.7) of w = u . . . u k for standard Lyndonloop words u ≥ · · · ≥ u k , then (2.49) implies that: u k ≤ · · · ≤ u < (cid:96) Together with the assumption that (cid:96) (cid:48) < (cid:96) , this violates Corollary 2.36 since:deg u + · · · + deg u k + deg (cid:96) (cid:48) = deg w + deg (cid:96) (cid:48) = deg (cid:96) (cid:48) − deg (cid:96) + deg (cid:96) (cid:48) = deg (cid:96) (cid:3) UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 19 Lyndon words and Weyl groups
In the present Section, we will show that the lexicographic order (1.16) on ∆ + × Z induced by (2.34) is a particular case of the construction of [36, 37] applied to areduced decomposition of a certain translation in the extended affine Weyl groupassociated to g . The reader who is interested in quantum groups, and prepared toaccept the proof of Theorem 3.14, may skip ahead to Section 4.3.1. Let us consider the affine root system of type g : (cid:98) ∆ = (cid:98) ∆ + (cid:116) (cid:98) ∆ − ⊂ (cid:98) Q The affine root system has one more simple root α besides the simple roots { α i } i ∈ I of the finite root system. Therefore, we may use formulas (2.1) for I replaced by: (cid:98) I = I (cid:116) a ij ) i,j ∈ (cid:98) I and the affine symmetrized Cartanmatrix ( d ij ) i,j ∈ (cid:98) I . There is a natural identification:(3.1) (cid:98) Q ∼ −→ Q × Z with α i (cid:55)→ ( α i , , α (cid:55)→ ( − θ, θ ∈ ∆ + is the highest root of the finite root system. Note that (0 , ∈ Q × Z is the minimal imaginary root of the affine root system. With this in mind, we havethe following explicit description of the affine root system in terms of finite roots: (cid:98) ∆ + = (cid:110) ∆ + × Z ≥ (cid:111) (cid:116) (cid:110) × Z > (cid:111) (cid:116) (cid:110) ∆ − × Z > (cid:111) (3.2) (cid:98) ∆ − = (cid:110) ∆ − × Z ≤ (cid:111) (cid:116) (cid:110) × Z < (cid:111) (cid:116) (cid:110) ∆ + × Z < (cid:111) (3.3)where Z ≥ , Z > , Z ≤ , Z < denote the obvious subsets of Z . Definition 3.2.
Let (cid:98) g be as in Definition 2.2, but using (cid:98) I instead of I . As opposed from the non-degenerate pairing on finite type root systems, the pairingon affine type root systems has a 1-dimensional kernel, which is spanned by theimaginary root. Explicitly, this implies the fact that:( α + θ, − ) = 0 ⇔ d j + (cid:88) i ∈ I θ i d ij = 0for all j ∈ I , where the positive integers { θ i } i ∈ I (called the “labels” of the corre-sponding extended Dynkin diagram) are defined via:(3.4) θ = (cid:88) i ∈ I θ i α i Using formula (2.3), this implies that the Cartan element:(3.5) c = h + (cid:88) i ∈ I θ i h i is central in (cid:98) g . Furthermore, we have the following relation between (cid:98) g and L g . , AND ALEXANDER TSYMBALIUK
Lemma 3.3.
There exists a Lie algebra isomorphism: (cid:98) g / ( c ) ∼ −→ L g determined by the formulas: e i (cid:55)→ e i ⊗ t e (cid:55)→ f θ ⊗ t f i (cid:55)→ f i ⊗ t f (cid:55)→ e θ ⊗ t − h i (cid:55)→ h i ⊗ t h (cid:55)→ − (cid:88) i ∈ I θ i h i ⊗ t for all i ∈ I , where e θ (resp. f θ ) is a root vector of degree θ (resp. − θ ). + are in 1-to-1 correspon-dence with reduced decompositions of the longest element of the finite Weyl group W associated to g . To define the latter explicitly, consider the coroot lattice:(3.6) Q ∨ = (cid:77) i ∈ I Z · α ∨ i where for any α ∈ ∆ + the corresponding coroot α ∨ is defined via:(3.7) α ∨ = 2 α ( α, α )The finite Weyl group W , i.e. the abstract Coxeter group associated to the Cartanmatrix ( a ij ) i,j ∈ I , acts on the coroot lattice Q ∨ as well as on the root lattice Q :(3.8) W (cid:121) Q ∨ and W (cid:121) Q via the following assignments:(3.9) s i ( µ ) = µ − ( α i , µ ) α ∨ i and s i ( λ ) = λ − ( λ, α ∨ i ) α i ∀ i ∈ I, µ ∈ Q ∨ , λ ∈ Q .3.5. We will also encounter the affine Weyl group, which is by definition thesemidirect product:(3.10) (cid:99) W = W (cid:110) Q ∨ defined with respect to the action (3.8). It is well-known that (cid:99) W is also the Coxetergroup associated to the Cartan matrix ( a ij ) i,j ∈ (cid:98) I . In other words, the affine Weylgroup is generated by the symbols { s i } i ∈ (cid:98) I defined by: s i = ( s i , , ∀ i ∈ Is = ( s θ , − θ ∨ )The affine analogue of the action W (cid:121) Q from the previous Subsection is:(3.11) (cid:99) W (cid:121) (cid:98) Q where the generators of the affine Weyl group act by the following formulas: s i ( λ, d ) = ( λ − ( λ, α ∨ i ) α i , d ) , ∀ i ∈ I (3.12) s ( λ, d ) = ( λ − ( λ, θ ∨ ) θ, d + ( λ, θ ∨ ))(3.13) UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 21 for all ( λ, d ) ∈ Q × Z (cid:39) (cid:98) Q , see (3.1). An important feature of the affine Weyl groupis that it contains a large commutative subalgebra:1 (cid:110) Q ∨ ⊂ (cid:99) W which acts on the affine root lattice (cid:98) Q (cid:39) Q × Z by translations:(3.14) (cid:98) µ ( λ, d ) = ( λ, d − ( λ, µ )) ∀ µ ∈ Q ∨ , λ ∈ Q, d ∈ Z . Here and henceforth, we write (cid:98) µ for the element 1 (cid:110) µ ∈ (cid:99) W .3.6. We will also need to consider the extended affine Weyl group, which is bydefinition the semidirect product:(3.15) (cid:99) W ext = W (cid:110) P ∨ Above, P ∨ is the coweight lattice:(3.16) P ∨ = (cid:77) i ∈ I Z · ω ∨ i where the fundamental coweights { ω ∨ i } i ∈ I are dual to the simple roots { α j } j ∈ I :(3.17) ( α j , ω ∨ i ) = δ ji In particular, Q ∨ is a finite index subgroup of P ∨ . It is well-known that:(3.18) (cid:99) W ext (cid:39) T (cid:110) (cid:99) W where the finite subgroup T of (cid:99) W ext is naturally identified with a subgroup ofautomorphisms of the Dynkin diagram of (cid:98) g . The semi-direct product (3.18) is suchthat: τ s i = s τ ( i ) τ, ∀ τ ∈ T , i ∈ (cid:98) I Finally, the action (3.11) extends to:(3.19) (cid:99) W ext (cid:121) (cid:98) Q via: τ ( α i ) = α τ ( i ) , ∀ τ ∈ T , i ∈ (cid:98) I We still have the following formula, akin to (3.14):(3.20) (cid:98) µ ( λ, d ) = ( λ, d − ( λ, µ )) ∀ µ ∈ P ∨ , λ ∈ Q, d ∈ Z , where (cid:98) µ denotes the element 1 (cid:110) µ ∈ (cid:99) W ext .3.7. Recall that the length of an element x ∈ (cid:99) W , denoted by l ( x ) ∈ N , is thesmallest number l ∈ N such that we can write:(3.21) x = s i − l . . . s i for various i − l , . . . , i ∈ (cid:98) I . Every factorization (3.21) with l = l ( x ) is called areduced decomposition of x . Given such a reduced decomposition, the terminalsubset (a priori, a multiset) of the affine root system is:(3.22) E x = (cid:110) s i s i − . . . s i k +1 ( α i k ) (cid:12)(cid:12)(cid:12) ≥ k > − l (cid:111) ⊂ (cid:98) ∆ , AND ALEXANDER TSYMBALIUK
It is well-known that E x is independent of the reduced decomposition of x , andconsists of the positive affine roots (all with multiplicity one) that are mapped tonegative ones under the action of x :(3.23) E x = (cid:110)(cid:101) λ ∈ (cid:98) ∆ + | x ( (cid:101) λ ) ∈ (cid:98) ∆ − (cid:111) In particular, we get the following description of the length of x :(3.24) l ( x ) = (cid:110)(cid:101) λ ∈ (cid:98) ∆ + | x ( (cid:101) λ ) ∈ (cid:98) ∆ − (cid:111) The aforementioned length function l : (cid:99) W → Z ≥ naturally extends to (cid:99) W ext via: l ( τ w ) = l ( w ) , ∀ τ ∈ T , w ∈ (cid:99) W Thus, the length l ( x ) of x ∈ (cid:99) W ext is the smallest number l such that we can write:(3.25) x = τ s i − l . . . s i for various i − l , . . . , i ∈ (cid:98) I and (uniquely determined) τ ∈ T . Given a reduceddecomposition of x ∈ (cid:99) W ext as in (3.25) with l = l ( x ), define E x via (3.22). We notethat E x is still described via (3.23) since τ acts by permuting negative affine roots.Therefore, E x is independent of the reduced decomposition of x and we still have:(3.26) l ( x ) = (cid:110)(cid:101) λ ∈ (cid:98) ∆ + | x ( (cid:101) λ ) ∈ (cid:98) ∆ − (cid:111) Remark . A restricted case of the discussion above is when (cid:99)
W , (cid:98) ∆ are replacedby W, ∆. In this case, applying (3.23) to the longest element w ∈ W yields E w = ∆ + . Furthermore, choosing a reduced decomposition w = s i − l . . . s i amounts to placing a total order on E w = ∆ + via:(3.27) α i < s i ( α i − ) < · · · < s i s i − . . . s i − l ( α i − l )According to [35], this total order of ∆ + is convex, and conversely, any convex orderof ∆ + arises in this way for a certain (unique) reduced decomposition of w . Wewill study the affine version of this picture in Subsection 3.10.Let us recall the element ρ ∈ Q defined by: ρ = 12 (cid:88) α ∈ ∆ + α The following result is standard ([23, Exercise 6.10]).
Proposition 3.9.
For any µ ∈ P ∨ such that ( α i , µ ) ∈ N for all i ∈ I : l ( (cid:98) µ ) = (2 ρ, µ ) Proof.
Applying formula (3.20) for the action of (cid:98) µ ∈ (cid:99) W ext on (cid:98) Q (cid:39) Q × Z , we seethat the only positive affine roots (cid:101) λ ∈ (cid:98) ∆ + that are mapped to negative ones are:(3.28) (cid:110) ( α, d ) | α ∈ ∆ + , ≤ d < ( α, µ ) (cid:111) Combining this with formula (3.26), we find l ( (cid:98) µ ) = (cid:88) α ∈ ∆ + ( α, µ ) = (2 ρ, µ ) UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 23 (cid:3) µ ∈ P ∨ such that ( α i , µ ) ∈ N for all i ∈ I . Let l = (2 ρ, µ ) bethe length of (cid:98) µ ∈ (cid:99) W ext (Proposition 3.9) and consider any reduced decomposition:(3.29) (cid:98) µ = τ s i − l s i − l . . . s i Extend i − l , . . . , i to a ( τ -quasiperiodic) bi-infinite sequence { i k } k ∈ Z via:(3.30) i k + l = τ ( i k ) , ∀ k ∈ Z To such a bi-infinite sequence (3.30), one assigns the following bi-infinite sequenceof affine roots:(3.31) β k = (cid:40) s i s i . . . s i k − ( − α i k ) if k > s i s i − . . . s i k +1 ( α i k ) if k ≤ β > β > β > . . . (3.32) β < β − < β − < . . . (3.33)give convex orders of the sets ∆ + × Z < and ∆ + × Z ≥ , respectively. Remark . The above exposition follows that of [6] as we consider µ ∈ P ∨ . Toreduce it to the setup of [2, 36, 37], where only elements of Q ∨ are treated, we notethat if r ∈ N is the order of τ , then rµ ∈ Q ∨ , s i − rl s i − rl . . . s i − s i is a reduceddecomposition of (cid:99) rµ , and the sequence { i k } k ∈ Z is periodic with period l ( (cid:99) rµ ) = rl . Remark . For any k ∈ Z , if β k = ( α, d ) and β k + l = ( α (cid:48) , d (cid:48) ), then:(3.34) β k + l = (cid:98) µ ( β k ) ⇒ α = α (cid:48) and d = d (cid:48) + ( α, µ )due to (3.20). This reveals a periodicity of the entire set ∆ + × Z , not just of its twohalves ∆ + × Z < and ∆ + × Z ≥ (it is also the reason for the minus sign in (3.31)).3.13. Recall the element ρ ∨ ∈ P ∨ ∩ Q ∨ defined by: ρ ∨ = (cid:88) i ∈ I ω ∨ i = 12 (cid:88) α ∈ ∆ + α ∨ The following is the main result of this Section.
Theorem 3.14.
There exists a reduced decomposition of (cid:99) ρ ∨ ∈ (cid:99) W ext such that: • the order (3.32) of the roots { ( α, d ) | α ∈ ∆ + , d < } matches the lexicographicorder of the standard Lyndon loop words (cid:96) ( α, − d ) via (1.16), • the order (3.33) of the roots { ( α, d ) | α ∈ ∆ + , d ≥ } matches the lexicographicorder of the standard Lyndon loop words (cid:96) ( α, − d ) via (1.16). The second bullet implies i = 1 (the smallest letter in I , given our chosen order).On the other hand, combining s i ρ ∨ = s τ ( i − l ) τ s i − l s i − l . . . s i = τ s i − l . . . s i withthe fact that l ( s j ρ ∨ ) > l ( ρ ∨ ) ∀ j ∈ I (a consequence of (3.26)), implies that i = 0. , AND ALEXANDER TSYMBALIUK
Proof of Theorem 3.14.
Consider the finite subset: L = (cid:110) ( α, d ) | α ∈ ∆ + , ≤ d < | α | (cid:111) of (cid:98) ∆ + , ordered via:(3.35) ( α, d ) < ( β, e ) ⇔ (cid:96) ( α, − d ) < (cid:96) ( β, − e )If ( α, d ) , ( β, e ) ∈ L with ( α, d ) < ( β, e ) and ( α + β, d + e ) ∈ (cid:98) ∆, then clearly( α + β, d + e ) ∈ L , as well as ( α, d ) < ( α + β, d + e ) < ( β, e ), due to Proposition 2.33.Furthermore, we claim that if (cid:101) λ, (cid:101) µ ∈ (cid:98) ∆ + with (cid:101) λ + (cid:101) µ ∈ L , then at least one of (cid:101) λ or (cid:101) µ belongs to L and is < (cid:101) λ + (cid:101) µ . This is obvious when (cid:101) λ = ( α, d ), (cid:101) µ = ( β, e ) with α, β ∈ ∆ + and d, e ≥
0. In the remaining case, we may assume (cid:101) λ = ( α + β, d ) , (cid:101) µ = ( − β, e ),so that α, β, α + β ∈ ∆ + and d ≥ , e >
0. Then d < d + e < | α | < | α + β | , so that (cid:101) λ ∈ L . It remains to verify (cid:101) λ < (cid:101) λ + (cid:101) µ , that is, (cid:96) ( α + β, − d ) < (cid:96) ( α, − d − e ). Since( α + β, − d ) = ( β, e ) + ( α, − d − e ), it suffices to prove (cid:96) ( β, e ) < (cid:96) ( α, − d − e ), due toProposition 2.33. But applying Proposition 2.25, we see that the exponent of thefirst letter in (cid:96) ( β, e ) is >
0, while the exponent of the first letter in (cid:96) ( α, − d − e ) is ≤
0, hence, indeed (cid:96) ( β, e ) < (cid:96) ( α, − d − e ).Invoking [35] (which also applies to finite subsets in affine root systems), we get: • there is a unique element w ∈ (cid:99) W such that L = E w • the order of L arises via a certain reduced decomposition of w , cf. (3.27).However, as noticed in our proof of Proposition 3.9, we have L = E (cid:99) ρ ∨ = (cid:110) β , β − , . . . , β − l (cid:111) There is a unique τ ∈ T such that τ − (cid:99) ρ ∨ ∈ (cid:99) W (note that τ = 1 since 2 ρ ∨ ∈ Q ∨ ).Then: L = E (cid:99) ρ ∨ = E τ − (cid:99) ρ ∨ Therefore, there exists a reduced decomposition (3.29) of (cid:99) ρ ∨ such that the orderedfinite sequence β < β − < · · · < β − l exactly coincides with L ordered via (3.35).The proof of Theorem 3.14 now follows by a simple combination of (3.34) andPropositions 2.25, 2.29. Indeed, let us split ∆ + × Z into the blocks: L N = (cid:110) ( α, d ) | α ∈ ∆ + , N | α | ≤ d < ( N + 1) | α | (cid:111) so that: (cid:71) N ≥ L N = ∆ + × Z ≥ = { β k } k ≤ (cid:71) N< L N = ∆ + × Z < = { β k } k> According to (3.34) and L = L = { β , . . . , β − l } , we have: L N = (cid:110) β − Nl , β − Nl − , . . . , β − ( N +1) l (cid:111) , ∀ N ∈ Z For any ( α, d ) ∈ L N , the exponent of the first letter in (cid:96) ( α, − d ) is − N , due toProposition 2.25 (and its proof). Therefore, for any ( α, d ) ∈ L M , ( β, e ) ∈ L N with M > N , we have (cid:96) ( α, − d ) > (cid:96) ( β, − e ). As for the affine roots from the same UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 25 block, consider β r − Nl , β s − Nl ∈ L N with 1 − l ≤ s < r ≤
0. If β r = ( α, d ) and β s = ( β, e ), then β r − Nl = ( α, d + N | α | ) and β s − Nl = ( β, e + N | β | ), due to (3.34).On the other hand, the words (cid:96) ( α, − d − N | α | ) and (cid:96) ( β, − e − N | β | ) are obtainedfrom (cid:96) ( α, − d ) and (cid:96) ( β, − e ), respectively, by decreasing each exponent by N , due toProposition 2.29. Since the latter operation obviously preserves the lexicographicorder, and (cid:96) ( α, − d ) < (cid:96) ( β, − e ) as a consequence of r > s , we obtain the requiredinequality (cid:96) ( α, − d − N | α | ) < (cid:96) ( β, − e − N | β | ). (cid:3) We actually have the stronger result that the order of ∆ + × Z given by:(3.36) · · · < β < β < β < β < β − < β − < . . . matches the lexicographic order of the standard Lyndon loop words (cid:96) ( α, − d ) (since (cid:96) ( α, − d ) < (cid:96) ( β, − e ) if d < ≤ e , itself a consequence of Proposition 2.25). Remark . We expect that a similar treatment can be done for any µ ∈ P ∨ such that ( α i , µ ) > i ∈ I . On the side of Lyndon loop words, this wouldrequire an analogue of Proposition 2.29 stating that (cid:96) ( α, d +( α, µ )) is obtained from (cid:96) ( α, d ) by adding ( α i , µ ) to all the exponents of letters i ∈ I . For this operation topreserve the property of words being Lyndon, one can replace the order (1.12) onloop letters { i ( d ) } d ∈ Z i ∈ I by: i ( d ) < j ( e ) if d ( α i ,µ ) > e ( α j ,µ ) or d ( α i ,µ ) = e ( α j ,µ ) and i < j We expect the contents of Sections 2 and 3 to carry through in this more generalsetup, but we make no claims in this regard.4.
Quantum groups and shuffle algebras
We will review the connection between Drinfeld-Jimbo quantum groups andshuffle algebras, following [15, 39, 41]. We will also recall the point of view of [26](see also [40]), which connects shuffle algebras with the notion of standard Lyndonwords. Then we develop a loop version of this treatment, and prove Theorem 1.5.4.1. Let us recall the notation of Subsection 2.1 which, as we have seen, cor-responds to a finite-dimensional simple Lie algebra g . Consider the q -numbers, q -factorials and q -binomial coefficients:[ k ] i = q ki − q − ki q i − q − i , [ k ]! i = [1] i . . . [ k ] i , (cid:18) nk (cid:19) i = [ n ]! i [ k ]! i [ n − k ]! i for any i ∈ I , where q i = q dii . Definition 4.2.
The Drinfeld-Jimbo quantum group associated to g is: U q ( g ) = Q ( q ) (cid:68) e i , f i , ϕ ± i (cid:69) i ∈ I (cid:46) relations (4.1) , (4.2) , (4.3) where we impose the following relations for all i, j ∈ I : (4.1) − a ij (cid:88) k =0 ( − k (cid:18) − a ij k (cid:19) i e ki e j e − a ij − ki = 0 , if i (cid:54) = j , AND ALEXANDER TSYMBALIUK (4.2) ϕ j e i = q d ji e i ϕ j , ϕ i ϕ j = ϕ j ϕ i as well as the opposite relations with e ’s replaced by f ’s, and finally the relation: (4.3) [ e i , f j ] = δ ji · ϕ i − ϕ − i q i − q − i If we let ϕ i = q h i i and take the limit q →
1, then U q ( g ) degenerates to U ( g ).4.3. Recall that U q ( g ) is a bialgebra with respect to the coproduct ([22, § ϕ i ) = ϕ i ⊗ ϕ i ∆( e i ) = ϕ i ⊗ e i + e i ⊗ f i ) = 1 ⊗ f i + f i ⊗ ϕ − i This bialgebra structure preserves the Q -grading induced by setting ([22, § e i = α i , deg ϕ i = 0 , deg f i = − α i Recall the triangular decomposition ([22, § U q ( g ) = U q ( n + ) ⊗ U q ( h ) ⊗ U q ( n − )where U q ( n + ) , U q ( h ) , U q ( n − ) are the subalgebras of U q ( g ) generated by the e i ’s, ϕ ± i ’s, f i ’s, respectively. We will also consider the following sub-bialgebras of U q ( g ): U q ( b + ) = U q ( n + ) ⊗ U q ( h ) U q ( b − ) = U q ( h ) ⊗ U q ( n − ) Remark . As an associative algebra, U q ( n + ) (resp. U q ( b + )) is generated by e i ’s(resp. e i , ϕ ± i ’s) with the defining relations (4.1) (resp. (4.1, 4.2)), see e.g. [22, § § (4.5) (cid:104)· , ·(cid:105) : U q ( b + ) ⊗ U q ( b − ) −→ Q ( q )where the word “bialgebra” means that it satisfies the following properties: (cid:104) a, bc (cid:105) = (cid:104) ∆( a ) , b ⊗ c (cid:105) (4.6) (cid:104) ab, c (cid:105) = (cid:104) b ⊗ a, ∆( c ) (cid:105) (4.7)for all applicable a, b, c . Then (4.5) is determined by the assignments: (cid:104) e i , f j (cid:105) = δ ji q − i − q i , (cid:104) ϕ i , ϕ j (cid:105) = q − d ij and the fact that (cid:104) a, b (cid:105) = 0 unless deg a + deg b = 0The quantum group U q ( g ) is the Drinfeld double of ( U q ( b + ) , U q ( b − ) , (cid:104)· , ·(cid:105) ), whichmeans that the multiplication map induces an isomorphism: U q ( b + ) ⊗ U q ( b − ) (cid:46) ( ϕ i ⊗ ϕ − i − ⊗ ∼ −→ U q ( g ) Henceforth, given two algebras
A, B over a ring K , a K -valued bilinear pairing A × B → K shall be rather denoted A ⊗ B → K (with ⊗ standing for ⊗ K ) to indicate its K -bilinear nature. UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 27 and that the commutation rule of the two factors is governed by the relation: (4.8) a b (cid:104) a , b (cid:105) = (cid:104) a , b (cid:105) b a for all a ∈ U q ( b + ) and b ∈ U q ( b − ). Here we use Sweedler notation ∆( a ) = a ⊗ a for the coproduct of Subsection 4.3 (a summation sign is implied in front of a ⊗ a ).4.6. Since the quantum group of Definition 4.2 is a q -deformation of the universalenveloping of the Lie algebra of Definition 2.2, it is natural that many featuresof the latter admit q -deformations as well. For example, let us recall the notionof standard Lyndon words from Subsections 2.3 - 2.11, and consider the following q -version of the construction of Definition 2.9. Definition 4.7. ( [26] ) For any word w , define e w ∈ U q ( n + ) by: e [ i ] = e i for all i ∈ I , and then recursively by: (4.9) e (cid:96) = [ e (cid:96) , e (cid:96) ] q = e (cid:96) e (cid:96) − q (deg (cid:96) , deg (cid:96) ) e (cid:96) e (cid:96) if (cid:96) is a Lyndon word with factorization (2.6) , and: (4.10) e w = e (cid:96) . . . e (cid:96) k if w is an arbitrary word with canonical factorization (cid:96) . . . (cid:96) k , as in (2.7) . We also define f w ∈ U q ( n − ) by replacing e ’s by f ’s in the Definition above. Thenwe have the following q -deformation of the PBW statement (2.15). Theorem 4.8.
We have: (4.11) U q ( n + ) = k ∈ N (cid:77) (cid:96) ≥···≥ (cid:96) k standard Lyndon words Q ( q ) · e (cid:96) . . . e (cid:96) k = (cid:77) w standard loop words Q ( q ) · e w The analogous result also holds with + ↔ − and e ↔ f . In Subsection 5.5, we will show that Theorem 4.8 is just a reformulation of theusual PBW theorem for U q ( n ± ), since e (cid:96) ’s are simply renormalizations of the rootvectors constructed in [30]. According to [34, Remark 2.4], formula (4.8) is equivalent to a more standard commutationrule appearing in the literature. We prefer our formula as it does not require us to define theantipode, which exists but will not be necessary in the present paper. , AND ALEXANDER TSYMBALIUK q -shuffle algebra interpretation of thequantum group U q ( n + ), due to [15, 39, 41], which we recall now. Definition 4.10.
Consider the Q ( q ) -vector space F with a basis given by words: (4.12) [ i . . . i k ] for arbitrary k ∈ N , i , . . . , i k ∈ I , and endow it with the following shuffle product: (4.13) [ i . . . i k ] ∗ [ j . . . j l ] = (cid:88) { ,...,k + l } = A (cid:116) B | A | = k, | B | = l q λ A,B · [ s . . . s k + l ] where in the right-hand side, if A = { a < · · · < a k } and B = { b < · · · < b l } , wewrite: (4.14) s c = (cid:40) i • if c = a • j • if c = b • and: (4.15) λ A,B = (cid:88) A (cid:51) a>b ∈ B d s a s b It is straightforward to see that ( F , ∗ ) is an associative algebra. If we set q = 1,then F coincides with the classical shuffle algebra on the alphabet I . The classicalshuffle algebra is actually a bialgebra, with coproduct defined by splitting words:∆ ([ i . . . i k ]) = k (cid:88) a =0 [ i . . . i a ] ⊗ [ i a +1 . . . i k ]But for generic q , the coproduct above is no longer multiplicative with respect to theshuffle product (4.13). To remedy this, we consider the extended shuffle algebra: F ext = F ⊗ Q ( q ) (cid:2) ϕ ± i (cid:3) i ∈ I with pairwise commuting ϕ i ’s, where the multiplication is governed by the rule:(4.16) ϕ j · [ i . . . i k ] = q (cid:80) ka =1 d jia [ i . . . i k ] · ϕ j It is straightforward to check that the assignment ∆( ϕ i ) = ϕ i ⊗ ϕ i and:(4.17) ∆ ([ i . . . i k ]) = k (cid:88) a =0 [ i . . . i a ] ϕ i a +1 . . . ϕ i k ⊗ [ i a +1 . . . i k ]is both coassociative and gives rise to a bialgebra structure on F ext . Remark . Our construction differs slightly from [15, 39], where F itself is en-dowed with a bialgebra structure by modifying the product on F ⊗ F in the spiritof [30, p. 3]. However, the two approaches are easily seen to be equivalent. We note that the formula (4.13) is worded differently from [26, formula (9)], but it is animmediate consequence of [26, formula (8)].
UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 29 U q ( n + ) Φ −→ F sending e i to [ i ] (as one just needs to check that relations (4.1) hold in F , dueto Remark 4.4). Moreover, it is easy to prove by induction on | deg x | (using thebialgebra pairing properties (4.6, 4.7)) that the map Φ is explicitly given by:(4.19) Φ( x ) = k ∈ N (cid:88) i ,...,i k ∈ I (cid:34) k (cid:89) a =1 ( q − i a − q i a ) (cid:35) (cid:68) x, f i . . . f i k (cid:69) · [ i . . . i k ]Because the bialgebra pairing (4.5) is non-degenerate and (cid:104) x, yϕ − (cid:105) = (cid:104) x, y (cid:105) for any x ∈ U q ( n + ) , y ∈ U q ( n − ) and ϕ − a product of ϕ − i ’s (which is a simple consequenceof the bialgebra pairing properties (4.6, 4.7)), (4.19) implies the injectivity of Φ.The image of the map Φ is described in [26, Theorem 5], which states that:(4.20) Im Φ = r ∈ N (cid:88) i ,...,i r ∈ I γ ( i . . . i r ) · [ i . . . i r ] where the constants γ ( i . . . i r ) ∈ Q ( q ) vanish for all but finitely many values of r and satisfy the following property:(4.21) − a ij (cid:88) k =0 ( − k (cid:18) − a ij k (cid:19) i γ w i . . . i (cid:124) (cid:123)(cid:122) (cid:125) k symbols j i . . . i (cid:124) (cid:123)(cid:122) (cid:125) − a ij − k symbols w (cid:48) = 0for any distinct i, j ∈ I and any words w, w (cid:48) .Comparing (4.2) with (4.16), it is easy to see that the algebra homomorphism (4.18)extends to a bialgebra homomorphism: U q ( b + ) Φ −→ F ext by sending ϕ i (cid:55)→ ϕ i .4.13. As in Subsection 2.3, we fix a total order on the set I , and consider theinduced lexicographic order on the set of all words (2.5). Definition 4.14. ( [26] ) A word w is called good if there exists an element: (4.22) w + (cid:88) v Proposition 4.15. ( [26, Lemma 21] ) A word is good if and only if it is standard. Above, we invoke the notion of standard words from Definition 2.12(a). Likewise,the standard Lyndon words from Definition 2.12(b) as well as the bijection (2.13)can also be characterized in terms of the map Φ, as follows. , AND ALEXANDER TSYMBALIUK Lemma 4.16. ( [26, Corollary 27, Theorem 36] ) For any α ∈ ∆ + , the leading wordof Φ( e (cid:96) ( α ) ) is (cid:96) ( α ) . Moreover, the word (cid:96) ( α ) is the smallest good word of degree α . e i ( z ) = (cid:88) k ∈ Z e i,k z k , f i ( z ) = (cid:88) k ∈ Z f i,k z k , ϕ ± i ( z ) = ∞ (cid:88) l =0 ϕ ± i,l z ± l and consider the formal delta function δ ( z ) = (cid:80) k ∈ Z z k . For any i, j ∈ I , set:(4.23) ζ ij (cid:16) zw (cid:17) = z − wq − d ij z − w We now recall the definition of the quantum loop group (new Drinfeld realization). Definition 4.18. The quantum loop group associated to g is: U q ( L g ) = Q ( q ) (cid:68) e i,k , f i,k , ϕ ± i,l (cid:69) i ∈ I,k ∈ Z ,l ∈ N (cid:46) relations (4.24) – (4.28) where we impose the following relations for all i, j ∈ I : (4.24) e i ( z ) e j ( w ) ζ ji (cid:16) wz (cid:17) = e j ( w ) e i ( z ) ζ ij (cid:16) zw (cid:17) (4.25) (cid:88) σ ∈ S (1 − a ij ) 1 − a ij (cid:88) k =0 ( − k (cid:18) − a ij k (cid:19) i · e i ( z σ (1) ) . . . e i ( z σ ( k ) ) e j ( w ) e i ( z σ ( k +1) ) . . . e i ( z σ (1 − a ij ) ) = 0 , if i (cid:54) = j (4.26) ϕ ± j ( w ) e i ( z ) ζ ij (cid:16) zw (cid:17) = e i ( z ) ϕ ± j ( w ) ζ ji (cid:16) wz (cid:17) (4.27) ϕ ± i ( z ) ϕ ± (cid:48) j ( w ) = ϕ ± (cid:48) j ( w ) ϕ ± i ( z ) , ϕ + i, ϕ − i, = 1 as well as the opposite relations with e ’s replaced by f ’s, and finally the relation: (4.28) [ e i ( z ) , f j ( w )] = δ ji δ (cid:0) zw (cid:1) q i − q − i · (cid:16) ϕ + i ( z ) − ϕ − i ( w ) (cid:17) Note that there is a unique algebra homomorphism: U q ( g ) (cid:44) −→ U q ( L g )sending e i (cid:55)→ e i, , f i (cid:55)→ f i, , ϕ ± i (cid:55)→ ϕ ± i, . UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 31 U q ( L g ) is a topological bialgebra with respect to the followingcoproduct ([8, formulas (5)–(7)]):(4.29) ∆ (cid:0) ϕ ± i ( z ) (cid:1) = ϕ ± i ( z ) ⊗ ϕ ± i ( z )(4.30) ∆ ( e i ( z )) = ϕ + i ( z ) ⊗ e i ( z ) + e i ( z ) ⊗ f i ( z )) = 1 ⊗ f i ( z ) + f i ( z ) ⊗ ϕ − i ( z )This bialgebra structure preserves the Q × Z -grading induced by setting:deg e i,k = ( α i , k ) , deg ϕ ± i,l = (0 , ± l ) , deg f i,k = ( − α i , k )for all applicable indices. Recall the triangular decomposition ([18, § U q ( L g ) = U q ( L n + ) ⊗ U q ( L h ) ⊗ U q ( L n − )where U q ( L n + ) , U q ( L h ) , U q ( L n − ) are the subalgebras of U q ( L g ) generated by the e i,k ’s, ϕ ± i,l ’s, f i,k ’s, respectively. We note that the following subalgebras of U q ( L g ): U q ( L b + ) = U q ( L n + ) ⊗ Q ( q ) (cid:2) ϕ ± i, , ϕ + i, , ϕ + i, , . . . (cid:3) i ∈ I U q ( L b − ) = Q ( q ) (cid:2) ϕ ∓ i, , ϕ − i, , ϕ − i, , . . . (cid:3) i ∈ I ⊗ U q ( L n − )are preserved by the coproduct ∆, and hence are sub-bialgebras of U q ( L g ).4.20. It is well-known ([16, Lemma 9.1], see also [10, § § (cid:104)· , ·(cid:105) : U q ( L b + ) ⊗ U q ( L b − ) −→ Q ( q )that satisfies (4.6, 4.7) and is determined by the properties: (cid:68) e i ( z ) , f j ( w ) (cid:69) = δ ji δ (cid:0) zw (cid:1) q − i − q i (4.34) (cid:68) ϕ + i ( z ) , ϕ − j ( w ) (cid:69) = ζ ij (cid:0) zw (cid:1) ζ ji (cid:0) wz (cid:1) (4.35)(the right-hand side of (4.35) is expanded in | z | (cid:29) | w | ) and the fact that: (cid:104) a, b (cid:105) = 0 unless deg a + deg b = (0 , ∈ Q × Z This pairing is known to be non-degenerate (cf. [16, Lemma 9.2], [17, Proposi-tion 9], [11, Theorem 1.4]), although we will provide an alternative argument below. Proposition 4.21. The pairing (cid:104)· , ·(cid:105) of (4.33) is non-degenerate in each argument. We will give a proof of this result in Subsection 6.16. , AND ALEXANDER TSYMBALIUK Definition 4.23. For any loop word w , define e w ∈ U q ( L n + ) , f w ∈ U q ( L n − ) by: e [ i ( d ) ] = e i,d and f [ i ( d ) ] = f i, − d for all i ∈ I , d ∈ Z , and then recursively by: e (cid:96) = [ e (cid:96) , e (cid:96) ] q = e (cid:96) e (cid:96) − q (hdeg (cid:96) , hdeg (cid:96) ) e (cid:96) e (cid:96) (4.36) f (cid:96) = [ f (cid:96) , f (cid:96) ] q = f (cid:96) f (cid:96) − q (hdeg (cid:96) , hdeg (cid:96) ) f (cid:96) f (cid:96) (4.37) if (cid:96) is a Lyndon loop word with factorization (2.6) , and: (4.38) e w = e (cid:96) . . . e (cid:96) k and f w = f (cid:96) . . . f (cid:96) k if w is an arbitrary loop word with canonical factorization (cid:96) . . . (cid:96) k , as in (2.7) . Note that deg e w = − deg f w = deg w for all loop words w . We have the followingresult, which is simultaneously an analogue of both (2.36, 2.37) and Theorem 4.8. Theorem 4.24. We have: U q ( L n + ) = k ∈ N (cid:77) (cid:96) ≥···≥ (cid:96) k standard Lyndon loop words Q ( q ) · e (cid:96) . . . e (cid:96) k = (cid:77) w standard loop words Q ( q ) · e w The analogous result also holds with + ↔ − and e ↔ f . The proof of the Theorem above will occupy most of Section 5, where we will deriveit from the PBW Theorem for the affine quantum group (in the Drinfeld-Jimbopresentation) constructed by [2, 4].4.25. We will now define a “loop” version of the shuffle algebra, which is to U q ( L g )as the shuffle algebra of Definition 4.10 is to U q ( g ). The careful reader will observea slight error in Definition 4.26, which will be remedied in Subsection 4.30, but weprefer this slightly imprecise approach in order to keep the exposition clear. TheDefinition below is equivalent to the main construction of [17, § Definition 4.26. Take the Q ( q ) -vector space F L with a basis given by loop words: (cid:104) i ( d )1 . . . i ( d k ) k (cid:105) for arbitrary k ∈ N , i , . . . , i k ∈ I , d , . . . , d k ∈ Z , and endow it with the followingshuffle product: (cid:104) i ( d )1 . . . i ( d k ) k (cid:105) ∗ (cid:104) j ( e )1 . . . j ( e l ) l (cid:105) = (cid:88) { ,...,k + l } = A (cid:116) B | A | = k, | B | = l (cid:88) π + ··· + π k + l =0 π ,...,π k + l ∈ Z γ A,B,π ,...,π k + l · (cid:104) s ( t + π )1 . . . s ( t k + l + π k + l ) k + l (cid:105) (4.39) UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 33 where in the right-hand side, if A = { a < · · · < a k } and B = { b < · · · < b l } , wewrite: (4.40) s c = (cid:40) i • if c = a • j • if c = b • , t c = (cid:40) d • if c = a • e • if c = b • and γ A,B,π ,...,π k + l are defined as the coefficients of the Taylor expansion: (4.41) (cid:89) A (cid:51) a>b ∈ B ζ s a s b (cid:16) z a z b (cid:17) ζ s b s a (cid:16) z b z a (cid:17) = (cid:88) π + ··· + π k + l =0 π ,...,π k + l ∈ Z γ A,B,π ,...,π k + l · z π . . . z π k + l k + l in the limit when | z a | (cid:29) | z b | for all a ∈ A, b ∈ B .Remark . (a) We note that in the inner sum of (4.39) the only terms whichappear with non-zero coefficient are those with π c ≤ c ∈ A and π c ≥ c ∈ B .(b) We also have γ A,B, ,..., = q λ A,B with λ A,B defined in (4.15).It is straightforward to see that ( F L , ∗ ) is an associative algebra, Q + × Z -gradedby (2.19), and we leave this check as an exercise to the interested reader. Proposition 4.28. There is a unique algebra homomorphism: (4.42) U q ( L n + ) Φ L −→ F L sending e i,d (cid:55)→ (cid:2) i ( d ) (cid:3) . The homomorphism Φ L is injective and is explicitly given by (4.43) Φ L ( x ) = k ∈ N (cid:88) i ,...,i k ∈ Id ,...,d k ∈ Z (cid:34) k (cid:89) a =1 ( q − i a − q i a ) (cid:35) (cid:68) x, f i , − d . . . f i k , − d k (cid:69) · (cid:104) i ( d )1 . . . i ( d k ) k (cid:105) for all x ∈ U q ( L n + ) , where the pairing is that of (4.33) . The Proposition above is straightforward, so we leave it as an exercise to the inter-ested reader (alternatively, it follows from Proposition 6.21 below). The injectivityfollows immediately from the non-degeneracy of (4.33), due to Proposition 4.21. Remark . We note that a version of the above construction of F L and thehomomorphism (4.42) (which would correspond in our notation to | I | = 1, butwith a more complicated ζ -factor) featured in [42, § F L = (cid:77) k ∈ Q + ,d ∈ Z F L k ,d , AND ALEXANDER TSYMBALIUK where we consider the following completions:(4.45) F L k ,d = (cid:88) d + ··· + d a bounded frombelow, for all a ∈{ ,...,k } c i ,...,i k ; d ,...,d k · (cid:104) i ( d )1 . . . i ( d k ) k (cid:105)(cid:124) (cid:123)(cid:122) (cid:125) has degree ( k ,d ) with arbitrary coefficients c i ,...,i k ; d ,...,d k ∈ Q ( q ). Proposition 4.31. The shuffle product (4.39) is well-defined on F L of (4.44, 4.45).Proof. We begin by showing that the operation w ∗ w (cid:48) of (4.39) extends to a well-defined operation on infinite linear combinations of the form:(4.46) (cid:88) deg w =( k ,d ) c w · w ∗ (cid:88) deg w (cid:48) =( k (cid:48) ,d (cid:48) ) c (cid:48) w (cid:48) · w (cid:48) where we have c w (cid:54) = 0 (resp. c (cid:48) w (cid:48) (cid:54) = 0) only if every prefix of w (resp. w (cid:48) ) has verticaldegree bounded from below by some fixed m ∈ Z . Take an arbitrary word v andconsider the set: S = (cid:110) ( w, w (cid:48) ) such that c w (cid:54) = 0 , c (cid:48) w (cid:48) (cid:54) = 0 and v appears as a summand in w ∗ w (cid:48) (cid:111) We need to show that S is finite, which would imply that the coefficient of v in theshuffle product (4.46) is well-defined. Let us assume for the purpose of contradictionthat S is infinite. Since the vertical degrees of arbitrary prefixes of w and w (cid:48) arebounded from below, this implies that one of these prefixes has arbitrarily largevertical degree. Without loss of generality, let us assume that we are talking aboutthe length a prefix of w . Thus, for any N ∈ N , there exists ( w, w (cid:48) ) ∈ S such thatthe vertical degree of w a | is at least N . However, since all the prefixes of w (cid:48) havevertical degree at least equal to the fixed constant m , then all terms in the shuffleproduct w ∗ w (cid:48) will have some prefix with vertical degree at least N + m . If N islarge enough, this contradicts the fact that v appears as a summand in w ∗ w (cid:48) .We now need to prove that the expression (4.46) is of the form (4.44, 4.45). The loopwords v that appear in the expression (4.46) also do appear in the shuffle products w ∗ w (cid:48) , where w and w (cid:48) are loop words of fixed degrees, such that every prefix of w and w (cid:48) has vertical degree bounded from below by some fixed m ∈ Z . Thus, anyloop word appearing in the shuffle product w ∗ w (cid:48) has degree deg w + deg w (cid:48) , whileany of its prefixes has vertical degree bounded from below by 2 m (an immediateconsequence of (4.39, 4.41)), which is precisely what we needed to prove. (cid:3) F L . However,there is a bialgebra structure on the extended shuffle algebra: F L, ext = F L ⊗ Q ( q ) (cid:2) ( ϕ + i, ) ± , ϕ + i, , ϕ + i, , . . . (cid:3) i ∈ I with pairwise commuting ϕ ’s, where the multiplication is governed by the rule: ϕ + j,e ∗ (cid:104) i ( d )1 . . . i ( d k ) k (cid:105) = (cid:88) π ,...,π k ≥ µ π ,...,π k · (cid:104) i ( d + π )1 . . . i ( d k + π k ) k (cid:105) ∗ ϕ + j,e − π −···− π k (4.47) UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 35 where ϕ + j,< = 0 and µ π ,...,π k are defined as the coefficients of the Taylor expansion: k (cid:89) r =1 ζ ji r ( w/z i ) ζ i r j ( z i /w ) = (cid:88) π ,...,π k ≥ µ π ,...,π k · z π . . . z π k k w π + ··· + π k It is straightforward to check that the right-hand side of (4.47) indeed lies in F L of (4.44, 4.45) tensored with Q ( q ) (cid:2) ( ϕ + i, ) ± , ϕ + i, , ϕ + i, , . . . (cid:3) i ∈ I , and that (4.47) ex-tends to the entire F L . It is also easy to check that the assignment∆( ϕ + i ( z )) = ϕ + i ( z ) ⊗ ϕ + i ( z )and(4.48) ∆ (cid:16)(cid:104) i ( d )1 . . . i ( d k ) k (cid:105)(cid:17) = k (cid:88) a =0 (cid:88) π a +1 ,...,π k ≥ (cid:104) i ( d )1 . . . i ( d a ) a (cid:105) ϕ + i a +1 ,π a +1 . . . ϕ + i k ,π k ⊗ (cid:104) i ( d a +1 − π a +1 ) a +1 . . . i ( d k − π k ) k (cid:105) is both coassociative and gives rise to a bialgebra structure on F L, ext . We notethat the coproduct (4.48) is topological, in the same sense as the coproduct (4.30).Finally, comparing (4.26) with (4.47) as well as (4.30) with (4.48), we see that thealgebra homomorphism (4.42) extends to a bialgebra homomorphism: U q ( L b + ) Φ L −→ F L, ext by sending ϕ + i,r (cid:55)→ ϕ + i,r .4.33. Define good loop words just like in Definition 4.14 (by replacing Φ with Φ L ). Proposition 4.34. Any subword of a good loop word is good.Proof. It is enough to prove that any prefix and suffix of a good loop word is good.To this end, assume that w is a good loop word of length k , which implies thatthere exists x ∈ U q ( L n + ) such that:Φ L ( x ) = w + (cid:88) v A loop word is good if and only if it can be written as: (cid:96) . . . (cid:96) k where (cid:96) ≥ · · · ≥ (cid:96) k are good Lyndon loop words.Proof. The “only if” statement is an immediate consequence of Proposition 2.7 andProposition 4.34. As for the “if” statement, suppose that we have good Lyndonloop words (cid:96) ≥ · · · ≥ (cid:96) k . By definition, there exist elements:(4.54) Φ L ( x r ) = (cid:96) r + (cid:88) v<(cid:96) r coefficient · v for various x r ∈ U q ( L n + ). We may assume that each x r is homogeneous, and thatso are the v ’s in (4.54), hence all of them have the same number of letters as (cid:96) r .But then the leading order term of Φ L ( x . . . x k ) is the leading word in the shuffleproduct (cid:96) ∗· · ·∗ (cid:96) k . By the obvious analogue of [26, Lemma 15], this shuffle producthas the leading order term equal to the concatenation (cid:96) . . . (cid:96) k . This exactly meansthat the latter concatenation is a good loop word, as we needed to show. (cid:3) UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 37 w consider:(4.55) U q ( L n − ) ≤ w = (cid:77) v ≤ w standard loop word Q ( q ) · f v which is finite-dimensional in any degree ∈ Q − × Z according to Corollary 2.31.For any loop word w , we also define:(4.56) U q ( L n + ) ≤ w ⊂ U q ( L n + )to consist of those elements x such that the leading order term of Φ L ( x ) is ≤ w .Invoking (4.43), we note that U q ( L n + ) ≤ w consists of those x ∈ U q ( L n + ) such that:(4.57) (cid:104) x, u f (cid:105) = 0 , ∀ u > w where for any loop word u = (cid:104) i ( d )1 . . . i ( d k ) k (cid:105) we set:(4.58) u f := f i , − d . . . f i k , − d k Proposition 4.37. The restriction of the pairing (4.33) to the subspaces: U q ( L n + ) ≤ w ⊗ U q ( L n − ) ≤ w −→ Q ( q ) is still non-degenerate in the first factor, i.e. (cid:104) x, −(cid:105) = 0 implies x = 0 .Proof. Assume x ∈ U q ( L n + ) ≤ w has the property that:(4.59) (cid:104) x, f v (cid:105) = 0for any standard loop word v ≤ w , and our goal is to show that x = 0. To this end,note that for any loop word v we have (by analogy with [26, Proposition 20]):(4.60) f v ∈ (cid:88) u ≥ v Q ( q ) · u f Since (cid:104) x, u f (cid:105) = 0 for all u > w by (4.57), we conclude:(4.61) (cid:104) x, f v (cid:105) = 0for any loop word v > w . By Theorem 4.24, the set { f v | v standard loop word } isa basis of U q ( L n − ), so relations (4.59) and (4.61) imply that: (cid:10) x, U q ( L n − ) (cid:11) = 0Thus x = 0 due to the non-degeneracy statement of Proposition 4.21. (cid:3) U q ( L n + ) ≤ w ≤ (cid:110) standard loop words ≤ w (cid:111) Note a slight imprecision in the inequality above: what we actually mean is thatthe dimension of the left-hand side in any fixed degree ( α, d ) ∈ Q + × Z is less thanor equal to the number of standard loop words of degree ( α, d ) (the latter numberis finite by Corollary 2.31). On the other hand, by the very definition of a goodloop word, we have:(4.63) dim U q ( L n + ) ≤ w = (cid:110) good loop words ≤ w (cid:111) The following Proposition establishes the fact that we have equality in (4.62). , AND ALEXANDER TSYMBALIUK Proposition 4.39. A loop word is standard if and only if it is good.Proof. Assume for the purpose of contradiction that there exists a good loop word w which is not standard, and choose it such that its degree ( α, d ) ∈ Q + × Z hasminimal | α | . This minimality, combined with Propositions 2.16 and 4.35, impliesthat w must be Lyndon. Therefore, we may write it as (2.6): w = (cid:96) (cid:96) where (cid:96) < w < (cid:96) are Lyndon loop words. By Proposition 4.34, (cid:96) and (cid:96) aregood Lyndon loop words, hence by the minimality of | α | , standard Lyndon loopwords. However, because of (4.62) and (4.63), there must exist a standard loop word v < w with deg v = deg w . Then let us consider the canonical factorization (2.7) v = (cid:96) (cid:48) . . . (cid:96) (cid:48) k where (cid:96) (cid:48) ≥ · · · ≥ (cid:96) (cid:48) k are standard Lyndon loop words. Because:deg (cid:96) + deg (cid:96) = deg w = deg v = deg (cid:96) (cid:48) + · · · + deg (cid:96) (cid:48) k Corollary 2.36 implies that (cid:96) (cid:48) ≥ (cid:96) . However, the only way this is compatible with: (cid:96) (cid:96) = w > v = (cid:96) (cid:48) . . . (cid:96) (cid:48) k is if (cid:96) (cid:48) = (cid:96) u for some loop word u that satisfies:(4.64) (cid:96) > u(cid:96) (cid:48) . . . (cid:96) (cid:48) k and deg (cid:96) = deg u + deg (cid:96) (cid:48) + · · · + deg (cid:96) (cid:48) k Because (cid:96) (cid:48) is standard, Proposition 2.15 implies that so is u . Therefore we maywrite u = (cid:96) (cid:48)(cid:48) . . . (cid:96) (cid:48)(cid:48) m for various standard Lyndon loop words (cid:96) (cid:48)(cid:48) ≥ · · · ≥ (cid:96) (cid:48)(cid:48) m . Formula(4.64) implies that (cid:96) > u , so (cid:96) > (cid:96) (cid:48)(cid:48) ≥ · · · ≥ (cid:96) (cid:48)(cid:48) m . However, we also have (cid:96) > w >v > (cid:96) (cid:48) ≥ · · · ≥ (cid:96) (cid:48) k , and so (4.64) contradicts Corollary 2.36.For the converse, let us prove by induction on | α | that for any standard loop word w of degree ( α, d ), there exists a linear combination:(4.65) (cid:88) v ≥ w coefficient · Φ L ( e v ) ∈ Q ( q ) ∗ · w + smaller wordsfor various coefficients in Q ( q ) with v being standard loop words, where we mayfurther assume that all summands have the same Q + × Z -degree ( α, d ). Claim . If (4.65) holds for two loop words w = (cid:96) and w (cid:48) = (cid:96) . . . (cid:96) k , where (cid:96) ≥ (cid:96) ≥ · · · ≥ (cid:96) k are all standard Lyndon, then (4.65) also holds for the concatenation ww (cid:48) .Let us first show how the Claim allows us to complete the proof of the Proposition.Since any standard loop word can be written as w = (cid:96) . . . (cid:96) k where (cid:96) ≥ · · · ≥ (cid:96) k are standard Lyndon loop words, then the Claim says that it suffices to prove (4.65)when w = (cid:96) is a standard Lyndon loop word. To this end, let us write:Φ L ( e (cid:96) ) = c · u + (cid:88) v (cid:96) , then u is a concatenation of standard Lyndon loop words UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 39 of length less than that of (cid:96) , to which we may apply the induction hypothesis.According to the Claim, we may thus use (4.65) for u to write:Φ L ( e (cid:96) ) − (cid:88) v ≥ u coefficient · Φ L ( e v ) = (cid:88) v(cid:96) coefficient · Φ L ( e v ) = (cid:88) v<(cid:96) coefficient · v Since Φ L is injective and { e v | v standard loop word } is a basis of U q ( L n + ) due toTheorem 4.24, the left-hand side of (4.66) is non-zero, hence so is the right-handside. This implies that there are good, hence standard, loop words of degree ( α, d )which are < (cid:96) . The latter contradicts Corollary 2.36, and so (4.66) is impossible.Claim 4.40 follows immediately from the two facts below (assume w, w (cid:48) , (cid:96) , . . . , (cid:96) k are as in the statement of the Claim):(1) the largest word which appears in the shuffle product w ∗ w (cid:48) is ww (cid:48) (2) e v e v (cid:48) is a linear combination of e t ’s with t ≥ ww (cid:48) , for all v ≥ w and v (cid:48) ≥ w (cid:48) satisfying deg v = deg w and deg v (cid:48) = deg w (cid:48) The first fact is proved as in [26, Lemma 15] (cf. our proof of Proposition 4.35).To prove the second fact, note that formula (5.76) (as we will see, for any ( α, d ) ∈ ∆ + × Z , our e (cid:96) ( α,d ) will be a scalar multiple of the element denoted by (cid:36) ( e − ( α, − d ) )later on) implies that for all standard Lyndon loop words (cid:96) < (cid:96) (cid:48) , we can write:(4.67) e (cid:96) e (cid:96) (cid:48) = a linear combination of e (cid:96) (cid:48) (cid:96) and various e m (cid:48)(cid:48) ...m (cid:48)(cid:48) t (cid:48)(cid:48) with (cid:96) (cid:48) > m (cid:48)(cid:48) ≥ · · · ≥ m (cid:48)(cid:48) t (cid:48)(cid:48) > (cid:96) standard Lyndon loop words. Consider the canonicalfactorizations (2.7): v = m . . . m t and v (cid:48) = m (cid:48) . . . m (cid:48) t (cid:48) where m ≥ · · · ≥ m t and m (cid:48) ≥ · · · ≥ m (cid:48) t (cid:48) are standard Lyndon loop words. It iselementary to prove that v ≥ w , deg v = deg w , and w being Lyndon imply thateither m > w , or that v = w . In the former case ( m > w ), (4.67) implies that(cf. the argument in the proofs of Lemmas 5.4, 5.15): e v e v (cid:48) = e m . . . e m t e m (cid:48) . . . e m (cid:48) t (cid:48) = a linear combination of e t ’sfor standard t with canonical factorization m (cid:48)(cid:48) . . . m (cid:48)(cid:48) t (cid:48)(cid:48) satisfying m (cid:48)(cid:48) ≥ m > w . Aresult of Melan¸con ([32]), which states that two words with canonical factorization(2.7) are in the relative order > if the largest Lyndon words in their canonicalfactorizations are in the relative order > , implies that t > ww (cid:48) , as we needed toshow. In the latter case ( v = w = (cid:96) ), we have two more possible situations: • if (cid:96) ≥ m (cid:48) , then e v e v (cid:48) = e vv (cid:48) and we are done since vv (cid:48) ≥ ww (cid:48) (as v ≥ w, v (cid:48) ≥ w (cid:48) and the loop words v, w are of the same length) • if m (cid:48) i > (cid:96) ≥ m (cid:48) i +1 for some i ∈ { , . . . , t (cid:48) } , then (4.67) implies that: e v e v (cid:48) = e (cid:96) e m (cid:48) . . . e m (cid:48) t (cid:48) = a linear combination of e t ’swhere t = m (cid:48)(cid:48) . . . m (cid:48)(cid:48) i (cid:48)(cid:48) m (cid:48) i +1 . . . m (cid:48) t (cid:48) satisfies m (cid:48)(cid:48) ≥ · · · ≥ m (cid:48)(cid:48) i (cid:48)(cid:48) ≥ m (cid:48) i +1 ≥ · · · ≥ m (cid:48) t (cid:48) and m (cid:48)(cid:48) > (cid:96) . Thus, the aforementioned result of Melan¸con implies that t > ww (cid:48) . (cid:3) , AND ALEXANDER TSYMBALIUK Proof of Theorem 1.5. The statement about the homomorphism Φ L is proved inSubsection 4.25. The classification of standard Lyndon loop words is accomplishedin (2.34). The construction of the root vectors (1.14) is done in Definition 4.23.Finally, the PBW statement (1.15) is the subject of Theorem 4.24, whose proof willbe completed in the next Section. (cid:3) Computer experiments (in all types, but for a particular order of the simple roots)suggest that the generalization of Lemma 4.16 to the loop case holds. Conjecture 4.42. For any ( α, d ) ∈ ∆ + × Z , the leading word of Φ L ( e (cid:96) ( α,d ) ) is (cid:96) ( α, d ) . Moreover, the word (cid:96) ( α, d ) is the smallest good loop word of degree ( α, d ) . Quantum affine and quantum loop: two presentations In the present Section, we will recall the general framework (due to Lusztig inthe finite case, and Beck and Damiani in the affine case) of PBW bases for quantumgroups, from which we will deduce Theorems 4.8 and 4.24. The former of these willbe immediate, while the latter will require some work to connect quantum loop andquantum affine groups, and will require the use of the results of Section 3.5.1. Consider any convex order ≤ of the set of positive roots ∆ + , as in Defini-tion 2.18. According to [35] (see also Remark 3.8) there is a unique reduced de-composition w = s i − l . . . s i of the longest element w of the Weyl group W suchthat the ordered set α i < s i ( α i − ) < · · · < s i . . . s i − l ( α i − l ) precisely recovers(∆ + , ≤ ). To this choice, one may associate ([30]) a collection of “root vectors”:(5.1) E ± β ∈ U q ( n ± )for all β ∈ ∆ + , via the following formula for all 0 ≥ k > − l :(5.2) if β = s i . . . s i k +1 ( α i k ) then (cid:40) E β := T − i . . . T − i k +1 ( e i k ) E − β := T − i . . . T − i k +1 ( f i k )where { T i } i ∈ I determine Lusztig’s braid group action [30] on U q ( g ) (cf. [22, § § U q ( n ± ) = k ∈ N (cid:77) γ ≤···≤ γ k ∈ ∆ + Q ( q ) · E ± γ . . . E ± γ k By analogy with (1.4), the formula above is called a PBW theorem for U q ( n ± ). Remark . (a) Due to [22, formula (9) of § U q ( n + ) and U q ( n − ) are intertwined by an algebra automorphism ω of U q ( g ):(5.4) ω : e i (cid:55)→ f i , f i (cid:55)→ e i , ϕ ± i (cid:55)→ ϕ ∓ i , ∀ i ∈ I (b) We note that formulas (5.2, 5.3) differ slightly from [22, § § T i instead of T − i to define the UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 41 root vectors, as well the opposite order of ∆ + in the PBW theorem for U q ( n + ). Torelate the exposition of [22] to ours, recall the algebra anti-involution τ of U q ( g ):(5.5) τ : e i (cid:55)→ e i , f i (cid:55)→ f i , ϕ ± i (cid:55)→ ϕ ∓ i , ∀ i ∈ I It can be easily verified ([22, formula (10) of § τ ◦ T i ◦ τ = T − i for any i ∈ I . Therefore, (5.3) is obtained from [22, formula (3) of § w = w − = s i s i − . . . s i − l , followed by τ .Likewise, [22, formula (2) of § U q ( n ± ) = k ∈ N (cid:77) γ ≥···≥ γ k ∈ ∆ + Q ( q ) · E ± γ . . . E ± γ k (c) Henceforth, we will use the following non-tautological equalities ([22, § E α i = e i , E − α i = f i , ∀ i ∈ I E ± α ’s can be written as a sum of the ordered products. How-ever, there is a restriction on the products that may appear, as in [2, Proposition 7](which takes its origins in the formulas of [28]):(5.8) E ± β E ± α − q ( α,β ) E ± α E ± β ∈ k ∈ N (cid:77) α<γ ≤···≤ γ k <βγ + ··· + γ k = α + β Q ( q ) · E ± γ . . . E ± γ k for any positive roots α < β . If we assume that α + β is also a positive root, andthat its decomposition as the sum of α and β is minimal in the sense that:(5.9) (cid:54) ∃ α (cid:48) , β (cid:48) ∈ ∆ + s.t. α < α (cid:48) < β (cid:48) < β and α + β = α (cid:48) + β (cid:48) then the sum in the right-hand side of (5.8) consists of a single term: (5.10) [ E ± β , E ± α ] q = E ± β E ± α − q ( α,β ) E ± α E ± β ∈ Q ( q ) ∗ · E ± ( α + β ) (the coefficient of E ± ( α + β ) in the right-hand side actually lies in Z [ q, q − ] ∗ , as shownin [31, Theorem 6.7(a)]). Therefore, one can recover the root vectors { E β } β ∈ ∆ + (resp. { E − β } β ∈ ∆ + ) as iterated q -commutators of the e i ’s (resp. f i ’s) times scalars,based solely on the chosen convex order of the set of positive roots ∆ + .We conclude this Subsection with another important corollary of formula (5.8). Lemma 5.4. For any α ≤ β ∈ ∆ + , let U ± q ([ α, β ]) be the subalgebra of U q ( n ± ) generated by { E ± γ | α ≤ γ ≤ β } . Then: (5.11) U ± q ([ α, β ]) = k ∈ N (cid:77) β ≥ γ ≥···≥ γ k ≥ α ∈ ∆ + Q ( q ) · E ± γ . . . E ± γ k Indeed, assume that there existed a decomposition α + β = γ + · · · + γ k for positive roots α < γ ≤ · · · ≤ γ k < β with k > 1. Then, as shown in the proof of Proposition 2.33, we can modifythe decomposition by clumping some of the γ r ’s together so as to ensure k = 2 (the resulting tworoots are still bounded by α and β , due to the convexity). This would contradict (5.9). , AND ALEXANDER TSYMBALIUK as well as: (5.12) U ± q ([ α, β ]) = k ∈ N (cid:77) α ≤ γ ≤···≤ γ k ≤ β ∈ ∆ + Q ( q ) · E ± γ . . . E ± γ k Proof. Let α = γ < γ < · · · < γ s = β be a complete list of positive roots γ ∈ ∆ + satisfying α ≤ γ ≤ β . First, let us note that the ordered monomials featuring inthe right-hand sides of (5.11) and (5.12) are linearly independent since they alreadyappeared as part of the basis of U q ( n ± ) in (5.6) and (5.3), respectively. Therefore,to prove (5.11) (resp. (5.12)), it suffices to show that any product E ± γ i . . . E ± γ ik can be reordered as a linear combination of such products with i ≥ · · · ≥ i k (resp. i ≤ · · · ≤ i k ). We prove this by induction primarily on M − m (where M = max i a and m = min i a ) and then secondarily on the total number of times M and m appear in the sequence i , . . . , i k . Indeed, formula (5.8) allows to move all the E ± γ M ’s to the left and all the E ± γ m ’s to the right (resp. all the E ± γ M ’s to the rightand all the E ± γ m ’s to the left), at the cost of gaining extra products E ± γ j . . . E ± γ jl to which the induction hypothesis applies. (cid:3) + , see Proposition 2.19. Proof of Theorem 4.8. Consider the anti-involution (cid:36) of U q ( g ) defined via: (cid:36) : e i (cid:55)→ f i , f i (cid:55)→ e i , ϕ ± i (cid:55)→ ϕ ± i for i ∈ I ; thus (cid:36) is a composition of (5.4, 5.5). Applying (cid:36) to (5.3), we obtain:(5.13) U q ( n ± ) = k ∈ N (cid:77) γ ≥···≥ γ k ∈ ∆ + Q ( q ) · (cid:36) ( E ∓ γ ) . . . (cid:36) ( E ∓ γ k )We claim that Theorem 4.8 follows from (5.13). To this end, it suffices to show:(5.14) e (cid:96) ( α ) ∈ Q ( q ) ∗ · (cid:36) ( E − α ) and f (cid:96) ( α ) ∈ Q ( q ) ∗ · (cid:36) ( E α )for any α ∈ ∆ + , where (cid:96) is the bijection (2.13). We prove (5.14) by induction onthe height of α . The base case α = α i (with i ∈ I ) is immediate, due to (5.7): e [ i ] = e i = (cid:36) ( f i ) = (cid:36) ( E − α i ) and f [ i ] = f i = (cid:36) ( e i ) = (cid:36) ( E α i )For the induction step, consider the factorization (2.6) of (cid:96) = (cid:96) ( α ): (cid:96) = (cid:96) (cid:96) Since factors of standard words are standard, we have (cid:96) = (cid:96) ( γ ) and (cid:96) = (cid:96) ( γ ) forsome γ , γ ∈ ∆ + such that α = γ + γ . By the induction hypothesis, we have: e (cid:96) k ∈ Q ( q ) ∗ · (cid:36) ( E − γ k ) and f (cid:96) k ∈ Q ( q ) ∗ · (cid:36) ( E γ k )for k ∈ { , } . However, by (the finite counterpart of) Proposition 2.37 and thedefinition (1.11), we note that γ < α < γ is a minimal decomposition in the senseof (5.9). Therefore, comparing (4.9) (and its f -analogue) with (5.10), we obtain: e (cid:96) = [ e (cid:96) , e (cid:96) ] q ∈ Q ( q ) ∗ · (cid:36) ([ E − γ , E − γ ] q ) = Q ( q ) ∗ · (cid:36) ( E − α ) f (cid:96) = [ f (cid:96) , f (cid:96) ] q ∈ Q ( q ) ∗ · (cid:36) ([ E γ , E γ ] q ) = Q ( q ) ∗ · (cid:36) ( E α )as we needed to prove. (cid:3) UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 43 Definition 5.7. Let U q ( (cid:98) g ) be as in Definition 4.2, but using (cid:98) I instead of I . Letting U q ( (cid:98) n + ) , U q ( (cid:98) h ) , U q ( (cid:98) n − ) be the subalgebras generated by the e i ’s, ϕ ± i ’s, f i ’s,respectively (with i ∈ (cid:98) I ), we obtain a triangular decomposition analogous to (4.4):(5.15) U q ( (cid:98) g ) = U q ( (cid:98) n + ) ⊗ U q ( (cid:98) h ) ⊗ U q ( (cid:98) n − )We will also consider the following sub-bialgebras of U q ( (cid:98) g ): U q ( (cid:98) b + ) = U q ( (cid:98) n + ) ⊗ U q ( (cid:98) h ) U q ( (cid:98) b − ) = U q ( (cid:98) h ) ⊗ U q ( (cid:98) n − )The algebra U q ( (cid:98) g ) is (cid:98) Q (cid:39) Q × Z -graded via:deg e = α = ( − θ, 1) deg e i = α i = ( α i , f = − α = ( θ, − 1) deg f i = − α i = ( − α i , ϕ = 0 = (0 , 0) deg ϕ i = 0 = (0 , i ∈ I , where θ is the highest root of ∆ + , and (cid:98) Q is identified with Q × Z via (3.1).Sending e i (cid:55)→ e i , f i (cid:55)→ f i , ϕ ± i (cid:55)→ ϕ ± i for i ∈ I yields an algebra homomorphism: U q ( g ) (cid:44) −→ U q ( (cid:98) g )of Q × Z -graded algebras, where the Z -grading on U q ( g ) is set to be trivial. Finally,we observe the fact that the element:(5.16) C = ϕ (cid:89) i ∈ I ϕ θ i i is central in U q ( (cid:98) g ), where the positive integers { θ i } i ∈ I were introduced in (3.4).Note that C of (5.16) is to c of (3.5) as { ϕ i } i ∈ (cid:98) I are to { h i } i ∈ (cid:98) I .5.8. Let us now recall, following [2], the affine version of the construction of theroot vectors from Subsection 5.1. Following Subsection 3.10, pick any µ ∈ P ∨ suchthat ( α i , µ ) > i ∈ I and consider (cid:98) µ = 1 (cid:110) µ ∈ (cid:99) W ext . Let l = l ( (cid:98) µ ) = (2 ρ, µ )be the length of (cid:98) µ (Proposition 3.9) and consider any reduced decomposition: (cid:98) µ = τ s i − l s i − l . . . s i as in (3.29) with (a uniquely determined) τ ∈ T . Following (3.30), we extend { i k | − l < k ≤ } to a ( τ -quasiperiodic) bi-infinite sequence { i k } k ∈ Z via: i k + l = τ ( i k ) , ∀ k ∈ Z By analogy with (3.31), we may construct the following set of positive affine roots:(5.17) ˜ β k = (cid:40) s i s i . . . s i k − ( α i k ) if k > s i s i − . . . s i k +1 ( α i k ) if k ≤ , AND ALEXANDER TSYMBALIUK which are related to the roots β k of (3.31) via:(5.18) ˜ β k = (cid:40) − β k if k > β k if k ≤ β < ˜ β − < ˜ β − < ˜ β − < · · · < ˜ β < ˜ β < ˜ β < ˜ β Remark . Formula (5.17) provides all real positive roots of the affine root system:(5.20) (cid:98) ∆ re , + = (cid:110) ∆ + × Z ≥ (cid:111) (cid:116) (cid:110) ∆ − × Z > (cid:111) ⊂ (cid:98) ∆ + Furthermore, (5.19) induces convex orders on the corresponding halves:∆ + × Z ≥ = (cid:110) ˜ β < ˜ β − < ˜ β − < . . . (cid:111) and ∆ − × Z > = (cid:110) · · · < ˜ β < ˜ β < ˜ β (cid:111) To have a complete theory, one also needs to deal with the imaginary roots: (cid:98) ∆ im , + = (cid:110) × Z > (cid:111) ⊂ (cid:98) ∆ + but they will not feature in the present paper.5.10. We may define the “root vectors”:(5.21) E ± ˜ β ∈ U q ( (cid:98) n ± )for all ˜ β ∈ (cid:98) ∆ re , + of (5.20) via the following analogue of (5.2):(5.22) E ˜ β k = (cid:40) T i . . . T i k − ( e i k ) if k > T − i . . . T − i k +1 ( e i k ) if k ≤ E − ˜ β k = (cid:40) T i . . . T i k − ( f i k ) if k > T − i . . . T − i k +1 ( f i k ) if k ≤ { T i } i ∈ (cid:98) I determine Lusztig’s affine braid group action [30] on U q ( (cid:98) g ). Remark . (a) The above construction applied to (cid:98) µ ∈ (cid:99) W ext is equivalent to thatof [2] applied to (cid:98) x ∈ (cid:99) W for a multiple x = rµ ∈ Q ∨ with r ∈ N , see Remark 3.11.(b) We also note that [2] defines root vectors E − ˜ β ∈ U q ( (cid:98) n − ) for ˜ β ∈ (cid:98) ∆ re , + via:(5.24) E − ˜ β := Ω( E ˜ β )where the Q -algebra anti-involution Ω of U q ( (cid:98) g ) is defined via:(5.25) Ω : e i (cid:55)→ f i , f i (cid:55)→ e i , ϕ ± i (cid:55)→ ϕ ∓ i , q (cid:55)→ q − , ∀ i ∈ (cid:98) I Formulas (5.23) and (5.24) agree, as Ω commutes with the affine braid group action:(5.26) Ω ◦ T i = T i ◦ Ω , ∀ i ∈ (cid:98) I UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 45 Due to [2, Proposition 7], the root vectors satisfy the natural analogue of (5.8): (5.27) E ± ˜ β E ± ˜ α − q (˜ α, ˜ β ) E ± ˜ α E ± ˜ β ∈ k ∈ N (cid:77) ˜ α< ˜ γ ≤···≤ ˜ γ k < ˜ β ˜ γ + ··· +˜ γ k =˜ α + ˜ β Q ( q ) · E ± ˜ γ . . . E ± ˜ γ k for any real positive affine roots ˜ α < ˜ β which both belong to either ∆ + × Z ≥ or∆ − × Z > . Therefore, due to the convexity of the corresponding orders of ∆ + × Z ≥ or ∆ − × Z > (Remark 5.9), we also have the following analogue of (5.10):(5.28) [ E ± ˜ β , E ± ˜ α ] q = E ± ˜ β E ± ˜ α − q (˜ α, ˜ β ) E ± ˜ α E ± ˜ β ∈ Q ( q ) ∗ · E ± (˜ α + ˜ β ) for any real positive affine roots ˜ α < ˜ β as above, which have the additional propertythat ˜ α + ˜ β is a positive affine root whose decomposition as the sum of ˜ α and ˜ β isminimal in the sense that:(5.29) (cid:54) ∃ ˜ α (cid:48) , ˜ β (cid:48) ∈ (cid:98) ∆ re , + s.t. ˜ α < ˜ α (cid:48) < ˜ β (cid:48) < ˜ β and ˜ α + ˜ β = ˜ α (cid:48) + ˜ β (cid:48) Remark . According to [14, Theorem 4.8], there is an explicit subring R of Q ( q )which admits a q = 1 specialization, such that the Lusztig R -form U R ( (cid:98) g ) of U q ( (cid:98) g )admits a natural PBW basis. Moreover, the q = 1 specialization gives rise to:(5.30) U R ( (cid:98) g ) / ( q − ∼ −→ U ( (cid:98) g ) with E ± ˜ β (cid:55)→ E ± ˜ β the latter denoting Chevalley generators of (cid:98) g , see [14, formulas (5.5, 5.7)]. Thisimplies that [ E ± ˜ β , E ± ˜ α ] q ∈ R ∗ · E ± (˜ α + ˜ β ) under the same assumptions as in (5.28). Amore detailed analysis of [14] shows that for real root vectors one can replace R with Z [ q, q − ], hence the following refinement of (5.28), under the same assumptions:(5.31) [ E ± ˜ β , E ± ˜ α ] q ∈ Z [ q, q − ] ∗ · E ± (˜ α + ˜ β ) As a consequence of (5.28), we obtain the following. Corollary 5.13. (a) The root vectors { E ± ˜ β } ˜ β ∈ ∆ + × Z ≥ can be obtained (up to non-zero scalars) as iterated q -commutators of the root vectors: { E ± ( α i ,d ) } d ≥ i ∈ I (b) The root vectors { E ± ˜ β } ˜ β ∈ ∆ − × Z > can be obtained (up to non-zero scalars) asiterated q -commutators of the root vectors: { E ± ( − α i ,d ) } d> i ∈ I and { E ± ( − α, } α ∈ ∆ + (c) The root vectors { E ± ( − α, } α ∈ ∆ + can be obtained (up to non-zero scalars) asiterated q -commutators of the root vectors: { E ± ( α i , } i ∈ I and E ± ( − θ, where θ ∈ ∆ + denotes the highest root as before. Parts (a) and (b) follow readily from the combinatorics of ∆ ± . Part (c) followsfrom the convexity [2, Corollary 4] of the entire PBW basis of U q ( (cid:98) n ± ). See Remark 5.16 for the unexpected ordering of the ˜ γ ’s when the sign ± is − . , AND ALEXANDER TSYMBALIUK Lemma 5.15. For all < s ≤ r or s ≤ r ≤ , let U ± q ([ s, r ]) be the subalgebra of U q ( (cid:98) n ± ) generated by { E ± ˜ β k | s ≤ k ≤ r } . Then: (5.32) U ± q ([ s, r ]) = (cid:77) n s ,n s +1 ,...,n r − ,n r ∈ Z ≥ Q ( q ) · E n s ± ˜ β s E n s +1 ± ˜ β s +1 . . . E n r − ± ˜ β r − E n r ± ˜ β r as well as: (5.33) U ± q ([ s, r ]) = (cid:77) n s ,n s +1 ,...,n r − ,n r ∈ Z ≥ Q ( q ) · E n r ± ˜ β r E n r − ± ˜ β r − . . . E n s +1 ± ˜ β s +1 E n s ± ˜ β s Remark . We should note right away that Lemma 5.15 has been implicitly usedin (5.27), since applying Ω of (5.25) to [2, Proposition 7] one actually obtains:(5.34) E − ˜ β E − ˜ α − q (˜ α, ˜ β ) E − ˜ α E − ˜ β ∈ (cid:77) ˜ β> ˜ γ ≥···≥ ˜ γ k > ˜ α ˜ γ + ··· +˜ γ k =˜ α + ˜ β Q ( q ) · E − ˜ γ . . . E − ˜ γ k under the same assumptions as in (5.27). Therefore, we need the equivalence of(5.32) and (5.33) to convert (5.34) into (5.27). Proof of Lemma 5.15. The fact that the ordered monomials featuring in the right-hand sides of (5.32) or (5.33) span U ± q ([ s, r ]) is proved exactly as in our proof ofLemma 5.4. Meanwhile, their linear independence follows from the usual PBWtheorem for U ( (cid:98) g ) in view of (5.30). (cid:3) We shall also need the limit cases of (5.32, 5.33) when ( s, r ) = ( −∞ , 0) or (1 , + ∞ ).To this end, let U ± q (+ ∞ ) and U ± q ( −∞ ) denote the subalgebras of U q ( (cid:98) n ± ) generatedby { E ± ˜ β k | k ≥ } and { E ± ˜ β k | k ≤ } , respectively. In accordance with (5.32, 5.33),each of these subalgebras admits a pair of opposite PBW decompositions: U ± q (+ ∞ ) = (cid:77) n ,n , ···∈ Z ≥ n + n + ··· < ∞ Q ( q ) · E n ± ˜ β E n ± ˜ β . . . = (cid:77) n ,n , ···∈ Z ≥ n + n + ··· < ∞ Q ( q ) · . . . E n ± ˜ β E n ± ˜ β (5.35) U ± q ( −∞ ) = (cid:77) n ,n − , ···∈ Z ≥ n + n − + ··· < ∞ Q ( q ) · E n ± ˜ β E n − ± ˜ β − . . . = (cid:77) n ,n − , ···∈ Z ≥ n + n − + ··· < ∞ Q ( q ) · . . . E n − ± ˜ β − E n ± ˜ β (5.36) UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 47 (cid:98) I instead of I ). Followingthe notations of Lemma 5.15, loc. cit. defines the subalgebra U ± q ( r ) of U q ( (cid:98) n ± ) via: U ± q ( r ) = (cid:40) U ± q ([ r, r ≤ U ± q ([1 , r ]) for r ≥ z of degree (cid:80) i ∈ (cid:98) I r i α i ∈ (cid:98) Q , we set:(5.37) ϕ deg( z ) := ϕ (cid:80) i ∈ (cid:98) I r i α i = (cid:89) i ∈ (cid:98) I ϕ r i i ∈ U q ( (cid:98) h )According to [5, Proposition 7.1.2] we have:(5.38) ∆ (cid:16) E ˜ β r (cid:17) = ϕ ˜ β r ⊗ E ˜ β r + E ˜ β r ⊗ (cid:88) ϕ deg( y ) x ⊗ y where the last sum is vacuous for r = 0 , − 1, while otherwise it is restricted by: x ∈ U + q ( r − 1) and y ∈ U q ( (cid:98) n + ) for r > x ∈ U q ( (cid:98) n + ) and y ∈ U + q ( r + 1) for r < op :∆ ◦ Ω = (Ω ⊗ Ω) ◦ ∆ op Therefore, applying Ω to (5.38, 5.39), we obtain:(5.40) ∆ (cid:16) E − ˜ β r (cid:17) = 1 ⊗ E − ˜ β r + E − ˜ β r ⊗ ϕ − ˜ β r + (cid:88) y ⊗ ϕ deg( y ) x where the last sum is vacuous for r = 0 , − 1, while otherwise it is restricted by: x ∈ U − q ( r − 1) and y ∈ U q ( (cid:98) n − ) for r > x ∈ U q ( (cid:98) n − ) and y ∈ U − q ( r + 1) for r < µ = ρ ∨ Our interest in the Drinfeld-Jimbo affine quantum groups of Definition 5.7 is mo-tivated by the following connection with Drinfeld’s new presentation of quantumloop groups of Definition 4.18 (due to [2, 3, 6]): Theorem 5.19. There exists an algebra isomorphism: (5.42) U q ( L g ) ∼ −→ U q ( (cid:98) g ) / ( C − determined by the following assignment for all i ∈ I and d ∈ Z : e i,d (cid:55)→ (cid:40) o ( i ) d E ( α i ,d ) if d ≥ − o ( i ) d E ( α i ,d ) ϕ − i if d < f i,d (cid:55)→ (cid:40) − o ( i ) d ϕ i E ( − α i ,d ) if d > o ( i ) d E ( − α i ,d ) if d ≤ where o : I → {± } is a map satisfying o ( i ) o ( j ) = − whenever a ij < . The inverse of (5.42) was provided (without a proof) earlier in [9, Theorem 3]. , AND ALEXANDER TSYMBALIUK Proof. The isomorphism (5.42) was proved in [3, Theorem 4.7] with respect to thefollowing seemingly different formula:(5.44) e i,d (cid:55)→ o ( i ) d T − d (cid:99) ω ∨ i ( e i ) , f i,d (cid:55)→ o ( i ) d T d (cid:99) ω ∨ i ( f i )Here, the aforementioned action of the affine braid group on U q ( (cid:98) g ) has been ex-tended to the extended affine braid group by adding automorphisms { T τ } τ ∈T : T τ : e i (cid:55)→ e τ ( i ) , f i (cid:55)→ f τ ( i ) , ϕ ± i (cid:55)→ ϕ ± τ ( i ) , ∀ τ ∈ T , i ∈ (cid:98) I which satisfy the following relations: T τ T i = T τ ( i ) T τ , ∀ τ ∈ T , i ∈ (cid:98) I Therefore, it remains for us to show that (5.43) is equivalent to (5.44) by proving:(5.45) T − d (cid:99) ω ∨ i ( e i ) = (cid:40) E ( α i ,d ) if d ≥ − E ( α i ,d ) ϕ − i if d < T d (cid:99) ω ∨ i ( f i ) = (cid:40) − ϕ i E ( − α i ,d ) if d > E ( − α i ,d ) if d ≤ i ∈ I . According to our proof of Theorem 3.14, there is − l < k ≤ β k = ( α i , E ( α i , = E ˜ β k = e i , E ( − α i , = E − ˜ β k = Ω( E ˜ β k ) = Ω( e i ) = f i thus verifying (5.45, 5.46) for d = 0. On the other hand, we note that:˜ β k − dl = ( α i , d ) , ∀ d > β k − dl = (cid:99) ρ ∨− τ s i − l . . . s i k − dl +1 ( α i k − dl ) = (cid:99) ρ ∨− s τ ( i − l ) . . . s τ ( i k − dl +1 ) ( α τ ( i k − dl ) ) = (cid:99) ρ ∨− s i . . . s i k − ( d − l +1 ( α i k − ( d − l ) = · · · = (cid:99) ρ ∨− d s i . . . s i k +1 ( α i k ) = (cid:99) ρ ∨− d ( ˜ β k ) = ( α i , d )with the last equality due to (3.20). Then, the same argument verifies: E ˜ β k − dl = T − (cid:99) ρ ∨ T τ T − i − l . . . T − i k − dl +1 ( e i k − dl ) = T − (cid:99) ρ ∨ T − τ ( i − l ) . . . T − τ ( i k − dl +1 ) ( e τ ( i k − dl ) ) = T − (cid:99) ρ ∨ T − i . . . T − i k − ( d − l +1 ( e i k − ( d − l ) = · · · = T − d (cid:99) ρ ∨ T − i . . . T − i k +1 ( e i k ) = T − d (cid:99) ρ ∨ ( E ˜ β k )To simplify the latter, we note that ρ ∨ = (cid:80) j ∈ I ω ∨ j and l ( (cid:99) ρ ∨ ) = (cid:80) j ∈ I l ( (cid:99) ω ∨ j ), due toProposition 3.9. Therefore, T (cid:99) ρ ∨ may be evaluated by using a reduced decompositionof (cid:99) ρ ∨ obtained as a concatenation of the reduced decompositions of (cid:99) ω ∨ j ’s, hence: T − d (cid:99) ρ ∨ = T − d (cid:99) ω ∨ i · j (cid:54) = i (cid:89) j ∈ I T − d (cid:99) ω ∨ j As T ± (cid:99) ω ∨ j ( e i ) = e i for j (cid:54) = i by [3, Corollary 3.2], and E ˜ β k = e i by (5.47), we get:(5.48) E ( α i ,d ) = E ˜ β k − dl = T − d (cid:99) ρ ∨ ( E ˜ β k ) = T − d (cid:99) ρ ∨ ( e i ) = T − d (cid:99) ω ∨ i ( e i ) UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 49 which proves (5.45) for d > 0. Furthermore, as Ω commutes with the extendedaffine braid group action (due to (5.26) and Ω ◦ T τ = T τ ◦ Ω for τ ∈ T ), we obtain: E ( − α i , − d ) = E − ˜ β k − dl = Ω( E ˜ β k − dl ) = Ω( T − d (cid:99) ω ∨ i ( e i )) = T − d (cid:99) ω ∨ i (Ω( e i )) = T − d (cid:99) ω ∨ i ( f i )which proves (5.46) for d < 0. For the remaining cases, let us note first that(5.49) ˜ β k + dl = ( − α i , d ) , ∀ d > d = 1 case of (5.49) follows from:(5.50) ( α i , − 1) = (cid:99) ρ ∨ ( α i , 0) = (cid:99) ρ ∨ ( ˜ β k ) = s i . . . s i l τ s i . . . s i k +1 ( α i k ) = s i . . . s i l s i l . . . s i l + k +1 ( α i k + l ) = s i . . . s i k + l − s i k + l ( α i k + l ) = − ˜ β k + l while the d > β k + dl = (cid:99) ρ ∨ τ − s i l +1 . . . s i k + dl − ( α i k + dl ) = (cid:99) ρ ∨ s τ − ( i l +1 ) . . . s τ − ( i k + dl − ) ( α τ − ( i k + dl ) )= (cid:99) ρ ∨ s i . . . s i k +( d − l − ( α i k +( d − l ) = · · · = (cid:99) ρ ∨ d − ˜ β k + l = (cid:99) ρ ∨ d − ( − α i , 1) = ( − α i , d )with the last equality due to (3.20). Using the same arguments as before, we obtain:(5.51) T (cid:99) ρ ∨ ( e i ) = T (cid:99) ρ ∨ ( E ˜ β k ) = T i . . . T i l T τ T − i . . . T − i k +1 ( e i k ) = T i . . . T i k + l − T i k + l ( e i k + l ) = T i . . . T i k + l − ( − f i k + l ϕ i k + l ) = − E − ˜ β k + l ϕ − i = − E ( α i , − ϕ − i where we used T j ( e j ) = − f j ϕ j (for any j ∈ (cid:98) I ) as well as (using notation (5.37)): T i . . . T i k + l − ϕ i k + l = ϕ s i ...s ik + l − ( α ik + l ) = ϕ ˜ β k + l = ϕ ( − α i , = ϕ − i with the last equality due to C = 1. Likewise, for any d > E ( α i , − d ) = E − ˜ β k + dl = T i . . . T i k + dl − ( f i k + dl ) = T (cid:99) ρ ∨ T − τ T i l +1 . . . T i k + dl − ( f i k + dl ) = T (cid:99) ρ ∨ T i . . . T i k +( d − l − ( f i k +( d − l ) = · · · = T d − (cid:99) ρ ∨ T i . . . T i k + l − ( f i k + l ) = − T d (cid:99) ρ ∨ ( e i ) ϕ i with the last equality due to (5.51) and C = 1. Combining (5.51, 5.52) with theequality T d (cid:99) ρ ∨ ( e i ) = T d (cid:99) ω ∨ i ( e i ), already established as part of (5.48), verifies (5.45) for d < 0. Then, we also get: T d (cid:99) ω ∨ i ( f i ) = T d (cid:99) ω ∨ i (Ω( e i )) = Ω( − E ( α i , − d ) ϕ − i ) = − ϕ i Ω( E ( α i , − d ) ) = − ϕ i E ( − α i ,d ) which verifies (5.46) for d > 0. This completes our proof of Theorem 5.19. (cid:3) Remark . The two formulas of (5.43) are compatible with each other, inthat (5.42) intertwines (5.25) with the following Q -algebra anti-involution Ω L of U q ( L g ):(5.53) Ω L : e i,k (cid:55)→ f i, − k , f i,k (cid:55)→ e i, − k , ϕ ± i,l (cid:55)→ ϕ ∓ i,l , q (cid:55)→ q − for any i ∈ I, k ∈ Z , l ∈ Z ≥ . , AND ALEXANDER TSYMBALIUK Remark . The isomorphism (5.42) can be upgraded to a generic C , if oneintroduces the corresponding central element in the definition of the quantum loopgroup of Definition 4.18 (which we choose not to do).Notably, the isomorphism (5.42) does not intertwine the triangular decompositions(4.32) and (5.15). In fact, if we think of U q ( L g ) and U q ( (cid:98) g ) / ( C − 1) as one and thesame algebra, then these two decompositions are “orthogonal” to each other, as thefollowing picture suggests. (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 ( (cid:98) n − ) { U q ( (cid:98) n + ) } U q ( L n − ) } U q ( L n + ) N − N Q + Q − F IGURE 1. The grading of U q ( L g ) (cid:39) U q ( (cid:98) g ) / ( C − 1) and its various subalgebrasThe axes in the picture above describe the two components of Q × Z , and the foursubalgebras marked by parentheses indicate the degrees in which elements of thesesubalgebras lie (although U q ( (cid:98) n + ) and U q ( (cid:98) n − ) also include elements on the positiveand negative horizontal axes, respectively, namely products of e i and f i for i ∈ I ).5.22. The picture in the previous Subsection suggests that we can further obtaintriangular decompositions of the “half” subalgebras in terms of “quarter” subalge-bras. Specifically, it was shown in [2, Lemma 5, Proof of Lemma 6] that: U + q ( L n − ) := U q ( L n − ) ∩ U q ( (cid:98) b + ) = (cid:110) subalgebra generated by e ˜ β k , k > (cid:111) (5.54) U + q ( L n + ) := U q ( L n + ) ∩ U q ( (cid:98) b + ) = (cid:110) subalgebra generated by e ˜ β k , k ≤ (cid:111) (5.55)where we define e ˜ β k in accordance with (5.43) via:(5.56) e ˜ β k = (cid:40) ϕ − hdeg( ˜ β k ) E ˜ β k if k > E ˜ β k if k ≤ We note that [2, Lemma 5] treats U q ( (cid:98) n + ) , U q ( (cid:98) n − ) in place of U q ( (cid:98) b + ) , U q ( (cid:98) b − ) and A > , A < inplace of U q ( L n + ) , U q ( L n − ), respectively. However, this does not affect the equalities (5.54, 5.55),since A > (resp. A < ) is obtained from U q ( L n + ) (resp. U q ( L n − )) by adding negative (resp. positive)imaginary root vectors as well as U q ( h ), as follows from [2, Proof of Lemma 6]. UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 51 Henceforth, given a homogeneous element z of degree (cid:0)(cid:80) i ∈ I r i α i , d (cid:1) ∈ Q × Z , set(5.57) ϕ ± hdeg( z ) := ϕ ± (cid:80) i ∈ I r i α i = (cid:89) i ∈ I ϕ ± r i i ∈ U q ( L h )As C = 1, we note that (5.37) and (5.57) agree under the identification (5.42):(5.58) ϕ hdeg( z ) = ϕ deg( z ) Formulas (5.27, 5.28) still hold when the E ˜ β k are replaced with the e ˜ β k , sincecommuting ϕ ’s simply produces powers of q . Likewise, the PBW decomposi-tions (5.35, 5.36) imply that the subalgebras above have the following PBW bases: U + q ( L n − ) = (cid:77) n ,n , ···∈ Z ≥ n + n + ··· < ∞ Q ( q ) · . . . e n ˜ β e n ˜ β (5.59) U + q ( L n + ) = (cid:77) n ,n − , ···∈ Z ≥ n + n − + ··· < ∞ Q ( q ) · . . . e n − ˜ β − e n ˜ β (5.60)Applying the anti-involutions (5.25, 5.53), compatible via Remark 5.20, we see thatthe analogous “quarter” subalgebras of U q ( (cid:98) b − ) have similar PBW decompositions: U − q ( L n − ) := U q ( L n − ) ∩ U q ( (cid:98) b − ) = (cid:77) n ,n − , ···∈ Z ≥ n + n − + ··· < ∞ Q ( q ) · e n − ˜ β e n − − ˜ β − . . . (5.61) U − q ( L n + ) := U q ( L n + ) ∩ U q ( (cid:98) b − ) = (cid:77) n ,n , ···∈ Z ≥ n + n + ··· < ∞ Q ( q ) · e n − ˜ β e n − ˜ β . . . (5.62)where we define:(5.63) e − ˜ β k = Ω( e ˜ β k ) = (cid:40) E − ˜ β k ϕ hdeg( ˜ β k ) if k > E − ˜ β k if k ≤ U q ( L n ± ). Proposition 5.23. The multiplication map induces a vector space isomorphism: (5.64) U + q ( L n − ) ⊗ U − q ( L n − ) ∼ −→ U q ( L n − ) Proof. The triangular decomposition (5.15) implies the injectivity of the map (5.64).It thus suffices to show that: U + q ( L n − ) ⊗ U − q ( L n − )is an algebra, since the fact that it contains all the generators of U q ( L n − ) (namely, f i,d ’s with i ∈ I, d ∈ Z , due to (5.43)) will imply the surjectivity of the map (5.64).By definition, U + q ( L n − ) and U − q ( L n − ) are subalgebras, so it remains to show that:(5.65) ba ∈ U + q ( L n − ) ⊗ U − q ( L n − )for any a ∈ U + q ( L n − ) and b ∈ U − q ( L n − ). Products such as (5.65) are governed byformula (4.8), with respect to the Drinfeld-Jimbo coproduct ∆ of U q ( (cid:98) g ). , AND ALEXANDER TSYMBALIUK In accordance with (5.38)–(5.41) and (5.58), for r ≥ s ≤ 0, we thus obtain:(5.66) ∆ (cid:16) e ˜ β r (cid:17) = 1 ⊗ e ˜ β r + e ˜ β r ⊗ ϕ − ˜ β r + (cid:88) ϕ − hdeg( x ) x ⊗ ϕ − hdeg( ˜ β r ) y with x ∈ U + q ( r − 1) and y ∈ U q ( (cid:98) n + )(5.67) ∆ (cid:16) e − ˜ β s (cid:17) = 1 ⊗ e − ˜ β s + e − ˜ β s ⊗ ϕ − ˜ β s + (cid:88) y ⊗ ϕ hdeg( y ) x with x ∈ U q ( (cid:98) n − ) and y ∈ U − q ( s + 1)where the sums of (5.66) and (5.67) are vacuous for r = 1 and s = 0, respectively.Therefore, for a ∈ U + q ( L n − ) and b ∈ U − q ( L n − ) we have∆( a ) = a ⊗ a where a ∈ U + q ( L n − ) and a ∈ U q ( (cid:98) b + )∆( b ) = b ⊗ b where b ∈ U − q ( L n − ) and b ∈ U q ( (cid:98) b − )Because the affine version of the pairing (4.5) pairs trivially elements that do notsit in opposite Q × Z -degrees, we have: (cid:104) a, b (cid:105) = ε ( a ) ε ( b )for all a ∈ U + q ( L n − ) and b ∈ U − q ( L n − ) (here, ε is the counit of the bialgebra U q ( (cid:98) g )).Therefore, (4.8) implies:(5.68) a b (cid:104) a , b (cid:105) = (cid:104) a , b (cid:105) b a = ε ( a ) ε ( b ) b a = ba (the latter identity is part of the counit property of ε ) thus implying (5.65). (cid:3) β k of (5.17) to β k of (3.31) via (5.18), so that the subalgebras U + q ( L n − ) and U − q ( L n − ) are generatedby { e − β k } k ≥ and { e − β k } k ≤ , respectively. Then, combining the above Propositionwith the PBW decompositions (5.59, 5.61), we obtain the PBW basis for U q ( L n − ):(5.69) U q ( L n − ) = (cid:77) ...,n − ,n ,n ,n , ···∈ Z ≥ ··· + n − + n + n + n + ··· < ∞ Q ( q ) · . . . e n − β e n − β e n − β e n − − β − . . . We note that the set {− β k } k ∈ Z exactly coincides with ∆ − × Z .For any r ≥ 1, let U + q ( L n − )[ r ] be the subalgebra of U + q ( L n − ) generated by e − β , . . . , e − β r . Likewise, for s ≤ 0, let U − q ( L n − )[ s ] be the subalgebra of U − q ( L n − )generated by e − β , . . . , e − β s , that is, the subalgebra U − q ([ s, U + q ( L n − )[ r ] = (cid:77) n ,n ,...,n r ∈ Z ≥ Q ( q ) · e n r − β r . . . e n − β e n − β (5.70) U − q ( L n − )[ s ] = (cid:77) n ,n − ,...,n s ∈ Z ≥ Q ( q ) · e n − β e n − − β − . . . e n s − β s (5.71)Just as in the proof of Proposition 5.23, we obtain the following analogue of (5.27). UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 53 Proposition 5.25. For any s ≤ < r , we have e − β s e − β r − q ( β s ,β r ) e − β r e − β s ∈ (cid:77) n r − ,...,n s +1 ∈ Z ≥ Q ( q ) · e n r − − β r − . . . e n − β e n − β . . . e n s +1 − β s +1 where the sum is finite as it is taken over all tuples n r − , . . . , n s +1 ∈ Z ≥ such that: n r − β r − + · · · + n β + n β + · · · + n s +1 β s +1 = β r + β s Proof. This follows from (5.68) applied to the pair a = e − β r and b = e − β s , combinedwith (5.66, 5.67) and the above PBW decompositions (5.70, 5.71). (cid:3) In the simplest case β r = ( α, − , β s = ( α i , Corollary 5.26. [ e ( − α i , , e ( − α, ] q ∈ Q ( q ) ∗ · e ( − α − α i , if α, α + α i ∈ ∆ + .Remark . The above Corollary is in contrast with the better known approachwhich recovers E ( − α, via q -commutators of E ( − θ, and e i , see Corollary 5.13(c).5.28. We shall now see that Theorem 4.24 is equivalent to the PBW decomposi-tion (5.69) applied to the reduced decomposition of (cid:99) ρ ∨ produced by Theorem 3.14.The key feature of this reduced decomposition is that the ordered set of roots:(5.72) · · · < β < β < β < β − < . . . coincides with ∆ + × Z ordered in accordance with the bijection (2.34) via:(5.73) · · · < (cid:96) ( β ) < (cid:96) ( β ) < (cid:96) ( β ) < (cid:96) ( β − ) < . . . where for any ( α, d ) ∈ ∆ + × Z we set:(5.74) ( α, d ) = ( α, − d ) Proof of Theorem 4.24. Our proof will closely follow that of Theorem 4.8. In par-ticular, we shall need the anti-involution (cid:36) of U q ( L g ) defined via: (cid:36) : e i,k (cid:55)→ f i,k , f i,k (cid:55)→ e i,k , ϕ ± i,l (cid:55)→ ϕ ± i,l for any i ∈ I , k ∈ Z , l ∈ N , which should viewed as the loop version of (cid:36) for U q ( g ).Applying (cid:36) to (5.69), we obtain:(5.75) U q ( L n + ) = k ∈ N (cid:77) γ ≥···≥ γ k ∈ ∆ + × Z Q ( q ) · (cid:36) ( e − γ ) . . . (cid:36) ( e − γ k )with the above order on ∆ + × Z being (5.72). On the other hand, combining thecompatibility of (5.72, 5.73) with Proposition 5.25 and formula (5.27), we obtain:[ e − β , e − β (cid:48) ] q ∈ k ∈ N (cid:77) (cid:96) ( β (cid:48) ) <(cid:96) ( γ ) ≤···≤ (cid:96) ( γ k ) <(cid:96) ( β ) γ + ··· + γ k = β + β (cid:48) Q ( q ) · e − γ . . . e − γ k , AND ALEXANDER TSYMBALIUK for any β, β (cid:48) ∈ ∆ + × Z such that β (cid:48) < β , or equivalently (cid:96) ( β (cid:48) ) < (cid:96) ( β ). Applyingthe anti-involution (cid:36) to the equation above, we get:(5.76) [ (cid:36) ( e − β (cid:48) ) , (cid:36) ( e − β )] q ∈ k ∈ N (cid:77) (cid:96) ( β ) >(cid:96) ( γ ) ≥···≥ (cid:96) ( γ k ) >(cid:96) ( β (cid:48) ) γ + ··· + γ k = β + β (cid:48) Q ( q ) · (cid:36) ( e − γ ) . . . (cid:36) ( e − γ k )under the same restrictions on β, β (cid:48) ∈ ∆ + × Z . In particular, if β + β (cid:48) ∈ ∆ + × Z and β, β (cid:48) are minimal in the sense:(5.77) (cid:54) ∃ α, α (cid:48) ∈ ∆ + × Z s.t. β (cid:48) < α (cid:48) < α < β and α + α (cid:48) = β + β (cid:48) we get (due to the convexity of Proposition 2.33):[ (cid:36) ( e − β (cid:48) ) , (cid:36) ( e − β )] q ∈ Q ( q ) · (cid:36) ( e − β − β (cid:48) )Using the arguments of Remark 5.12, the formula above can be further refined to:(5.78) [ (cid:36) ( e − β (cid:48) ) , (cid:36) ( e − β )] q ∈ Z [ q, q − ] ∗ · (cid:36) ( e − β − β (cid:48) )We claim that Theorem 4.24 follows from (5.75). To this end, it suffices to show:(5.79) e (cid:96) ( β ) ∈ Q ( q ) ∗ · (cid:36) ( e − β )for any β = ( α, d ) ∈ ∆ + × Z . We prove (5.79) by induction on the height of α . Thebase case α = α i (with i ∈ I ) is immediate, due to (5.43, 5.56, 5.63): e [ i ( d ) ] = e i,d = (cid:36) ( f i,d ) = ± (cid:36) ( e ( − α i ,d ) )For the induction step, consider the factorization (2.6) of (cid:96) = (cid:96) ( α, d ): (cid:96) = (cid:96) (cid:96) Since factors of standard loop words are standard, we have (cid:96) = (cid:96) ( γ , d ) and (cid:96) = (cid:96) ( γ , d ) for some ( γ , d ) , ( γ , d ) ∈ ∆ + × Z such that α = γ + γ , d = d + d .By the induction hypothesis, we have: e (cid:96) k ∈ Q ( q ) ∗ · (cid:36) ( e ( − γ k ,d k ) )for k ∈ { , } . However, we note that ( γ , d ) < ( α, d ) < ( γ , d ) is a minimaldecomposition in the sense of (5.77), according to Proposition 2.37. Therefore,comparing (4.36) with (5.78), we obtain: e (cid:96) = [ e (cid:96) , e (cid:96) ] q ∈ Q ( q ) ∗ · (cid:36) ([ e ( − γ ,d ) , e ( − γ ,d ) ] q ) = Q ( q ) ∗ · (cid:36) ( e ( − α,d ) )as we needed to prove. (cid:3) Feigin-Odesskii shuffle algebras In the present Section, we will connect the loop shuffle algebra F L with thetrigonometric degeneration of the Feigin-Odesskii elliptic shuffle algebra associatedwith g , with the goal of establishing Theorem 1.8. UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 55 g . Consider the vector space of color-symmetric ratio-nal functions:(6.1) V = (cid:77) k = (cid:80) i ∈ I k i α i ∈ Q + Q ( q )( . . . , z i , . . . , z ik i , . . . ) Sym i ∈ I The index i ∈ I will be called the color of the variables z i , . . . , z ik i . The termcolor-symmetric (as well as the superscript “Sym” in the formula above) refers torational functions which are symmetric in the variables of each color separately.We make the vector space V into a Q ( q )-algebra via the following shuffle product:(6.2) F ( . . . , z i , . . . , z ik i , . . . ) ∗ G ( . . . , z i , . . . , z il i , . . . ) = 1 k ! · l ! · Sym F ( . . . , z i , . . . , z ik i , . . . ) G ( . . . , z i,k i +1 , . . . , z i,k i + l i , . . . ) (cid:89) i,j ∈ I (cid:89) a ≤ k i ,b>k j ζ ij (cid:18) z ia z jb (cid:19) In (6.2), Sym denotes symmetrization with respect to the:(6.3) ( k + l )! := (cid:89) i ∈ I ( k i + l i )!permutations that permute the variables z i , . . . , z i,k i + l i for each i independently. Definition 6.2. ( [10] , inspired by [13] ) The positive shuffle algebra A + is the sub-space of V consisting of rational functions of the form: (6.4) R ( . . . , z i , . . . , z ik i , . . . ) = r ( . . . , z i , . . . , z ik i , . . . ) (cid:81) unordered { i (cid:54) = i (cid:48) }⊂ I (cid:81) ≤ a (cid:48) ≤ k i (cid:48) ≤ a ≤ k i ( z ia − z i (cid:48) a (cid:48) ) where r is a symmetric Laurent polynomial that satisfies the wheel conditions: (6.5) r ( . . . , z ia , . . . ) (cid:12)(cid:12)(cid:12) ( z i ,z i ,z i ,...,z i, − aij ) (cid:55)→ ( w,wq i ,wq i ,...,wq − aiji ) , z j (cid:55)→ wq − aiji = 0 for any distinct i, j ∈ I .Remark . Because of (6.5), any r as in (6.4) is actually divisible by: unordered (cid:89) { i (cid:54) = i (cid:48) }⊂ I : a ii (cid:48) =0 1 ≤ b (cid:48) ≤ k i (cid:48) (cid:89) ≤ b ≤ k i ( z ib − z i (cid:48) b (cid:48) )Therefore, rational functions R satisfying (6.4, 6.5) can only have simple poles onthe diagonals z ib = z i (cid:48) b (cid:48) with adjacent i, i (cid:48) ∈ I , that is, such that a ii (cid:48) < Proposition 6.4. A + is closed under the product (6.2) , and is thus an algebra. , AND ALEXANDER TSYMBALIUK A + is graded by k = (cid:80) i ∈ I k i α i ∈ Q + that encodes the numberof variables of each color, and by the total homogeneous degree d ∈ Z . We write:deg R = ( k , d )and say that A + is Q + × Z -graded. We will denote the graded pieces by: A + = (cid:77) k ∈ Q + A k and A k = (cid:77) d ∈ Z A k ,d We define the negative shuffle algebra as A − = ( A + ) op . It is graded by Q − × Z ,where a rational function in k variables of homogeneous degree d is assigned degree( − k , d ), when viewed as an element of A − . We will denote the graded pieces by: A − = (cid:77) − k ∈ Q − A − k and A − k = (cid:77) d ∈ Z A − k ,d Proposition 6.6. ( [10] ) There exist unique algebra homomorphisms: (6.6) U q ( L n + ) Υ −→ A + and U q ( L n − ) Υ −→ A − determined by Υ( e i,d ) = z di ∈ A α i ,d and Υ( f i,d ) = z di ∈ A − α i ,d , respectively. Proposition 6.7. The maps Υ of (6.6) are injective.Proof. We will prove the required statement for U q ( L n + ), as taking the opposite ofboth algebras yields the statement for U q ( L n − ). Let us consider the ring A = Q [[ (cid:126) ]],its fraction field F = Q (( (cid:126) )), and define: U A ( L n + ) and U F ( L n + )by replacing Q ( q ) in Definition 4.18 with A and F , respectively. Similarly, letus define A + A and A + F by replacing Q ( q ) with A and F in the definition of A + ,respectively (more precisely, by requiring r of (6.4) to have coefficients in A or F ,respectively). Then we have a commutative diagram: U A ( L n + ) Υ A −−−−→ A + A (cid:121) (cid:121) U F ( L n + ) Υ F −−−−→ A + F where the horizontal maps are defined by analogy with Υ (just over different coef-ficient rings). Note that the right-most map is injective, but the left-most map isnot necessarily so, due to the fact that U A ( L n + ) might have A -torsion. Claim . The map Υ F is injective.Let us first show how Claim 6.8 allows us to complete the proof of the Proposition.The assignment q = e (cid:126) gives us vertical maps which make the following diagramcommute: U q ( L n + ) Υ −−−−→ A + (cid:121) (cid:121) U F ( L n + ) Υ F −−−−→ A + F UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 57 We need to show that the top map is injective. Since the claim tells us that thebottom map is injective, then it suffices to show that the left-most map is injective.The latter claim follows from the fact that U q ( L n + ) (respectively U F ( L n + )) is a free Q ( q ) (respectively F ) module with a basis given by ordered products of the rootvectors { e ( α,d ) } d ∈ Z α ∈ ∆ + . In the case of U q ( L n + ), this follows from the correspondingresult for the affine quantum group via Theorem 5.19, following our discussion fromSection 5. Explicitly, it is obtained by applying the anti-involution Ω L to the PBWdecomposition (5.69), in view of Remark 5.20 and formula (5.63). In the case of U F ( L n + ), one simply does the same proof, replacing the field Q ( q ) by F everywhere.Let us now prove Claim 6.8. Consider any x ∈ U F ( L n + ) such that Υ F ( x ) = 0, andour goal is to prove that x = 0. We may write: x = ( y ) (cid:126) k for some k ∈ N and y ∈ U A ( L n + ), and assume for the purpose of contradiction that ( y ) (cid:54) = 0. The fact that Υ F ( x ) = 0 and the injectivity of the map A + A → A + F impliesthat Υ A ( y ) = 0. By [11, Corollary 1.4], this implies that: y ∈ ∞ (cid:92) n =0 (cid:126) n · U A ( L n + )Thus, for all n ≥ 0, there exists y n ∈ U A ( L n + ) such that y = (cid:126) n y n . Passing thisequality through the map , we have for all n ≥ ( y ) = (cid:126) n · ( y n )However, because y and y n ’s lie in U A ( L n + ), their images under will lie in thefree A -submodule of U F ( L n + ) spanned by ordered products of the root vectors e ( α,d ) (this statement uses the fact that the generators e i,d of U A ( L n + ) are amongthe e ( α,d ) ’s, due to (5.43), together with the fact that the structure constants ofarbitrary products of e ( α,d ) ’s lie in Z [ q, q − ] ⊂ A , cf. Remark 5.12). Therefore,there exist uniquely determined constants c d ,...,d k α ,...,α k ∈ A such that: ( y ) = k ∈ N (cid:88) ( α ,d ) ≥···≥ ( α k ,d k ) c d ,...,d k α ,...,α k · e ( α ,d ) . . . e ( α k ,d k ) But if in (6.7) we take n larger than the leading power of (cid:126) in all the c d ,...,d k α ,...,α k whichappear as coefficients of ( y ), we obtain a contradiction. (cid:3) Remark . In type A (both finite and affine), a proof of Proposition 6.7 wasprovided in [34, Theorem 1.1]. In general simply laced types (both finite and affine),a proof of injectivity follows from [45, Theorem 2.3.2(b) combined with formula(2.39)], using the framework of K -theoretic Hall algebras of quivers, see [42]. , AND ALEXANDER TSYMBALIUK A ≥ = A + ⊗ Q ( q ) (cid:2) ( ϕ + i, ) ± , ϕ + i, , ϕ + i, , . . . (cid:3) i ∈ I (6.8) A ≤ = A − ⊗ Q ( q ) (cid:2) ( ϕ − i, ) ± , ϕ − i, , ϕ − i, , . . . (cid:3) i ∈ I (6.9)with pairwise commuting ϕ ’s, where the multiplication is governed by the rule:(6.10) ϕ ± j ( w ) ∗ R ± ( . . . , z ia , . . . ) = R ± ( . . . , z ia , . . . ) ∗ ϕ ± j ( w ) · (cid:89) i ∈ I k i (cid:89) a =1 ζ ji ( w/z ia ) ± ζ ij ( z ia /w ) ± for any R ± ∈ A ± k , where the ζ -factors are expanded as power series in non-negativepowers of w ∓ . Above, as before, we encode all ϕ ’s into the generating series:(6.11) ϕ ± i ( w ) = ∞ (cid:88) d =0 ϕ ± i,d w ± d Our reason for defining the extended shuffle algebras is that they admit coproducts. Proposition 6.11. ( [11] , see also [33, 34] ) There exist bialgebra structures on A ≥ and A ≤ , with coproduct determined by: (6.12) ∆( ϕ ± i ( z )) = ϕ ± i ( z ) ⊗ ϕ ± i ( z ) and the following assignments for all R ± ∈ A ± k : ∆( R + ) = (cid:88) l = (cid:80) i ∈ I l i α i ∈ Q + , l i ≤ k i (cid:104)(cid:81) a>l i i ∈ I ϕ + i ( z ia ) (cid:105) ∗ R + ( z i,a ≤ l i ⊗ z i,a>l i ) (cid:81) i,i (cid:48) ∈ I (cid:81) a (cid:48) >l i (cid:48) a ≤ l i ζ i (cid:48) i ( z i (cid:48) a (cid:48) /z ia )(6.13) ∆( R − ) = (cid:88) l = (cid:80) i ∈ I l i α i ∈ Q + , l i ≤ k i R − ( z i,a ≤ l i ⊗ z i,a>l i ) ∗ (cid:104)(cid:81) a ≤ l i i ∈ I ϕ − i ( z ia ) (cid:105)(cid:81) i,i (cid:48) ∈ I (cid:81) a (cid:48) >l i (cid:48) a ≤ l i ζ ii (cid:48) ( z ia /z i (cid:48) a (cid:48) )(6.14) Remark . To think of (6.13) as a well-defined tensor, we expand the right-handside in non-negative powers of z ia /z i (cid:48) a (cid:48) for a ≤ l i and a (cid:48) > l i (cid:48) , thus obtaining aninfinite sum of monomials. In each of these monomials, we put the symbols ϕ + i,d tothe very left of the expression, then all powers of z ia with a ≤ l i , then the ⊗ sign,and finally all powers of z ia with a > l i . The resulting expression will be a powerseries, and therefore lies in a completion of A ≥ ⊗ A + . The same argument appliesto (6.14), still using non-negative powers of z ia /z i (cid:48) a (cid:48) for a ≤ l i and a (cid:48) > l i (cid:48) , andkeeping all the ϕ − i,d to the very right.The following is straightforward. Proposition 6.13. The maps (6.6) extend to bialgebra homomorphisms: (6.15) U q ( L b + ) Υ −→ A ≥ and U q ( L b − ) Υ −→ A ≤ by sending ϕ ± i,d ∈ U q ( L b + ) , U q ( L b − ) to the same-named ϕ ± i,d ∈ A ≥ , A ≤ . UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 59 A ≥ and A ≤ . As a first step towarddefining it, we start with the following result. Let Dz = dz πiz . Proposition 6.15. There exists a unique bialgebra pairing: (6.16) (cid:104)· , ·(cid:105) : A ≥ ⊗ U q ( L b − ) −→ Q ( q ) satisfying (4.35) as well as: (cid:68) R, f i , − d . . . f i k , − d k (cid:69) = k (cid:89) a =1 ( q − i a − q i a ) − (cid:90) | z |(cid:28)···(cid:28)| z k | R ( z , . . . , z k ) z − d . . . z − d k k (cid:81) ≤ a
Proposition 6.17. The pairing (6.16) is non-degenerate in the first argument: (cid:68) R, − (cid:69) = 0 ⇒ R = 0 for any R ∈ A ≥ .Proof. Because of (6.8), elements of A ≥ are linear combinations of R · ϕ + , where: R ∈ A + and ϕ + ∈ Q ( q ) (cid:2) ( ϕ + i, ) ± , ϕ + i, , ϕ + i, , . . . (cid:3) i ∈ I As a consequence of the bialgebra pairing properties (4.6, 4.7), it is easy to see that: (cid:68) Rϕ + , xϕ − (cid:69) = (cid:68) R, x (cid:69) · (cid:68) ϕ + , ϕ − (cid:69) for any x ∈ U q ( L n − ) and ϕ − a product of ϕ − i,d ’s. Thus the non-degeneracy of thepairing (6.16) is a consequence of the non-degeneracy of its restriction:(6.20) (cid:104)· , ·(cid:105) : A + ⊗ U q ( L n − ) −→ Q ( q )(indeed, the pairing between ϕ ’s is easily seen to be non-degenerate, due to theexplicit formula (4.35)). However, the non-degeneracy of (6.20) in the first argu-ment is an immediate consequence of formula (6.17): if R is a non-zero rationalfunction, then we simply choose an arbitrary order of its variables | z | (cid:28) · · · (cid:28) | z k | ,and consider the leading order term of R when expanded as a power series in thisparticular order. On one hand, this leading order term must be non-zero, but onthe other hand, it is of the form in the right-hand side of (6.17). (cid:3) We note that the pairings (4.33) and (6.16) are compatible, in the sense that:(6.21) (cid:68) x, y (cid:69) = (cid:68) Υ( x ) , y (cid:69) for all x ∈ U q ( L b + ) and y ∈ U q ( L b − ). Indeed, both sides of (6.21) define bialgebrapairings: U q ( L b + ) ⊗ U q ( L b − ) −→ Q ( q )which coincide on the generators, thus must be equal as a consequence of (4.6, 4.7).Combining (6.21) with Propositions 6.7, 6.17, we thus obtain the non-degeneracystatement of Proposition 4.21 (strictly speaking, we obtain the aforementionednon-degeneracy statement only in the first argument, but the case of the secondargument is treated by simply switching the roles of + and − everywhere).6.18. Once Theorem 1.8 will be proved, Proposition 6.15 can be construed as theexistence of a bialgebra pairing (which is non-degenerate by Proposition 4.21): (cid:104)· , ·(cid:105) : A ≥ ⊗ A ≤ −→ Q ( q )Hence, we may construct the Drinfeld double:(6.22) A := A ≥ ⊗ A ≤ / ( ϕ + i, ⊗ ϕ − i, − ⊗ UANTUM LOOP GROUPS AND SHUFFLE ALGEBRAS VIA LYNDON WORDS 61 Theorem 6.19. There exists a bialgebra isomorphism: (6.23) U q ( L g ) Υ −→ A which maps e i,d (cid:55)→ z di ∈ A + , f i,d (cid:55)→ z di ∈ A − , ϕ ± i,r (cid:55)→ ϕ ± i,r A + ι −→ F L given by the following formula:(6.25) ι ( R ) = (cid:88) i ,...,i k ∈ Id ,...,d k ∈ Z (cid:34) k (cid:89) a =1 ( q − i a − q i a ) (cid:35) (cid:68) R, f i , − d . . . f i k , − d k (cid:69) · (cid:104) i ( d )1 . . . i ( d k ) k (cid:105) for all R ∈ A + k , where k = | k | . Because of (6.17), we have the explicit formula: ι ( R ) = (cid:88) i ,...,i k ∈ Id ,...,d k ∈ Z (cid:104) i ( d )1 . . . i ( d k ) k (cid:105) · (cid:90) | z |(cid:28)···(cid:28)| z k | R ( z , . . . , z k ) z − d . . . z − d k k (cid:81) ≤ a