On the dimension of the Fomin-Kirillov algebra and related algebras
aa r X i v : . [ m a t h . QA ] J a n ON THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA ANDRELATED ALGEBRAS
CHRISTOPH BÄRLIGEA
Abstract.
Let E m be the Fomin-Kirillov algebra, and let B S m be the Nichols-Woronowiczalgebra model for Schubert calculus on the symmetric group S m which is a quotient of E m ,i.e. the Nichols algebra associated to a Yetter-Drinfeld S m -module defined by the set ofreflections of S m and a specific one-dimensional representation of a subgroup of S m . It is afamous open problem to prove that E m is infinite dimensional for all m ≥ . In this work,as a step towards a solution of this problem, we introduce a subalgebra of B S m , and prove,under the assumption of finite dimensionality of B S m , that this subalgebra admits uniqueintegrals in a strong sense, and we relate these integrals to integrals in B S m . The techniqueswe use rely on braided differential calculus as developed in [8, 9, 28], and on the notion ofintegrals for Hopf algebras as introduced in [36]. Contents
1. 0Introduction 12. 0Coxeter groups 43. 0Disjoint systems in Coxeter groups 64. 0Nichols algebras and braided differential calculus 85. 0Terminology concerning monomials 136. 0NilCoxeter algebra 167. 0Inversion of braided partial derivatives 178. 0Consequences of the generalized braided Leibniz rule 199. 0Integrals for Hopf algebras 2210. Invariance of integrals 2711. Disjoint systems and integrals 2812. Coproducts of integrals 2913. Reduction of monomials 3114. Hypothetical elements 34References 351.
Introduction
We fix once and for all an arbitrary ground field k . We will always work over this field. Wedenote by S m the symmetric group on m letters. We denote by R the root system of type A n Date : January 8, 2020.2010
Mathematics Subject Classification.
Primary 20G42; Secondary 16T05, 20F55.
Key words and phrases.
Braided differential calculus, Nichols-Woronowicz algebra model for Schubertcalculus on finite Coxeter groups, Fomin-Kirillov algebra, integrals for Hopf algebras.The research was partially supported by the German Research Foundation (DFG) under the projectnumber 345815019. where n = m − . Fomin-Kirillov introduced in [14, Definition 2.1] a quadratic k -algebra E m ,which is now called the Fomin-Kirillov algebra, defined by generators and relations where wehave generators x α for each α ∈ R and homogeneous relations x − α = − x α x α = 0 for all α ∈ R , x α x γ = x γ x α for all α, γ ∈ R such that α and γ are orthogonal, x α x α + γ + x α + γ x γ − x γ x α = 0 for all α, γ ∈ R such that α and γ span a root subsystem of R of type A with base { α, γ } .There are plenty of motivations to consider the Fomin-Kirillov algebra. Let us mention atleast two of them which are already present in [14]. • The divided difference operators ∂ α acting either on the polynomial ring k [ x , . . . , x m ] or the coinvariant algebra S S m of S m satisfy the above relations where ∂ α plays therole of x α , and the author does not know any other relations between them unlessrelations generated from the above. In other words, the algebra E m projects onto thesubalgebra of k [ x , . . . , x m ] or S S m generated by ∂ α where α runs through R . Theideas of the Schubert calculus of divided difference operators will be prevalent in thispaper, cf. Section 8. For an elementary introduction to them, we refer to [23, 25, 29]. • Recall that over the complex numbers the coinvariant algebra S S m is canonicallyisomorphic to the cohomology ring of the complete flag variety SL m ( C ) /B where B is a Borel subgroup of SL m ( C ) ([14, Theorem 6.1]). Because of this geometricinterpretation, the coinvariant algebra is sometimes also called Borel’s algebra (atleast when considered over the complex numbers). In [14, Theorem 7.1], it was provedthat S S m can be canonically embedded into E m via Dunkl elements. In this way, theFomin-Kirillov algebra can be thought of as a noncommutative model for Schubertcalculus, and can be useful to prove something about the latter.In [14, Conjecture 2.2], the authors ask about the dimension of E m considered as a k -vectorspace. The following conjecture is nowadays well-known. Conjecture 1.1.
The Fomin-Kirillov algebra E m is infinite dimensional for all m ≥ . Remark 1.2.
It is known from [32, Example 6.4] that E m is finite dimensional for all m ≤ . Setup of the paper.
Let ( W, S ) be a finite Coxeter system. Let B W be the Nichols-Woronowiczalgebra model for Schubert calculus on W as it has been studied in [9]. The algebra B W isa Nichols algebra associated to a Yetter-Drinfeld W -module defined by the set of reflections T of W and a specific one-dimensional representation of a subgroup of W . It is a braided Z ≥ -graded Hopf algebra in the Yetter-Drinfeld category over W . Over the symmetric group S m , the algebra B S m is known to be a quotient of E m , and conjecturally isomorphic to E m . In order to prove Conjecture 1.1, it therefore suffices to prove the following conjecture.
Conjecture 1.3.
The algebra B S m is infinite dimensional for all m ≥ . The same situation arises for all simply laced Weyl groups, and a more general conjecture can be formu-lated in this setting as Pierre-Emmanuel Chaput brought to our attention.
N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 3
From now on, we will work with B W whenever we can prove our results in this generality,and specialize to B S m whenever needed. This has the advantage over working directly with E m that the statements can be formulated in a clear and type independent way. Moreover,the algebras B W and in particular B S m enjoy desirable properties which are only conjecturallyknown for E m and which we want to use in our proofs, cf. Remark 9.16, Corollary 9.9(1): Mostsignificantly, just as every Nichols algebra associated to a finite dimensional Yetter-Drinfeldmodule, they admit a nondegenerate Hopf duality pairing. (cid:3) While we cannot prove neither Conjecture 1.1 nor Conjecture 1.3, we want, in this paper,to suppose finite dimensionality of B W and draw significant consequences which eventuallymight lead to a contradiction for B S m whenever m ≥ . More specifically, we want to proposea proof of the following theorem. Theorem 1.4 (Theorem 14.8) . Suppose that B S m is finite dimensional. Let B ′ S m be thesubalgebra of B S m generated by x α for all α ∈ R \ ± ∆ , where ∆ is the set of simple roots,and where we still denote by x α for each α ∈ R the image of the canonical generators of E m under the natural projection E m → B S m . Further, let B ′ top S m be the nonzero component of B ′ S m of largest Z ≥ -degree. Then, every one-dimensional left or right ideal in B ′ S m equals B ′ top S m . Inparticular, we know that B ′ top S m is a one-dimensional Z ≥ -graded two-sided ideal in B ′ S m . Remark 1.5.
In the setup of Theorem 1.4, its content can be paraphrased by saying thatthe subalgebra B ′ S m admits unique integrals up to scalar multiple, cf. [36, Section 2], althoughit is in general not a braided Hopf subalgebra of B S m . Intuition of the paper.
Whenever the algebra B W is finite dimensional, we expect certaincommutativity relations to hold, either on the nose under correspondent additional assump-tions, or up to multiplication with a nonzero element in B W . We make this intuition precisein Lemma 11.2: (1) ⇒ (3) , Theorem 12.3 and Corollary 9.14. In the setup of Theorem 1.4,such a commutativity relation is visible because every nonzero element P ∈ B ′ top S m gives riseto a nonzero integral P x w o = ± x w o P in B S m by multiplication with x w o from the left or theright (cf. Lemma 10.2(3),(4), Theorem 14.6(2)), where w o is the longest element of W , andwhere x w o is the basis element indexed by w o of the standard basis of the nilCoxeter algebraof W which embeds into B S m by [9, Theorem 6.1(i),(ii)]. (cid:3) References.
Although it did not become apparent by what we said up to now, the resultsin this paper rely on braided differential calculus as developed in [9, 28]. Even if many ofthe original ideas are due to [28], we will often refer to [8] for similar statements because thelatter paper is written in a generality and language which is more suitable for us.
Organization.
In Section 2-6, we setup common terminology and notation which will beused from thereon. In Section 7–9, we deepen some aspects of the braided differential calculusdeveloped in [9, 28]. Section 10-12 are not strictly necessary to be read to understand theproof of Theorem 1.4, but they serve as a good illustration of the principle of commutativityexplained in the paragraph “Intuition of the paper” . Section 13–14 finally contain the proofof Theorem 1.4. While we work mostly in the generality of B W as explained in the paragraph “Setup of the paper” , it should be noted that some of our results, in particular Theorem 12.3,work even in greater generality, i.e. for Nichols algebras associated to more general Yetter-Drinfeld modules. We point out the details concerning these generalizations in Subsection 4.5. CHRISTOPH BÄRLIGEA
Context.
It is a recurrent theme in the literature to ask which groups admit a finite dimen-sional Nichols algebra. This question has been studied in particular for symmetric groups,alternating groups and dihedral groups [1, 2, 3, 5]. This paper can be seen in line with theseworks. Especially, a positive or negative solution of Conjecture 1.3 would supplement theclassification theorem [1, Theorem 1.1] of finite dimensional Nichols algebras over symmetricgroups. It should be noted that there exists a general theory which allows, among otherthings, to treat the question of dimensionality of a Nichols algebra whenever it is associatedto a reducible
Yetter-Drinfeld module over a Hopf algebra with invertible antipode, see forexample [16, 17, 18, 19, 27]. This theory is based on combinatorics on Lyndon words overthe alphabet given by an index set of the irreducible components of the underlying Yetter-Drinfeld module. However, whenever the underlying Yetter-Drinfeld module is irreducible ,the theory of Andruskiewitsch, Heckenberger, Schneider et al. gives no information accordingto the author’s understanding. Thus, it cannot be suitable to proof Conjecture 1.3.
Acknowledgment.
The support of the German Research Foundation (DFG) is gratefullyacknowledged. The author thanks the Beijing International Center for Mathematical Re-search and Professor Xiaobo Liu for accepting him as a Boya Postdoctoral Fellow of PekingUniversity. 2.
Coxeter groups
We fix once and for all a finite Coxeter system ( W, S ) . We assume throughout that W = 1 or equivalently that S = ∅ . We denote by m ( s, s ′ ) the necessarily finite order of ss ′ in W (cf. [20, Proposition 5.3]). We denote by T = S w ∈ W wSw − the set of all reflections of W .We denote by h the geometric representation of W as in [20, Section 5.3], i.e. h is a realvector space of dimension | S | with basis given by the set of simple roots ∆ , and each simpleroot β corresponds bijectively to a simple reflection s β in S which acts on h via the formula s β ( x ) = x − B ( x, β ) β , where B is the W -invariant scalar product on h uniquely determinedby the assignment B ( β, β ′ ) = − cos πm ( s β , s β ′ ) on simple roots β, β ′ (cf. [12, Chapter V, § 4, n o
8, Theorem 2] and [20, Proposition 5.3]).Recall that the geometric representation h of W is faithful by [20, Corollary 5.4].To the situation above, we attach a root system R in h and a partial order “ ≤ ” on h considered as an abelian group as in [20, Section 5.4]. The root system R is a root system inthe weak sense of [20, Section 1.2], i.e. it is a finite subset of nonzero vectors in h satisfyingthe axioms(R1) R ∩ R α = {− α, α } for all α ∈ R ,(R2) w ( R ) = R for all w ∈ W ,in particular, it is in general not crystallographic in the sense of [20, Section 2.9]. We referto Axiom (R1) by saying that R is reduced. We define the set of positive roots R + and theset of negative roots R − by the equations R + = { α ∈ R | α > } and R − = { α ∈ R | α < } .Note that we have R = R + ∪ R − where the union is obviously disjoint (cf. [20, Theorem 5.4]).For a subset Θ of R , we denote by − Θ the set of roots − Θ = {− α | α ∈ Θ } . With thisnotation, we have for example R − = − R + . Because R is reduced, note that R + and R − are in bijection with T . Each time, the bijection is given by the assignment α s α where s α is the reflection associated to α as in [20, Section 5.7], which acts on h via the formula N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 5 s α ( x ) = x − B ( x, α ) α analogously as in the case of a simple root α = β . This formula clearlyimplies s α = s − α for all α ∈ R .We denote by ℓ the length function on W as defined in [20, Section 5.2]. We denote by w o the longest element of W , i.e. the unique element in W of maximum length (cf. [20,Section 5.6, Exercise 2]). We denote by “ ≤ ” the (strong) Bruhat order on W as defined in[20, Section 5.9]. Remark 2.1.
Whenever W is a simply laced Weyl group or equivalently whenever R isa simply laced root system, every root α written as a linear combination of simple rootshas integral coefficients, and we define the height of α , in formulas ht( α ) , as the sum ofthose coefficients (cf. [20, Section 3.20, p. 83]). In this situation, we define further a W -invariant scalar product on h given by ( − , − ) = 2 B ( − , − ) which has the property that ( α, γ ) ∈ {− , , } for all non-proportional roots α, γ and that ( α, α ) = 2 for all roots α . Remark 2.2.
In this remark, we want to recall some basic facts about the Bruhat order on W which we will use from now on without reference. For a positive root α , we have by [20,Proposition 5.7] the following equivalences (to be read from top to bottom): ws α < w ⇐⇒ ℓ ( ws α ) < ℓ ( w ) ⇐⇒ w ( α ) < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ws α > w ⇐⇒ ℓ ( ws α ) > ℓ ( w ) ⇐⇒ w ( α ) > If α = β is a simple root, it follows from the strong exchange condition [20, Theorem 5.8]that the above equivalences are further equivalent to: ⇐⇒ there exists a reduced expres-sion of w which ends with s β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⇐⇒ no reduced expression of w ends with s β By [20, Section 5.2, Equation (L1), Exercise 5.9], for a positive root α , there also exist leftanalogues of the above two lines of equivalences as follows: s α w < w ⇐⇒ ℓ ( s α w ) < ℓ ( w ) ⇐⇒ w − ( α ) < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s α w > w ⇐⇒ ℓ ( s α w ) > ℓ ( w ) ⇐⇒ w − ( α ) > If α = β is a simple root, it follows further that the above equivalences are equivalent to: ⇐⇒ there exists a reduced expres-sion of w which starts with s β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⇐⇒ no reduced expression of w starts with s β Remark 2.3 (The center of W ) . Let ( W, S ) = Q i ( W i , S i ) be the decomposition of ( W, S ) intoirreducible components where ( W i , S i ) is an irreducible finite Coxeter system correspondingto the i th connected component of the Coxeter diagram of ( W, S ) (cf. [20, Proposition 6.1]).The geometric representation h of W decomposes as a direct sum L i h i into irreduciblecomponents where h i is the geometric representation of W i (cf. [12, Chapter V, § 4, n o W decomposes into a product Q i Z i where Z i isthe center of W i . For each i , the group Z i embeds into Z / Z where the i th embedding is anisomorphisms if and only if w o acts as minus the identity on h i (in which case Z i = { , w o,i } where w o,i is the longest element of W i and where Z i is trivial otherwise). This description ofthe center of W was discussed in [7, Lemma 4.5]. The center of W is relevant for this paperbecause of Lemma 9.11. CHRISTOPH BÄRLIGEA Disjoint systems in Coxeter groups
In this section, we introduce the notion of disjoint systems in Coxeter groups. In the formwe define it, it is only suitable for finite Coxeter groups. This notion will be present in manyof our considerations.
Definition 3.1.
Let w ∈ W . We denote by T w the subset of R + defined by T w = { α ∈ R + | s α ∈ wSw − } . Fact 3.2.
Let w be an element in the centralizer of w o . Then, we have T w o w = T ww o = T w . Proof.
By assumption on w , the equality T w o w = T ww o is clear. The equality T ww o = T w follows because w o is an involution and because w o Sw o = S by [20, Section 5.6, Exercise 2]. (cid:3) Remark 3.3.
Fact 3.2 has a converse if ( W, S ) is irreducible. In this case, if w, w ′ ∈ W aresuch that T w = T w ′ , it follows that w = w ′ or w = w ′ w o . This is clear from [20, Section 5.6,Exercise 2] and the existence of a unique highest root for irreducible Coxeter systems (i.e. aroot θ such that B ( θ , β ) ≥ for all β ∈ ∆ ). Definition 3.4. • We say that D is a disjoint system if D is contained in the centralizer of w o and ifthe union S w ∈ D wSw − is disjoint. • If D is a disjoint system, we call the cardinality of D the order of D . • We say that a disjoint system D is complete if the order of D is equal to | T || S | , inother words, if D is a disjoint system such that T = S w ∈ D wSw − where the union isdisjoint. • We say that a disjoint system D is normalized if ∈ D . • If we want to emphasize the group W with respect to which a disjoint system isdefined, we explicitly speak about a disjoint system in W . Remark 3.5.
Note that we allow the empty set as disjoint system of order zero in any W . Remark 3.6. If D is a disjoint system, then any subset of D is also a disjoint system. Remark 3.7.
We remark that the integer | T || S | ∈ Z > is called the Coxeter number of W (cf. [20, Proposition 3.18]). Note that half the Coxeter number of W appears in the definitionof a complete disjoint system. If a complete disjoint system D exists, then the Coxeter numberof W is even and half the Coxeter number of W is equal to the order of D . Remark 3.8.
Let D be a disjoint system of order r . Let w , . . . , w s ∈ D . Then, the set D ′ = ( D \ { w , . . . , w s } ) ∪ { v , . . . , v s } where v i = w o w i = w i w o for all ≤ i ≤ s is also a disjoint system of order r . This follows fromFact 3.2 and because the centralizer of w o is a subgroup of W which contains the involution w o . The difference between D and disjoint systems D ′ derived from D in this way will alwaysbe irrelevant for our applications in this paper. Lemma 3.9.
Let D be a disjoint system of order r . Let v be an element in the centralizerof w o . Then, the set vD is also a disjoint system of order r . N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 7
Proof.
Let the notation be as in the statement. Let D ′ = vD for short. It is clear that D ′ has the same cardinality as D , that D ′ is contained in the centralizer of w o (becausethis centralizer is a subgroup of W ), and that the union S w ∈ D vwSw − v − is disjoint as atranslate of the disjoint union S w ∈ D wSw − . By replacing w with v − w in the translateddisjoint union, we find that the union S w ∈ D ′ wSw − is also disjoint. Thus, we know that D ′ is a disjoint system of order r . (cid:3) Corollary 3.10 (Normalization of disjoint systems) . Let D be a disjoint system of order r .For all v ∈ D , the set v − D is a normalized disjoint system of order r . In particular, if acomplete disjoint system exists, then there exists also a normalized complete disjoint system.Proof. The particular case is immediate from the more general statement. Let D be a disjointsystem of order r . Let v ∈ D and let D ′ = v − D . Note that v and v − are both containedin the centralizer of w o , by assumption, and because this centralizer is a subgroup of W . Ifwe apply Lemma 3.9 to D and v − , we find that D ′ is a disjoint system of order r . But D ′ is also normalized by definition. (cid:3) Remark 3.11.
The most urgent combinatorial question raised by the notion of disjointsystems is as follows: If the Coxeter number of W is even (which is necessary by Remark 3.7),does there always exist a complete disjoint system? Remark 3.12.
Let D be a normalized disjoint system. Then, every element w ∈ D \ { } satisfies wSw − ∩ S = ∅ . If D is a normalized disjoint system in S m , elements of D \ { } aretherefore a special example of so-called permutations without rising or falling successions.These permutations in general are subject to combinatorial studies and enumerations. Werefer to [34] for a list of literature. Disjoint systems in the symmetric group.
For the symmetric group S m , we write per-mutations either in one-line notation or in cycle notation where the difference between thetwo notations is visible by the absence or presence of parenthesis. Sometimes, we also writepermutations in S m as bijections of { , . . . , m } . Remark 3.13.
Recall that the longest element of S m is simply given by m ( m − · · · .Hence, the centralizer of the longest element of S m is given by permutations σ such that σ ( i ) + σ ( j ) = m + 1 for all ≤ i, j ≤ m where i + j = m + 1 . Example 3.14 (Complete disjoint system in S ) . For this example, let us assume that W = S and that the notation from Section 2 is realized for S . Let us consider the elements w = 241635 and w = 315264 in S . By Remark 3.13, we see that both w and w lie in thecentralizer of the longest element of S . Further, the union w Sw − ∪ w Sw − = { (24) , (14) , (16) , (36) , (35) } ∪ { (13) , (15) , (25) , (26) , (46) } is disjoint and equal to T \ S . Hence, we conclude that { w , w , } is a normalized completedisjoint system of order three in S . We have illustrated this disjoint system in Figure 1.Note that the nontrivial elements of this disjoint system satisfy the additional relations w − = w , w − = w , w − w = w = w w o , w − w = w = w w o , We want to thank Richard Stanley for pointing out the reference [34] in a comment to our question onMathOverflow [37].
CHRISTOPH BÄRLIGEA β β β β β Figure 1.
For this figure, we assume that W = S and that the notation from Section 2 is realizedfor S . Let w and w be the elements of S as defined in Example 3.14. The barycentric graph illus-trates all positive roots in R + where β , β , β , β , β denote all simple roots in ∆ with the labelingas in [12, Plate I]. The positive roots labeled with and correspond each up to multiplication with w o from the right to a permutation in S (in this case to w , w w o and w , w w o , cf. Remark 3.3)because they form a diagram with straight lines isomorphic to the Dynkin diagram of type A wherewe possibly allow reflection along the horizontal bottom line below the graph. They correspond toelements in the centralizer of the longest element of S because their diagrams are symmetric withrespect to the vertical line in the middle of the graph. They correspond to permutations withoutrising or falling succession because none of them is simple. If we denote by T , T the set of positiveroots labeled with , , respectively, then we have T = T w and T = T w . and that as a consequence the partition of T = w Sw − ∪ w Sw − ∪ S is invariant underconjugation with w , w . This example shows in particular that a complete disjoint systemin S exists. 4. Nichols algebras and braided differential calculus
In this section, we recall some basic facts about Nichols algebras and braided differentialcalculus. Braided differential calculus is the calculus of braided partial derivatives actingon a Nichols algebra associated to a finite dimensional Yetter-Drinfeld module, and definedin terms of a nondegenerate Hopf duality pairing. For more details concerning the generaltheory presented in this section, we refer to [4, 6, 8, 9, 10, 22, 28, 30]. More specifically, werefer for • Subsection 4.1 to [6, Subsection 1.1–1.3], [9, Subsection 2.1], [10, Subsection 5.1, 5.2], • Subsection 4.2 to [6, Subsection 2.1], [9, Section 2, 3], [10, Sibsection 5.3–5.8], • Subsection 4.3 to [8, Section 3], [28, Subsection 2.3], • Subsection 4.4 to [8, Subsection 4.1, 4.2], [9, Section 4].
We fix once and for all a Hopfalgebra H over k with invertible antipode. We denote by HH YD the Yetter-Drinfeld cate-gory over H . Recall that the objects in HH YD are Yetter-Drinfeld H -modules, i.e. k -vectorspaces V which are simultaneously left H -modules and left H -comodules which satisfy thecompatibility condition ( hv ) ( − ⊗ ( hv ) (0) = h (1) v ( − S ( h (3) ) ⊗ h (2) v (0) for all h ∈ H and all v ∈ V . Here and everywhere else where it is suitable, we use sumlessSweedler notation for coproducts and coactions. In Section 12, we will be however obliged N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 9 to choose specific decompositions of coproducts in which case we renounce to use (sumless)Sweedler notation. By [40], the above compatibility condition for a Yetter-Drinfeld H -module V can be equivalently expressed as h (1) v ( − ⊗ h (2) v (0) = ( h (1) v ) ( − h (2) ⊗ ( h (1) v ) (0) for all h ∈ H and all v ∈ V . The morphisms in HH YD are the obvious ones, i.e. those whichpreserve the H -module and H -comodule structure. The Yetter-Drinfeld category over H isa braided monoidal category where the braiding Ψ of two Yetter-Drinfeld H -modules V and W is given by Ψ( v ⊗ w ) = v ( − w ⊗ v (0) with inverse Ψ − ( w ⊗ v ) = v (0) ⊗ S − ( v ( − ) w for all v ∈ V and all w ∈ W . Just as in the previous sentence, we usually suppress the componentsof the braiding Ψ . Note that every finite dimensional Yetter-Drinfeld H -module V is rigidin the sense of [6, Subsection 1.1(d)] or [30, Definition 9.3.1], i.e. it admits a left dual, oneisomorphic copy of which we denote by V ∗ , namely the linear dual of V equipped with thestructure of a Yetter-Drinfeld H -module uniquely determined by the formulas ( hf )( v ) = f ( S ( h ) v ) ,f ( − f (0) ( v ) = S − ( v ( − ) f ( v (0) ) for all h ∈ H , v ∈ V , f ∈ V ∗ .Whenever A is a braided Z ≥ -graded Hopf algebra in HH YD , we denote by A m the componentof A of Z ≥ -degree m , i.e. we always use superscripts to indicate Z ≥ -graded components,and we follow this convention even on the level of elements in A . For two braided Hopfalgebras A and B in HH YD , we say according to [10, Subsection 5.3] or [30, Definition 1.4.3]that a morphism h− , −i : A ⊗ B → k in the Yetter-Drinfeld category over H is a Hopf dualitypairing between A and B if it satisfies the axioms h φψ, x i = (cid:10) φ, x (2) (cid:11)(cid:10) ψ, x (1) (cid:11) , h φ, xy i = (cid:10) φ (2) , x (cid:11)(cid:10) φ (1) , y (cid:11) which entail the properties h , x i = ǫ ( x ) , h φ, i = ǫ ( φ ) , h S ( φ ) , x i = h φ, S ( x ) i by [13], [22, Section III.9], [30, Proposition 1.3.1], and where everywhere in the two previousdisplayed equations φ, ψ ∈ A and x, y ∈ B . For a braided Hopf algebra A in HH YD , we call aHopf duality pairing between A and A simply a Hopf duality pairing between A and itself.For two braided Z ≥ -graded Hopf algebras A and B in HH YD , we say that a Hopf dualitypairing h− , −i between A and B respects the Z ≥ -grading if h− , −i restricted to A m ⊗ B m ′ is zero unless m = m ′ . Remark 4.1.
Let h− , −i be a Hopf duality pairing between A and B where A and B aretwo braided Z ≥ -graded Hopf algebras. Because h− , −i is a morphism in the Yetter-Drinfeldcategory over H , we have the following invariance properties h hx, y i = h x, S − ( h ) y i and h x, hy i = h S ( h ) x, y i where h ∈ H , x ∈ A , y ∈ B . In this subsection, we do not need Coxeter groups. We will use temporarily W with another meaning,and continue with the original setup from the previous sections in Subsection 4.4. Remark 4.2.
Every Yetter-Drinfeld H -module V also carries a natural structure of a right H -module given by vh = S − ( h ) v for all h ∈ H and all v ∈ V . This structure is natural as itgives rise to an equivalence of categories as discussed in [4, Proposition 2.2.1, Item (1): (i) ⇔ (iii) ]. We will equip from now on every V with this additional structure, and consider every h ∈ H either as plain element in H or as homothety in End k V acting from the left or theright on V , where the direction will be clear from context or will be indicated by evaluationon elements in V or placeholders ( − ) .Whenever A is a braided Hopf algebra in HH YD , we consider elements in A sometimes asendomorphisms of A given by multiplication from the left or the right and acting correspond-ingly. It is finally convenient to consider the antipode of such an A (and its inverse if existent)as endomorphisms of A acting either on the left or the right. Each time, for elements in A considered as endomorphisms as well as for the antipode of A (and its inverse if existent),the acting direction will be clear from context or will be indicated by evaluation on elementsin A or placeholders ( − ) .Let A be a braided Hopf algebra in HH YD . With the conventions in the two previousparagraphs in mind, we have ( − ) zh = ( − ) h (2) ( zh (1) ) and hz ( − ) = ( h (1) z ) h (2) ( − ) for all h ∈ H and all z ∈ A . We will use the first of the two formulas above in the proof ofTheorem 8.1. Let V be a Yetter-Drinfeld H -module. We denote by B ( V ) the Nichols-Woronowicz algebra of V . We call it simplythe Nichols algebra of V from now on, or the Nichols algebra associated to V . By [6,Definition 2.1, Proposition 2.2], this is a braided Z ≥ -graded Hopf algebra in HH YD uniquelydetermined up to isomorphism by the axioms:(N1) B ( V ) is connected, i.e. B ( V ) = k ,(N2) B ( V ) is generated as an algebra in Z ≥ -degree one,(N3) the Yetter-Drinfeld H -module consisting of all primitive elements in B ( V ) equals B ( V ) which is in turn equal to V .The explicit construction of B ( V ) is discussed in [6, Subsection 2.1], [9, Subsection 3.1], [10,Subsection 5.7]. We assume for the rest of this subsection that V is finite dimensional. Recallthe important property that B ( V ) comes equipped with a nondegenerate Hopf duality pairing h− , −i between B ( V ∗ ) and B ( V ) which respects the Z ≥ -grading and which is uniquelydetermined by the property that its restriction to V ∗ ⊗ V equals the evaluation pairing of V . Using this pairing, one defines a right action y ←− D y of the algebra B ( V ) on B ( V ∗ ) viabraided right partial derivatives ←− D x and a left action y ∗ −→ D y ∗ of the algebra B ( V ∗ ) on B ( V ) via braided left partial derivatives −→ D y ∗ , where ( z ∗ ) ←− D y = z ∗ (1) h z ∗ (2) , y i , −→ D y ∗ ( z ) = h y ∗ , z (1) i z (2) Here and in what follows,
End k and the word “endomorphism” without further specification refer toendomorphisms of vector spaces. N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 11 for y, z ∈ B ( V ) and y ∗ , z ∗ ∈ B ( V ∗ ) . We will be mostly concerned with braided right partialderivatives in this work. If it is clear from context whether left or right is meant or if it doesnot matter, we simply speak about braided partial derivatives. We record the formulas h x ∗ , yx i = (cid:10) ( x ∗ ) ←− D y , x (cid:11) , h x ∗ y ∗ , x i = (cid:10) x ∗ , −→ D y ∗ ( x ) (cid:11) , where x, y ∈ B ( V ) and x ∗ , y ∗ ∈ B ( V ∗ ) , which will be used in several of our considerationswithout further reference, e.g. in the proof of Proposition 9.10(1). Having the conventions inRemark 4.2 in mind, braided partial derivatives are uniquely determined by the multiplica-tivity of the actions they define and the formulas z ∗←− D v = ( z ∗ ) ←− D v + ←− D v (0) (cid:0) S − ( v ( − ) z ∗ (cid:1) , −→ D v ∗ z = −→ D v ∗ ( z ) + z (0) −→ D S − ( z ( − ) v ∗ , where v ∈ V , v ∗ ∈ V ∗ , z ∈ B ( V ) , z ∗ ∈ B ( V ∗ ) . The first of these formulas is called thebraided Leibniz rule and we refer to it under this name from now. The braided Leibniz ruleis of utmost importance for this work and will be often in use. The second of these formulasis used only in Remark 4.3 which follows and in the proof of Lemma 5.9(8), and is referencedalso as braided Leibniz rule. Let us finally mention the following formulas ←− D hy = S ( h (1) ) ←− D y S ( h (2) ) and −→ D hy ∗ = S − ( h (1) ) −→ D y ∗ S − ( h (2) ) where h ∈ H , y ∈ B ( V ) , y ∗ ∈ B ( V ∗ ) , which are a straight forward generalization of [8,Remark 3.16]. Remark 4.3 (Embedding the tensor square into endomorphisms [24, Lemma 1]) . Let V bea finite dimensional Yetter-Drinfeld H -module. As a consequence of the nondegeneracy ofthe Hopf duality pairing between B ( V ∗ ) and B ( V ) , we have according to [24, Lemma 1] twoembeddings of vector spaces B ( V ) ⊗ B ( V ∗ ) ֒ −−→ End k B ( V ∗ ) , B ( V ) ⊗ B ( V ∗ ) ֒ −−→ End k B ( V ) defined on pure tensors by the assignments y ⊗ ξ ←− D y ξ ,y ⊗ ξ y −→ D ξ and extended linearly. It follows from the braided Leibniz rule that the images of theseembeddings inherit the structures of a Z -graded algebra in HH YD whenever the antipode on H is an involution, where the algebra structure is inherited from End k B ( V ∗ ) and End k B ( V ) ,where the structure of a Yetter-Drinfeld H -module is inherited from B ( V ) ⊗ B ( V ∗ ) , and wherethe Z -grading is given by the way the operators in the image manipulate the Z ≥ -degree whenapplied to elements in B ( V ∗ ) and B ( V ) , cf. [8, Remark 2.16], i.e. the Z -degree of ←− D y ξ is givenby m − m ′ and that of y −→ D ξ by m ′ − m whenever ξ and y are homogeneous of Z ≥ -degree m and m ′ .The original context of the embeddings of the tensor square into endomorphisms is coho-mology and quantum cohomology of G/B where G is a reductive linear algebraic group and B is a Borel subgroup of G , cf. [9, Thoerem 5.4], [24, Theorem 1], [31, Lemma 1.3]. We re-mark that there is a formal analogy between this construction and similar construction whichevolve from generalizing Drinfeld’s quantum double, cf. [22, Chapter IX], [30, Chapter 6, 7]. We fix once and for all a group Γ . Wedenote the Yetter-Drinfeld category over the group algebra k Γ by ΓΓ YD , and we call its objectsYetter-Drinfeld Γ -modules. Let V be a Yetter-Drinfeld Γ -module. We always denote by V g the component of V of Γ -degree g . We assume for the rest of this subsection that V is finitedimensional and that its support in the sense of [32, Section 4, p. 8], i.e. the set of all g ∈ Γ such that V g = 0 , consists of involutions. Let b be a homogeneous basis of V with respect tothe Γ -grading, and let b ∗ be its dual basis of V ∗ , which is of course again homogeneous withrespect to the Γ -grading. By sending a member of b to its corresponding dual member in b ∗ and extending linearly, we define an isomorphism V ∼ = V ∗ of Yetter-Drinfeld Γ -modules,which in turn induces an isomorphism B ( V ) ∼ = B ( V ∗ ) of braided Z ≥ -graded Hopf algebras.Upon identification of V ∗ with V and B ( V ∗ ) with B ( V ) along these isomorphisms inducedby b , the Hopf duality pairing between B ( V ∗ ) and B ( V ) becomes a symmetric Hopf dualitypairing between B ( V ) and itself whose restriction to V ⊗ V has identical representation matrixwhen represented with respect to b . In the current situation, one introduces according to[8, Section 3], [28, Subsection 2.3] two Z ≥ -graded endomorphisms ρ, ¯ S of B ( V ) which areuniquely determined by the requirement that(S1) ρ is the identity in Z ≥ -degree zero and one,(S2) ρ is an anti-algebra homomorphism, i.e. we have ρ ( xy ) = ρ ( y ) ρ ( x ) for all x, y ∈ B ( V ) ,(S3) ¯ S restricted to B ( V ) m is given by ( − m ρ S .As in Remark 4.2, it is convenient to consider the maps ρ and ¯ S (as well as the antipode of B ( V ) and its inverse) as endomorphisms of B ( V ) acting either on the left or the right, where,each time, the acting direction will be clear from context or will be indicated by evaluationon elements in B ( V ) or placeholders ( − ) . W . Let V W bethe Yetter-Drinfeld W -module defined as the quotient of the free vector space with basis ([ α ]) α ∈ R by its vector subspace span k { [ α ] + [ − α ] | α ∈ R } , equipped with the W -action wx α = x w ( α ) for all w ∈ W and all α ∈ R , where x α always denotes the image of [ α ] in V W ,and equipped with the W -grading given by assigning the W -degree s α to x α for all α ∈ R .The Yetter-Drinfeld W -module V W has support T and is of dimension | R + | where a canonicalhomogeneous basis of V W with respect to the W -grading is given by ( x α ) α ∈ R + . Hence, all theassumptions of Subsection 4.3 are satisfied for V W . As a consequence of Subsection 4.3, we canand will from now on identify V ∗ W with V W and B ( V ∗ W ) with B ( V W ) along the isomorphisms V W ∼ = V ∗ W , B ( V W ) ∼ = B ( V ∗ W ) induced by ( x α ) α ∈ R + as it was proposed in [9, Subsection 4.4].Following [9], we set B W = B ( V W ) = B ( V ∗ W ) for short and call B W the Nichols-Woronowiczalgebra model for Schubert calculus on W . Remark 4.4.
From now on, we work mostly with the Nichols algebra B W , or even with thespecial case B S m whenever indicated, while we point out some generalizations in the nextsubsection. In this situation, we use shortcuts for braided partial derivatives, namely, wewrite ←− D α = ←− D x α and −→ D α = −→ D x α for all α ∈ R . We want to point out some generalizations which might be relevantfor further work on the dimension of Nichols algebras in general:
N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 13 • Lemma 5.9(3)–(8) works for any B ( V ) where V is a finite dimensional Yetter-Drinfeld Γ -module whose support consists of involutions once we choose a homogeneous basisof V with respect to the Γ -grading and do the identifications as in Subsection 4.3.Lemma 5.9(1),(2), however, make use of the nilpotent relation in B W which is notevident for arbitrary B ( V ) as in the previous sentence. • Section 7 without its unnumbered subsection, i.e. Proposition 7.1 and Corollary 7.2,work for any B ( V ) where V is a finite dimensional Yetter-Drinfeld Γ -module whosesupport consists of involutions once we choose a homogeneous basis of V with respectto the Γ -grading and do the identifications as in Subsection 4.3. • Corollary 9.13(2),(3) except the part “ x ∼ gx ” works for any finite dimensional B ( V ) where V is a Yetter-Drinfeld Γ -module whose support consists of involutions. We donot have to make the exception “ x ∼ gx ” if V is in addition link-indecomposable inthe sense of [32, Section 4, p. 8], i.e. if in addition the support of V generates Γ . • Corollary 9.13(5),(6) works for any finite dimensional connected braided Z ≥ -gradedHopf algebra A in HH YD which is generated in Z ≥ -degree one once we choose a basisof A . • The principle of concrete commutativity in Theorem 12.3 works for arbitrary finitedimensional Nicholas algebras B ( V ) associated to a Yetter-Drinfeld Γ -module V uponchoice of a basis of V .5. Terminology concerning monomials
In this section, we want to setup a common language concerning monomials to speak aboutcertain situations which will arise throughout the paper. All statements in this section areeither trivial or immediate consequences of the braided Leibniz rule.
Definition 5.1.
Let V be a Yetter-Drinfeld H -module. Let ( x α ) α ∈ I be a basis of V . Let γ ∈ I . Let Θ ⊆ I . • We say that M ∈ B ( V ) is a monomial if there exist λ ∈ k and α , . . . , α m ∈ I suchthat M = λx α · · · x α m . • We say that a monomial M ∈ B ( V ) starts with γ if there exist λ ∈ k and α , . . . , α m ∈ I such that M = λx α · · · x α m and such that α = γ . • We say that a monomial M ∈ B ( V ) ends with γ if there exist λ ∈ k and α , . . . , α m ∈ I such that M = λx α · · · x α m and such that α m = γ . • We say that a monomial M ∈ B ( V ) starts with Θ if it is a monomial which startswith δ for some δ ∈ Θ . • We say that a monomial M ∈ B ( V ) ends with Θ if it is a monomial which ends with δ for some δ ∈ Θ . • We say that a monomial M ∈ B ( V ) does only involve Θ if there exist λ ∈ k and α , . . . , α m ∈ Θ such that M = λx α · · · x α m . • We say that an element z ∈ B ( V ) starts with γ if it can be written as a sum ofmonomials which start with γ . • We say that an element z ∈ B ( V ) ends with γ if it can be written as a sum ofmonomials which end with γ . • We say that an element z ∈ B ( V ) starts with Θ if it can be written as a sum ofmonomials which start with Θ . • We say that an element z ∈ B ( V ) ends with Θ if it can be written as a sum ofmonomials which end with Θ . • We say that an element z ∈ B ( V ) does only involve Θ if it can be written as a sumof monomials which do only involve Θ . Remark 5.2.
Let the notation be as in Definition 5.1. Note that we allow every scalar asa monomial in Definition 5.1. Consequently, every scalar is also a monomial in B ( V ) whichdoes only involve Θ for any Θ ⊆ I . Conversely, every element in B ( V ) which does onlyinvolve ∅ must be a scalar. Remark 5.3.
Let the notation be as in Definition 5.1. Note that a monomial in B ( V ) whichstarts with γ / ends with γ / starts with Θ / ends with Θ / does only involve Θ is in particularan element in B ( V ) which starts with γ / ends with γ / starts with Θ / ends with Θ / doesonly involve Θ . In that sense, Definition 5.1 is consistent. Remark 5.4.
Let the notation be as in Definition 5.1. Note that a monomial or element in B ( V ) starts with γ / ends with γ if and only if it starts with { γ } / ends with { γ } . Remark 5.5.
Let the notation be as in Definition 5.1. Note that an element z ∈ B ( V ) startswith γ if and only if it can be written as x γ z ′ for some z ′ ∈ B ( V ) . Similarly, an element z ∈ B ( V ) ends with γ if and only if it can be written as z ′ x γ for some z ′ ∈ B ( V ) . Remark 5.6.
Let the notation be as in Definition 5.1. Note that a monomial M ∈ B W startswith γ / ends with Θ / does only involve Θ if and only if ρ ( M ) is a monomial in B W whichends with γ / ends with Θ / does only involve Θ . Consequently, an element z ∈ B W startswith γ / ends with Θ / does only involve Θ if and only if ρ ( z ) is an element in B W which endswith γ / ends with Θ / does only involve Θ . This is clear from [8, Proposition 3.7(2),(6)]and the definition of ρ . Remark 5.7.
Le the notation be as in Definition 5.1. Note that every monomial in B ( V ) ishomogeneous with respect to the Z ≥ -grading. More specifically, let V be a Yetter-Drinfeld Γ -module and let ( x α ) α ∈ I be a basis of V homogeneous with respect to the Γ -grading. Inthe situation of the previous sentence, every monomial in B ( V ) is even homogeneous withrespect to the Z ≥ -grading and the Γ -grading. Remark 5.8.
Whenever we work with B W as introduced in and subject to the identificationsin Subsection 4.4, we consider the canonical homogeneous basis of V W with the respect tothe W -grading given by ( x α ) α ∈ R + , and the terminology in Definition 5.1 will always refer tothis basis and the index set I = R + . Lemma 5.9.
Let Θ ⊆ R + .(1) Let z ∈ B W be an element which starts with γ for all γ ∈ Θ . Let ξ ∈ B W be anelement which ends with Θ . Then, we have ξz = 0 .(2) Let z ∈ B W be an element which ends with γ for all γ ∈ Θ . Let ξ ∈ B W be an elementwhich starts with Θ . Then, we have zξ = 0 .(3) Let z ∈ B W . We have ( z ) ←− D α = 0 for all α ∈ Θ if and only if ( z ) ←− D ξ = 0 for all ξ ∈ B W which start with Θ .(4) Let z ∈ B W . We have −→ D α ( z ) = 0 for all α ∈ Θ if and only if −→ D ξ ( z ) = 0 for all ξ ∈ B W which end with Θ .(5) Let z , . . . , z m ∈ B W be such that ( z ) ←− D α = · · · = ( z m ) ←− D α = 0 for all α ∈ Θ . Then,we have ( z · · · z ) ←− D ξ = 0 for all ξ ∈ B W which start with Θ .(6) Let z ∈ B W be an element which does only involve R + \ Θ . Let ξ ∈ B W be an elementwhich starts with Θ . Then, we have ( z ) ←− D ξ = 0 . N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 15 (7) Let z , z ∈ B W be such that ( z ) ←− D α = 0 for all α ∈ Θ . Then, we have ( z z ) ←− D ξ = z ( z ) ←− D ξ for all ξ ∈ B W which do only involve Θ .(8) Let z , z ∈ B W be such that −→ D α ( z ) = 0 for all α ∈ Θ and such that z is homogeneousof W -degree g . Then, we have −→ D ξ ( z z ) = z −→ D g − ξ ( z ) for all ξ ∈ B W which do onlyinvolve Θ .Proof of Item (1). For the proof of the desired vanishing, we can assume that ξ is a monomialwhich ends with Θ . In that case, we may write ξ as ξ ′ x α for some α ∈ Θ and some monomial ξ ′ ∈ B W . By assumption, the element z ∈ B W starts with α . Hence, it can be written as x α z ′ for some z ′ ∈ B W by Remark 5.5. We conclude that ξz = ξ ′ x α z ′ is zero since x α = 0 by[8, Example 4.4]. (cid:3) Proof of Item (2).
For the proof this item, one can argue analogously as in the proof ofItem (1). Alternaftively, one can apply Item (1) to ρ ( z ) and ρ ( ξ ) and apply Remark 5.6 andthe definition of ρ . (cid:3) Proof of Item (3).
The implication from right to left is obvious because every x α where α ∈ Θ is a monomial which starts with Θ . For the other implication, it suffices to prove the vanishingif ξ is an arbitrary but fixed monomial which starts with Θ . In that case, we may write ξ as x α ξ ′ for some α ∈ Θ and some monomial ξ ′ ∈ B W . It follows that ( z ) ←− D ξ = ( z ) ←− D α ←− D ξ ′ = 0 because ( z ) ←− D α vanishes by assumption. (cid:3) Proof of Item (4).
For the proof this item, one can argue analogously as in the proof ofItem (3). Alternatively, one can apply Item (3) to ¯ S ( z ) and ρ ( ξ ) . The result follows this wayusing [8, Proposition 3.7(6), Proposition 4.2, Remark 4.3] and Remark 5.6. (cid:3) Proof of Item (5).
By Item (3), we may assume that ξ = x α for some α ∈ Θ . The vanishingof ( z · · · z m ) ←− D α then follows from the braided Leibniz rule. (cid:3) Proof of Item (6).
For the proof of the desired vanishing, we may assume that z is a monomialwhich does only involve R + \ Θ and further, by Item (3), that ξ = x α for some α ∈ Θ . Bythis assumption, there exist λ ∈ k and α , . . . , α m ∈ R + \ Θ such that z = λx α · · · x α m . Ifwe apply Item (5) to z i = x α i for all ≤ i ≤ m and to ξ , the result follows. (cid:3) Proof of Item (7),(8).
For the proof of these items, we may assume that ξ is a monomialwhich does only involve Θ . By induction on the Z ≥ -degree of the monomial ξ , we canfurther assume that its Z ≥ -degree is one. But in that case, the desired equations followagain from the braided Leibniz rule. (cid:3) Lemma 5.10.
Let ξ ∈ B W be a nonzero homogeneous element of some Z ≥ -degree m andsome W -degree w . Then, the parity of ℓ ( w ) equals the parity of m .Proof. To prove the lemma, we can clearly assume that ξ is a nonzero monomial of some Z ≥ -degree m and some W -degree w . If we assume this from now on, we can find λ ∈ k × and α , . . . , α m ∈ R + such that ξ = λx α · · · x α m . Then, we necessarily have w = s α · · · s α m .Every reflection has odd length. Thus, by the deletion condition [20, Corollary 5.8], we seethat ℓ ( w ) has the same parity as m . (cid:3) NilCoxeter algebra
The nilCoxeter algebra was first introduced in [15] and studied in the context of thesymmetric group. Schubert polynomials and Stanley symmetric functions are treated in [15]as coefficients of certain expressions in the nilCoxeter algebra. Independently from this, weadopt here the point of view developed in [9] where the nilCoxeter algebra of W is canonicallyembedded into B W for any W .Let N W be the subalgebra of B W generated by x β where β runs through ∆ . As it wasproved in [9, Theorem 6.3(i),(ii)], these generators of N W satisfy the so-called nilCoxeterrelations, i.e. the homogeneous relations x β = 0 for all β ∈ ∆ and x β x β ′ x β · · · | {z } = x β ′ x β x β ′ · · · | {z } m ( s β ,s β ′ ) factors on each side for all β, β ′ ∈ ∆ , and all other relations between them are generated from those. Therefore,the algebra N W is called the nilCoxeter algebra of W in the sense of [15, Section 2]. Let s β · · · s β m be a reduced expression of some w ∈ W . Then, we define x w = x β · · · x β m . Thiselement is well-defined, i.e. independent of the choice of the reduced expression of w , becauseof the nilCoxeter relations and because any two reduced expressions of w can be connected bya sequence of braid moves by the Word Property [11, Theorem 3.3.1(ii)]. With this definition,the family ( x w ) w becomes a basis of N W . We call this basis the standard basis of N W . Wefinally remark that by definition x = 1 is a member of the standard basis of N W .Following Liu’s coproduct approach [28, Proposition 2.7], see also [8, Definition 5.1], weintroduce for all v, w ∈ W uniquely determined elements x w/v ∈ B W such that ∆( x w ) = P v x w/v ⊗ x v . By definition, we then have x w/v = 0 for all v w , and further x w/ = x w and x w/w = 1 for all w ∈ W . In line with Remark 4.4, we use further shortcuts, namely, we write ←− D w = ←− D x w and ←− D w/v = ←− D x w/v for all v, w ∈ W . Fact 6.1.
Let u, v, w ∈ W . Let y = wx v .(1) The element y is nonzero and homogeneous of Z ≥ -degree ℓ ( v ) and of W -degree wvw − . In particular, if v = w o and if w is an element in the centralizer of w o ,then the element y is nonzero and homogeneous of Z ≥ -degree ℓ ( w o ) and of W -degree w o .(2) We have w o x v = ( − ℓ ( v ) x w o vw o . If w is an element in the centralizer of w o , this meansthat w o y = ( − ℓ ( v ) wx w o vw o . If v and w are both elements in the centralizer of w o ,we see in particular that w o y = ( − ℓ ( v ) y . If v = w o and if w is an element in thecentralizer of w o , we see further that w o y = ( − ℓ ( w o ) y .(3) We have h x u , x v i = δ u,v − . If v is an involution, e.g. if v = w o , we have in particularthat h y, y i = 1 .(4) We have ρ ( x v ) = x v − . If v is an involution, e.g. if v = w o , it follows in particularthat ρ ( y ) = y . Remark 6.2.
The proof of Item (1) and its content are plain. We will use Item (1) withoutreference from now on.
Remark 6.3.
Note that it follows from Item (3) that the family ( x w ) w is a basis of N W . Remark 6.4. If v = w o and if w is an element in the centralizer of w o , then Item (3) says inparticular that h y, y i = 1 which is precisely the meaning of Corollary 8.5 applied to a disjointsystem of order one. In this sense, there is some overlap between Item (3) and Corollary 8.5. N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 17
Proof of Item (2).
In the proof of this item, we repeatedly use the fact that w o is an involu-tion. We only have to prove the first formula in the statement of Item (2). The special casesare immediate from it. To this end, recall that ℓ ( v ) = ℓ ( w o vw o ) because of [20, Section 1.8,Equation (2), Section 5.2, Equation (L1)], so that s β · · · s β m is a reduced expression of v ifand only if s w o ( β ) · · · s w o ( β m ) is a reduced expression of w o vw o by [20, Section 5.6, Exercise 2].In the transition from w o x v to x w o vw o , we therefore pick up a sign for each of the ℓ ( v ) factorsof x v because x − α = − x α for all α ∈ R . These observations prove the desired formula. (cid:3) Proof of Item (3).
We only have to prove the first formula in the statement of Item (3). Thespecial case follows because h− , −i is a morphism in the Yetter-Drinfeld category over W .But the first formula was already proved in [8, Proposition 6.4]. (cid:3) Proof of Item (4).
It is clear that s β · · · s β m is a reduced expression of v if and only if s β m · · · s β is a reduced expression of v − . The formula ρ ( x v ) = x v − follows from this.The special case follows then by [8, Proposition 3.7(1)]. (cid:3) Lemma 6.5.
Let w ∈ W . Let y = wx w o . The element y ∈ B W is a monomial which startswith γ for all γ ∈ T w . The element y ∈ B W is also a monomial which ends with γ for all γ ∈ T w and a monomial which does only involve T w .Proof. Let the notation be as in the statement. The statement in the last sentence is clearfrom the statement in the second last sentence because of Remark 5.6 and Fact 6.1(4). Weprove the statement in the second last sentence now. Let γ ∈ T w be fixed but arbitrary.Then, there exists a unique β ∈ ∆ and a unique sign ǫ such that w ( β ) = ǫγ . We may write x w o = x β x s β w o and thus y = ǫx γ ( wx s β w o ) . In this form, the monomial y is visibly a monomialwhich starts with γ . (cid:3) Corollary 6.6.
Let w ∈ W . Let y = wx w o . Then, we have zy = 0 for every element z ∈ B W which ends with T w and yz = 0 for every element z ∈ B W which starts with T w .Proof. This is clear from Lemma 5.9(1),(2) and Lemma 6.5. (cid:3)
Lemma 6.7.
Let v, w ∈ W be such that v = w o , then wx w o /v is a monomial which startswith T w .Proof. Let v, w ∈ W be such that v = w o . By [8, Example 5.4, Theorem 6.11] and because w o is an involution, we know that wx w o /v = w ¯ S ( x vw o ) . Since v = w o , we can find β ∈ ∆ such that s β vw o < vw o . Further, there exists a unique γ ∈ T w and a unique sign ǫ such that w ( β ) = ǫγ .By [8, Proposition 3.7(4)] and by what was said, we see that wx w o /v = ǫx γ ( s γ w ¯ S ( x s β vw o )) .From this equality and [8, Remark 3.8], we see that wx w o /v is indeed a monomial which startswith γ , and this γ lies in T w by definition. (cid:3) Inversion of braided partial derivatives
By inversion of braided partial derivatives, we mean the third formula in Proposition 7.1which shows how ρ and ←− D ξ where ξ ∈ B W commute when considered as endomorphisms of B W acting from the right. Such a formula has relevance for the vanishing in Corollary 7.2which is used in the proof of Lemma 14.3. Further, in this section, we work out a variantof the Nichols-Zoeller theorem for B W , namely Corollary 7.5, based on Corollary 7.4, wherethe latter result is used in the proof of Corollary 9.13(4). Proposition 7.1.
Let ξ ∈ B W be homogeneous of W -degree g . Then, we have S ←− D ξ = g − S −→ D S ( ξ ) , −→ D ξ S − = ←− D S − ( ξ ) S − g − ,ρ ←− D ξ = ←− D ¯ S ( ξ ) ρg . Proof.
Let us recall the equivalent formulas ∆ S = ( S ⊗ S )Ψ∆ , Ψ − ( S − ⊗ S − )∆ = ∆ S − from [30, Equation (9.39) on page 477] where we suppress the components of the braiding Ψ .The first of those formulas can be put in words by saying that the antipode in a braided Hopfalgebra is a braided anti coalgebra homomorphism. If we use suggestive Sweedler notationand plug in the braiding of WW YD , we can equally well write these formulas as S ( z ) (1) ⊗ S ( z ) (2) = g (1) S ( z (2) ) ⊗ S ( z (1) ) , S − ( z ) (1) ⊗ S − ( z ) (2) = S − ( z (2) ) ⊗ g − S − ( z (1) ) where z is an arbitrary element in B W and where g (1) , g (2) denote the W -degree of z (1) , z (2) ,respectively.With the help of the previous two formulas and the basic properties of h− , −i , we compute ( S ( z )) ←− D ξ = g (1) S ( z (2) ) (cid:10) S ( z (1) ) , ξ (cid:11) = (cid:10) S ( ξ ) , z (1) (cid:11) g − S ( z (2) ) = g − S −→ D S ( ξ ) ( z ) , −→ D ξ S − ( z ) = (cid:10) ξ, S − ( z (2) ) (cid:11) g − S − ( z (1) ) = g S − ( z (1) ) (cid:10) z (2) , S − ( ξ ) (cid:11) = ( z ) ←− D S − ( ξ ) S − g − . This proves the first two formulas in the statement.To prove the third formula, we can restrict the operators in the formula to a gradedcomponent B mW of some Z ≥ -degree m . Once the equality is proved for arbitrary but fixed m , it will be valid everywhere on B W . Let ǫ = ( − m . Because ρ and ¯ S are involutions by [8,Proposition 3.7(6)], we see from its definition that ¯ S restricted to B mW is given by ǫρ S = ǫ S − ρ .With the help of [8, Proposition 3.7(1), Proposition 4.2, Remark 2.16, Remark 4.3], this lastformula for ¯ S restricted to B mW and the second formula in the statement which was alreadyjustified, we compute ρ ←− D ξ = S − ¯ S ←− D ξ ǫ = ǫ ¯ S −→ D ρ ( ξ ) S − = ←− D S − ρ ( ξ ) S − ¯ S gǫ = ←− D ¯ S ( ξ ) ρg where this computation takes place restricted to B mW . By what said before, this completesthe proof. (cid:3) Corollary 7.2.
Let b ∈ B W . For all α ∈ R + , we have ( b ) ←− D α = 0 if and only if ( ρ ( b )) ←− D α = 0 . Proof.
The third formula in Proposition 7.1 applied to ξ = x α for some α ∈ R + becomes ρ ←− D α = ←− D α ρs α because ¯ S is the identity in Z ≥ -degree one by definition. The claimed equivalence is im-mediate from this formula and the fact that the operator ρ is invertible by [8, Proposi-tion 3.7(6)]. (cid:3) N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 19
A variant of the Nichols-Zoeller theorem.Theorem 7.3.
Let ξ ∈ B W be a homogeneous element of W -degree g . Then, we have S − ( ξ ) = ( − ℓ ( g ) g − S ( ξ ) .Proof. For the proof of the desired formula, we may assume that ξ is a monomial of W -degree g . We assume this from now on and proceed by induction on the Z ≥ -degree of ξ . The caseof Z ≥ -degree equal to zero, i.e. ξ being a scalar multiple of , being obvious, we assumefurther that the Z ≥ -degree of ξ is > . In that case, we may write ξ as x α ξ ′ for some α ∈ R + and some monomial ξ ′ ∈ B W . With the help of [30, Equation (9.39) on page 477] and theinduction hypothesis, we compute S − ( ξ ) = S − ( x α ξ ′ )= S − ( ξ ′ )( g − s α S − ( x α ))= ( − ℓ ( g )+1 ( g − s α S ( ξ ′ ))( g − s α S ( x α ))= ( − ℓ ( g ) g − (( s α S ( ξ ′ )) S ( x α )= ( − ℓ ( g ) g − S ( x α ξ ′ )= ( − ℓ ( g ) g − S ( ξ ) (cid:3) Corollary 7.4.
Let ξ ∈ B W be a homogeneous element of W -degree g . Then, we have S ( ξ ) = ( − ℓ ( g ) gξ .Proof. This is an immediate corollary of Theorem 7.3. (cid:3)
Corollary 7.5 (A variant of the Nichols-Zoeller theorem [33, Theorem 10.5.6, Corollary10.5.7(a)]) . Let e be the exponent of W , i.e. the least common multiple of the orders of allelements of W as in [39] . Then, we have S e = 1 .Proof. Indeed, it follows from Corollary 7.4 that S e ( ξ ) = ( − e · ℓ ( g ) g e ξ for all homogeneouselements ξ ∈ B W of W -degree g . By definition of e , we know that e is even, and furtherthat e is the smallest positive integer N such that g N = 1 for all g ∈ W . It follows that ( − e · ℓ ( g ) g e = 1 for all g ∈ W , and consequently S e = 1 – as claimed. (cid:3) Consequences of the generalized braided Leibniz rule
The general braided Leibniz rule appears in the context of the symmetric group, divideddifference operators and skew divided difference operators in the papers [23, 28, 29], forexample. We adopt here the point of view developed in [8] where a generalized braidedLeibniz rule was introduced for braided partial derivatives acting on B W . We give a selectivelist of consequences of this rule. Theorem 8.1 (Generalized braided Leibniz rule [8, Theorem 5.14], [23, Proposition 3(ii)],[29, Chapter 2, Theorem 2.18]) . Let v, w, w ′ ∈ W . Let z ∈ B W . Then, we have z ←− D w ′ x w/v = X v ≤ u ≤ w ←− D w ′ x u/v (cid:0) ( z ) ←− D w ′ x w/u w ′ uv − w ′− (cid:1) where we can equally well take the sum over all u ∈ W .Proof. Note that x w/v = 0 for all v w by definition. This shows that we can equally wellsum over all u ∈ W in the formula in the statement and everywhere else (in this proof) where similar situations arise. Let v, w, w ′ ∈ W be arbitrary. Suppose that the claimed formula isproved for w ′ = 1 , then we find for arbitrary w ′ that P u ←− D w ′ x u/v (cid:0) ( z ) ←− D w ′ x w/u w ′ uv − w ′− (cid:1) = P u ←− D w ′ x u/v (cid:0) ( zw ′ ) ←− D x w/u uv − w ′− (cid:1) = P u w ′←− D x u/v (cid:0) ( zw ′ ) ←− D x w/u uv − (cid:1) w ′− = w ′ ( zw ′ ) ←− D x w/v w ′− = z ←− D w ′ x w/v where we used Remark 4.2, [8, Remark 3.16] and the assumption for w ′ = 1 . This meansthat it suffices to prove the claimed formula for w ′ = 1 . We now do so.Let x, y ∈ B W . Evaluating once ( x ( yz )) ←− D w and twice (( xy ) z ) ←− D w with the help of [8,Theorem 5.14], we find the equality X v ′ ( x ) ←− D v ′ (cid:0) ( yz ) ←− D w/v ′ v ′ (cid:1) = X u,v ′ ( x ) ←− D v ′ (cid:0) ( y ) ←− D u/v ′ v ′ (cid:1)(cid:0) ( z ) ←− D w/u u (cid:1) . If we plug x = x v − into this equality, it becomes(1) X v ′ x v − /v ′− (cid:0) ( yz ) ←− D w/v ′ v ′ (cid:1) = X u,v ′ x v − /v ′− (cid:0) ( y ) ←− D u/v ′ v ′ (cid:1)(cid:0) ( z ) ←− D w/u u (cid:1) . in view of [8, Proposition 8.1]. We now prove by induction on ℓ ( v ) that ( yz ) ←− D w/v v = X u (( y ) ←− D u/v v (cid:1)(cid:0) ( z ) ←− D w/u u (cid:1) which suffices to finish the proof of the theorem because we can multiply with v − from theright. If v = 1 , the desired formula is simply [8, Theorem 5.14] by [8, Example 5.4]. Supposethat ℓ ( v ) > and that the induction hypothesis is satisfied for all v ′ of length < ℓ ( v ) . Withthe help of Equation (1) and [8, Example 5.4] we find that ( yz ) ←− D w/v v + X v ′ Let v, w ∈ W . Let y = wx w o . Then, we have ←− D y z ←− D wx v = ←− D y (cid:0) ( z ) ←− D wx v (cid:1) for all z ∈ B W . The expression ( zw ′ ) ←− D x w/v before the last equality in this equation has to be understood as the compo-sition of two endomorphisms of B W acting from the right, namely the composition of multiplication fromthe right with zw ′ = w ′− z and ←− D x w/v , and should not be confused with the endomorphism given by rightmultiplication with ( zw ′ ) ←− D x w/v . Except maybe in the proof of Corollary 8.3 where we explicitly make it clearin writing, this is the only instance where the conventions in Remark 4.2 lead to ambiguity. Everywhere elsethe absence or presence of parenthesis and arrows makes the meaning clear. N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 21 Proof. Let the notation be as in the statement. Let z ∈ B W be fixed but arbitrary. ByTheorem 8.1, we have ←− D y z ←− D wx v = X u ←− D y ( wx u ) (cid:0) ( z ) ←− D wx v/u wuw − (cid:1) . Since wx u is a monomial which starts with T w whenever u = 1 , we see from Corollary 6.6that y ( wx u ) vanishes for all u = 1 . In view of [8, Example 5.4], the above sum thereforereduces to the right side of the claimed formula in the statement of the corollary. (cid:3) Corollary 8.3. Let v, w ∈ W . Let b ∈ B W be such that ( b ) ←− D α = 0 for all α ∈ T w . Then, wehave b ←− D wx wo/v = ←− D wx wo/v ( wvw o w − b ) . Proof. Let the notation be as in the statement. By Lemma 5.9(3), Lemma 6.7 and [8,Example 5.4], we see that ( b ) ←− D wx wo/u vanishes for all u = w o and evaluates as b for u = w o .This means that if we apply Theorem 8.1 to the left side of the claimed formula in thestatement of the corollary, in the corresponding sum over u only one term associated to u = w o survives and this term is equal to the right side of the claimed formula in thestatement of the corollary. Note also that bww o v − w − = wvw o w − b considered as equalityof elements in B W because w o is an involution. This completes the proof. (cid:3) Corollary 8.4. Let D be a disjoint system of order two. Let w , w be some ordering of theelements of D . Let y = w x w o and let y = w x w o . Then, we have y ←− D y = ←− D y y ( − ℓ ( w o ) .Proof. Let the notation be as in the statement. By Lemma 6.5, the monomial y does onlyinvolve T w ⊆ R + \ T w . Thus, by Lemma 5.9(6), we know that ( y ) ←− D α = 0 for all α ∈ T w .Therefore, Corollary 8.3 applies to v = 1 , w , b = y and we obtain the claimed formulabecause of Fact 6.1(2). (cid:3) Corollary 8.5. Let D be a disjoint system of order r . Let w , . . . , w r be some ordering ofthe elements of D . Let y i = w i x w o for all ≤ i ≤ r . Then, we have (cid:10) y π (1) · · · y π ( r ) , y σ (1) · · · y σ ( r ) (cid:11) = ( − ( r − r + ℓ ( σπ − ) ) · ℓ ( w o ) for all permutations σ, π ∈ S r . In particular, it follows that y σ (1) · · · y σ ( r ) is nonzero for allpermutations σ ∈ S r .Proof. Let the notation be as in the statement. The particular case is obvious from theclaimed equation. We prove this equation. Let σ, π ∈ S r be arbitrary. If we change theindices of the ordering of D from , . . . , r to π (1) , . . . , π ( r ) and replace σ by σπ − , we seethat we can assume directly in the beginning that π = 1 . We will assume π = 1 from nowon.Let us define indices i j , . . . , i jr − j such that { i j < · · · < i jr − j } = { , . . . , r } \ { σ (1) , . . . , σ ( j ) } for all ≤ j ≤ r − . In order to prove the corollary, it suffices to prove the formula(2) (cid:10) y i j · · · y i jr − j , y σ ( j +1) · · · y σ ( r ) (cid:11) = ( − n j +1 · ℓ ( w o ) (cid:10) y i j +11 · · · y i j +1 r − j − , y σ ( j +2) · · · y σ ( r ) (cid:11) where n j +1 = r − σ ( j + 1) − { ≤ i < j + 1 | σ ( i ) > σ ( j + 1) } and where ≤ j ≤ r − . Indeed, once Equation (2) is established, we apply it iterativelyfor all ≤ j ≤ r − and we find that the bracket in the statement of the corollary is equalto minus one to the power of P rj =1 n j times ℓ ( w o ) . But the parity of the previous sum isprecisely the parity of ( r − r + ℓ ( σ ) . In this way, we see that it indeed suffices to proveEquation (2).To prove Equation (2), let us fix some arbitrary ≤ j ≤ r − . Let ≤ k ≤ r − j be suchthat i jk = σ ( j + 1) . By definition, we know that n j +1 = r − j − k . From this and repeatedapplication of Corollary 8.4 to suitable subsets of D with two elements (cf. Remark 3.6), wesee that (cid:0) y i j · · · y i jr − j (cid:1) ←− D y σ ( j +1) = ( − n j +1 · ℓ ( w o ) (cid:0) y i j · · · y i jk (cid:1) ←− D y σ ( j +1) y i jk +1 · · · y i jr − j . If we use additionally Lemma 5.9(6),(7), 6.5 and Fact 6.1(3), we see that the above quantityequals ( − n j +1 · ℓ ( w o ) y i j +11 · · · y i j +1 r − j − . But this shows everything we have to show in order to justify Equation (2) and completesthe proof of the corollary. (cid:3) Corollary 8.6. Let w ∈ W . Let y = wx w o . Let x , x , x ∈ B W be such that ( x ) ←− D α = ( x ) ←− D α = ( ww o w − x ) ←− D α = ( x ) ←− D α = 0 for all α ∈ T w . Then, we have ( x yx x ) ←− D y = ( x ( ww o w − x ) yx ) ←− D y = x ( ww o w − ( x x )) . Proof. The claimed equalities follow directly from Fact 6.1, Lemma 5.9(5),(7), Corollary 8.3and the assumptions. (cid:3) Corollary 8.7. Let w be an element in the centralizer of w o . Let y = wx w o . Let x , x , x ∈ B W be such that ( x ) ←− D α = ( x ) ←− D α = ( x ) ←− D α = 0 for all α ∈ T w . Then, we have ( x yx x ) ←− D y = ( x ( w o x ) yx ) ←− D y = x ( w o ( x x )) . Proof. Let the notation be as in the statement. The result follows from application of Corol-lary 8.6. We simply have to verify that ( w o x ) ←− D α vanishes for all α ∈ T w . By Fact 3.2 and [8,Remark 3.16] this vanishing is equivalent to the vanishing ( x ) ←− D α = 0 for all α ∈ T w whichis part of the assumption. (cid:3) Integrals for Hopf algebras In this section, we study integrals for Hopf algebras with a view towards our applicationsto Nichols algebras. For background material on integrals, we refer to the foundational works[26, 35, 36], to the more recent reference [4] and to the pedagogical text book [30]. For therelation between integrals and Frobenius algebras, we refer to [21, 26]. Definition 9.1 ([4, Equation (2.3.6)], [36, Section 2]) . Let A be a braided Hopf algebra in HH YD . Let x ∈ A . • We say that x is a left integral in A if zx = ǫ ( z ) x for all z ∈ A . We say that x is anonzero left integral in A if x is nonzero and if x is a left integral in A . • We say that x is a right integral in A if xz = ǫ ( z ) x for all z ∈ A . We say that x is anonzero right integral in A if x is nonzero and if x is a right integral in A . N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 23 • We say that x is an integral in A if x is a left and right integral in A . We say that x is a nonzero integral in A if x is nonzero and if x is an integral in A . Remark 9.2. Let A be a braided Hopf algebra in HH YD . The presence of a nonzero leftintegral / right integral / integral in A gives rise to a one-dimensional left ideal / right ideal/ two-sided ideal in A . Remark 9.3. If A is a classical Hopf algebra with trivial braiding, then there exists a nonzeroleft integral in A if and only if there exists a nonzero right integral in A if and only if A is finitedimensional. This theorem is proved in [36, Corollary 2.7, Equivalence (2 . ⇔ (2 . ]. Ingeneral, it is known that for a braided Hopf algebra in HH YD of finite dimension the two-sidedideal of all left integrals as well as the two-sided ideal of all right integrals are one-dimensional(cf. [4, Section 2.3]). However, we do not know the literature well enough to decide whetherthe presence of a nonzero left or right integral in a braided Hopf algebra in HH YD impliesfinite dimensionality (just as it is known in the classical case by [36, loc. cit.]). Therefore, wewill from now on follow the following convention: If we speak about a nonzero left integral/ right integral / integral in a braided Hopf algebra A in HH YD , we implicitly assume that A is finite dimensional. Since we certainly use finite dimensionality of A in the presence of anykind of nonzero integrals in A , we do not lose anything from this assumption. Remark 9.4. To handle infinite dimensional braided Hopf algebras A in HH YD , and to intro-duce sensible notions of integrals for them which extend the notions in Definition 9.1, peopleoften consider left integrals / right integrals / integrals in A ∗ instead of A where A ∗ is thealgebra dual of A considered as a coalgebra, cf. [30, Section 1.7] and [35]. We mention it butwe will not need this approach in this work. Proposition 9.5. Let A be a connected braided Z ≥ -graded Hopf algebra in HH YD . Let x bea nonzero left or right integral in A .(1) The element x is homogeneous of some Z ≥ -degree m . The integer m satisfies A m = k x and A m ′ = 0 for all m ′ > m . In particular, every left or right integral in A is anintegral in A .(2) An element x ′ is an integral in A if and only if zx ′ = 0 for all homogeneous elements z ∈ A of Z ≥ -degree > if and only if x ′ z = 0 for all homogeneous elements z ∈ A of Z ≥ -degree > . If for this sentence, in addition, the algebra A is generated in Z ≥ -degree one, then an element x ′ is an integral in A if and only if zx ′ = 0 forall homogeneous elements z ∈ A of Z ≥ -degree one if and only if x ′ z = 0 or allhomogeneous elements z ∈ A of Z ≥ -degree one.(3) For a nonzero element x ′ ∈ A , there exist homogeneous elements y, y ′ ∈ A with respectto the Z ≥ -grading such that yx ′ and x ′ y ′ are nonzero integrals in A . Remark 9.6. Recall that a braided Z ≥ -graded Hopf algebra A in HH YD as in the statementof Proposition 9.5 is connected if and only if A = k . Proof of Proposition 9.5. Let the notation be as in the statement. Since A is finite dimen-sional, there exists an integer m such that A m = 0 and such that A m ′ = 0 for all m ′ > m .By the assumptions on A , every element in A m is an integral in A . By the uniqueness ofleft or right integrals in A up to scalar or the uniqueness of right integrals in A up to scalar(cf. [4, Section 2.3 or Proposition 3.2.2] or [6, Lemma 1.12]), we conclude that A m is theone-dimensional two-sided ideal of all integrals in A . Item (1) follows from this. Item (2)follows from Item (1) and the assumptions on A . Let us prove Item (3). Let x ′ be a nonzero element in A . Let m ′ be the smallest integersuch that in the decomposition P m ′′ x ′ m ′′ of x ′ as a sum of homogeneous elements x ′ m ′′ of Z ≥ -degree m ′′ the element x ′ m ′ is nonzero. By Item (1), it suffices to prove the statementof Item (3) for x ′ m ′ instead of x ′ . Hence, we may and will assume right in the beginningthat x ′ is homogeneous of some Z ≥ -degree m ′ . We now perform an induction on m − m ′ . If m = m ′ , then x ′ is a nonzero integral in A by Item (1) and we can set y = y ′ = 1 . If m > m ′ ,then x ′ is a nonzero element in A which is not a nonzero integral in A . By Item (2), we canfind homogeneous elements y , y ′ ∈ A of Z ≥ -degree > such that y x ′ and x ′ y ′ are bothnonzero. By our assumption on x ′ and our choice of y , y ′ , we know that y x ′ and x ′ y ′ areboth nonzero homogeneous elements of Z ≥ -degree > m ′ . By the induction hypothesis, thereexists homogeneous elements y , y ′ ∈ A with respect to the Z ≥ -grading such that y y x ′ and x ′ y ′ y ′ are nonzero integrals in A . The elements y = y y and y ′ = y ′ y ′ are homogeneouselements of A with respect to the Z ≥ -grading as required. (cid:3) Remark 9.7. In the rest of this work, we will only be concerned with connected braided Z ≥ -graded Hopf algebras in HH YD . As we see from Proposition 9.5(1), for a connected braided Z ≥ -graded Hopf algebra A in HH YD , we do not need to make a distinction between left orright integrals in A and integrals in A . Therefore, we will not further make the distinctionin our statements. From now on, we will state our results only for bona fide integrals. Remark 9.8. Let A be a a connected braided Z ≥ -graded Hopf algebra in HH YD . In view ofthe convention in Remark 9.3 and Proposition 9.5(1), we have the equivalence that A is finite dimensional if and only if there exists a nonzero integral in A which we will use from now on without reference. Corollary 9.9. Let A and B be braided Z ≥ -graded Hopf algebras in HH YD . Assume that A or B is connected and that there exists a nondegenerate Hopf duality pairing h− , −i between A and B which respects the Z ≥ -grading. Then, both Hopf algebras A and B are connected.Moreover, there exists a nonzero integral in A if and only if there exists a nonzero integralin B . Let x ∗ be a nonzero integral in A and let x be a nonzero integral in B .(1) We have h x ∗ , x i 6 = 0 .(2) Let x ′ ∈ B be a homogeneous element with respect to the Z ≥ -grading such that h x ∗ , x i = h x ∗ , x ′ i . Then, it follows that x = x ′ . Let x ∗′ ∈ A be a homogeneouselement with respect to the Z ≥ -grading such that h x ∗ , x i = h x ∗′ , x i . Then, it followsthat x ∗ = x ∗′ .Proof. Let the notation be as in the statement. By assumption, we have A ∼ = B , so that A is connected if and only if B is. Because of the presence of a nondegenerate pairingbetween A and B , we know that A is finite dimensional if and only if B is. From this andProposition 9.5(1), it follows that there exists a nonzero integral in A if and only if thereexists a nonzero integral in B .Item (1) follows from Proposition 9.5(1) and the assumptions on h− , −i . Item (2) followsfrom Item (1) and again from Proposition 9.5(1) and the assumptions on h− , −i . (cid:3) Proposition 9.10. Let V be a finite dimensional Yetter-Drinfeld H -module. Let x be anonzero integral in B ( V ) and let x ∗ be a nonzero integral in B ( V ∗ ) .(1) Every nonzero element y ∈ B ( V ) satisfies ( x ∗ ) ←− D y = 0 and every nonzero element y ∗ ∈ B ( V ∗ ) satisfies −→ D y ∗ ( x ) = 0 . N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 25 (2) Let x ′ ∈ B ( V ) m and y ∗ ∈ B ( V ∗ ) m ∗ for some m, m ∗ ∈ Z ≥ such that y ∗ is nonzeroand such that −→ D y ∗ ( x ) = −→ D y ∗ ( x ′ ) . Then, we have x = x ′ . Let x ∗′ ∈ B ( V ∗ ) m ∗ and y ∈ B ( V ) m for some m, m ∗ ∈ Z ≥ such that y is nonzero and such that ( x ∗ ) ←− D y = ( x ∗′ ) ←− D y .Then, we have x ∗ = x ∗′ .Proof. Let x and x ∗ be as in the statement. Let us prove Item (1). Let y be a nonzeroelement in B ( V ) and let y ∗ be a nonzero element in B ( V ∗ ) . By Proposition 9.5(3), thereexist homogeneous elements x ′ ∈ B ( V ) and x ∗′ ∈ B ( V ∗ ) with respect to the Z ≥ -grading suchthat yx ′ is a nonzero integral in B ( V ) and such that x ∗′ y ∗ is a nonzero integral in B ( V ∗ ) .From Corollary 9.9(1), it follows that = h x ∗ , yx ′ i = (cid:10) ( x ∗ ) ←− D y , x ′ (cid:11) , = h x ∗′ y ∗ , x i = (cid:10) x ∗′ , −→ D y ∗ ( x ) (cid:11) . The result in Item (1) is clear from these equations.Let us prove Item (2). Let the notation be as in the first sentence of the statement ofItem (2). By Proposition 9.5(1), [8, Remark 2.16] and Item (1) of this proposition, we notethat x ′ is nonzero and that m is the Z ≥ -degree of x . Again, by Proposition 9.5(1), we seethat x ′ is a nonzero integral in B ( V ) and that there exists a nonzero scalar λ such that x ′ = λx . It follows that −→ D y ∗ ( x ) = λ −→ D y ∗ ( x ) . By Item (1), we conclude that λ = 1 and thus x = x ′ . The rest of the statement of Item (2) can be proved analogously. In the case of afinite dimensional Yetter-Drinfeld Γ -module whose support consists of involutions, the restof the statement of Item (2) follows equally well from the first sentence of the statement ofItem (2) by application of ¯ S , Proposition 9.5(1) and [8, Proposition 3.7(6), Proposition 4.2,Remark 4.3]. (cid:3) Lemma 9.11. Let A be a connected braided Z ≥ -graded Hopf algebra in ΓΓ YD . Let x be anonzero integral in A . Then, the element x is homogeneous of some Γ -degree which is centralin Γ and there exists a unique character Γ → k × , g λ g such that gx = λ g x for all g ∈ Γ .If for this sentence, in addition, Γ is generated by involutions, then the character as in theprevious sentence satisfies λ g ∈ {− , } for all g ∈ Γ .Proof. Let the notation be as in the statement. We know from Proposition 9.5(1) that x ishomogeneous of some Γ -degree h . For every g ∈ Γ , the element gx is a nonzero integral in A by Proposition 9.5(1). Hence, for every g ∈ Γ , there exists a unique nonzero scalar λ g suchthat gx = λ g x . By comparing Γ -degrees in the previous equation, we see that ghg − = h forall g ∈ Γ . In other words, the element h is central in Γ . It is clear that g λ g defines acharacter Γ → k × which is uniquely determined by the property that gx = λ g x for all g ∈ Γ .If in addition Γ is generated by involutions, we must have λ g ∈ {− , } for all g ∈ Γ because g λ g is multiplicative and because λ g = 1 for every involution g ∈ Γ . (cid:3) Definition 9.12. For a vector space, we define an equivalence relation ∼ by declaring twovectors x and x ′ to be equivalent, in formulas x ∼ x ′ , if there exists a sign ǫ such that x = ǫx ′ .We use the same symbol ∼ for this equivalence relation regardless on which vector space itis defined. Corollary 9.13. Let x be a nonzero integral in B W .(1) The element x is homogeneous of some Z ≥ -degree m and some W -degree w . Theparity of ℓ ( w ) equals the parity of m .(2) For all g ∈ W , we have x ∼ gx . Moreover, we have x ∼ ρ ( x ) ∼ S ( x ) ∼ ¯ S ( x ) . (3) In the sense of Item (2), we define signs ǫ ρ , ǫ S , ǫ ¯ S such that x = ǫ ρ ρ ( x ) , x = ǫ S S ( x ) , x = ǫ ¯ S ¯ S ( x ) . Then, we have ǫ ρ = ǫ ¯ S and ǫ S = ( − m where m is defined as in Item (1).(4) With w defined as in Item (1), we further have wx = ( − ℓ ( w ) x .(5) An element x ′ is an integral in B W if and only if x α x ′ = 0 for all α ∈ R + if and onlyif x ′ x α = 0 for all α ∈ R + .(6) Let x ′ be a nonzero element of B W . Then, there exist monomials M, M ′ ∈ B W such that M x ′ and x ′ M ′ are nonzero integrals in B W . In particular, there exists amonomial which is a nonzero integral in B W .Proof. Let the notation be as in the statement. Let us prove Item (6) first. By Proposi-tion 9.5(3), there exist homogeneous elements y, y ′ ∈ B W with respect to the Z ≥ -gradingsuch that yx ′ and x ′ y ′ are nonzero integrals in B W . Let us assume further that y and y ′ arechosen as in the proof of Proposition 9.5(3), i.e. with respect to the nonzero component of x ′ of smallest Z ≥ -degree. We can write y respectively y ′ as a sum P i M i respectively P i M ′ i of monomials M i , M ′ i all of equal Z ≥ -degree equal to the Z ≥ -degree of y and y ′ . Then, wehave P i M i x ′ = 0 and P i x ′ M ′ i = 0 and consequently M i x ′ = 0 and x ′ M ′ j = 0 for some i, j .By Proposition 9.5(1),(3) and our choice of y, y ′ , M i , M ′ j , we know that M i x ′ and x ′ M ′ j arenonzero integrals in B W . Thus, the monomials M = M i and M ′ = M ′ j are as required. Theparticular case follows by application to x ′ = 1 .Let us prove Item (1). By Proposition 9.5(1) and Lemma 9.11, we can define m and w asin the statement. The rest of the item follows by application of Lemma 5.10 to x .The first sentence of Item (2) follows from Lemma 9.11. The morphisms ρ and ¯ S are Z ≥ -graded involutions by [8, Proposition 3.7(1),(6)]. Hence, it follows from Proposition 9.5(1)that x ∼ ρ ( x ) ∼ ¯ S ( x ) . Using these last relations, Proposition 9.5(1) or Item (1) of thiscorollary, and the definition of ¯ S , we finally see that S ( x ) ∼ ρ ¯ S ( x ) ∼ ρ ( x ) ∼ x .Let us prove Item (3). By definition of ¯ S , it is clear that ǫ ¯ S = ( − m ǫ ρ ǫ S where m is definedas in Item (1). Hence, it suffices to prove that ǫ ρ = ǫ ¯ S . By [8, Proposition 3.7(6), 3.10], wesee that h x, x i = h ρ ( x ) , ¯ S ( x ) i = ǫ ρ ǫ ¯ S h x, x i . In view of this equation, Corollary 9.9(1) completes the proof of Item (3).Let us prove Item (4). In view of Item (2), we know that x = S ( x ) . And further, with w defined as in Item (1), that S ( x ) = ( − ℓ ( w ) wx by Corollary 7.4. The result in Item (4)follows from these observations. Finally, Item (5) is immediate from Proposition 9.5(2)because ( x α ) α ∈ R + is a basis of V W . (cid:3) Corollary 9.14 (Abstract commutativity) . Let w ∈ W . Let y = wx w o . Let x , x , x ∈ B W be homogeneous elements with respect to the Z ≥ -grading such that x yx x is a nonzerointegral in B W and such that ( x ) ←− D α = ( x ) ←− D α = ( ww o w − x ) ←− D α = ( x ) ←− D α = 0 for all α ∈ T w . Then, we have x yx x = x ( ww o w − x ) yx .Proof. By assumption, the element x ( ww o w − x ) yx is homogeneous with respect to the Z ≥ -grading. Further, the element y is nonzero. Therefore, the result follows from Corol-lary 8.6 and Proposition 9.10(2). (cid:3) Corollary 9.15. Let w be an element in the centralizer of w o . Let y = wx w o . Let x , x , x ∈ B W be homogeneous elements with respect to the Z ≥ -grading such that x yx x is a nonzero N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 27 integral in B W and such that ( x ) ←− D α = ( x ) ←− D α = ( x ) ←− D α = 0 for all α ∈ T w . Then, we have x yx x = x ( w o x ) yx .Proof. By assumption, the element x ( w o x ) yx is homogeneous with respect to the Z ≥ -grading. Further, the element y is nonzero. Therefore, the result follows from Corollary 8.7and Proposition 9.10(2). (cid:3) Remark 9.16. Whenever we apply Corollary 9.9(1) to B W or one of the results derivedwith its help, e.g. Corollary 9.14, 9.15, we use a consequence of the fact that B W admits a nondegenerate Hopf duality pairing between B W and itself.10. Invariance of integrals Under invariance properties of integrals, we understand formulas which show that a nonzerointegral in B W is invariant under certain operators which lie in the image of the embeddingsof the tensor square into endomorphisms as in Remark 4.3 or are composites thereof. Suchproperties can be derived manifoldly using the results in Section 8. In this section, we presenta selection of such. Lemma 10.1. Let x ∈ B W be an element such that xx α = 0 for some α ∈ R . Then, itfollows that ( x ) ←− D α x α = x .Proof. Let the notation be as in the statement. The braided Leibniz rule implies in particularthat ←− D α x α = 1 − x α ←− D α . The result follows by application of the previous operator to x becauseof the assumed vanishing. (cid:3) Lemma 10.2 (Invariance properties) . Let x be a nonzero integral in B W .(1) Let α ∈ R . Then, we have x = ( x ) ←− D α x α . (2) Let α ∈ R . Let ǫ be the sign such that s α x = ǫx α (which exists by Corollary 9.13(2)).Then, we have x = − ǫx α ( x ) ←− D α . (3) Let w ∈ W . Let y = wx w o . Then, we have x = ( x ) ←− D y y . (4) Let w be an element in the centralizer of w o . Let y = wx w o . Let ǫ be the sign suchthat w o x = ǫx (which exists by Corollary 9.13(2)). Then, we have x = ( − ℓ ( w o ) ǫy ( x ) ←− D y . (5) Let D be a disjoint system of order two. Let w , w be some ordering of the elementsof D . Let y = w x w o and let y = w x w o . Then, we have x = ( − ℓ ( w o ) y y ( x ) ←− D y y ,x = ( x ) ←− D y y y y . Proof of Item (1). The first item follows from Lemma 10.1 and Proposition 9.5(2). (cid:3) Proof of Item (2),(3),(4). Let the notation be as in the statement. By Proposition 9.5(1)and [8, Remark 2.16], we know that the terms on the right side of the displayed equations inthe three times we want to proof here are all homogeneous with respect to the Z ≥ -grading.We see that Proposition 9.10(2) applies. Thus, it suffices to prove that ( x ) ←− D α = − ǫ (cid:0) x α ( x ) ←− D α (cid:1) ←− D α , ( x ) ←− D y = (cid:0) ( x ) ←− D y y (cid:1) ←− D y . ( x ) ←− D y = ( − ℓ ( w o ) ǫ (cid:0) y ( x ) ←− D y (cid:1) ←− D y , where the symbols appearing in the three equalities are supposed to be defined as in thethree corresponding items. The first of these three equalities follows directly from the braidedLeibniz rule and [8, Remark 3.16], the second from Fact 6.1(3) and Corollary 8.2, and thethird from Fact 6.1(2),(3), Corollary 6.6, 8.3 and [8, Remark 3.16]. (cid:3) Proof of Item (5). Let the notation be as in the statement. With the help of Fact 6.1(2),Lemma 5.9(6),(7), Corollary 8.4, 9.15, Item (3),(4) and [8, Remark 3.16], we compute that ( − ℓ ( w o ) y y ( x ) ←− D y y = ( − ℓ ( w o ) ǫy ( x ) ←− D y y y = ( − ℓ ( w o ) ǫ ( y ( x ) ←− D y ) ←− D y y = ( x ) ←− D y y = x , ( x ) ←− D y y y y = ( − ℓ ( w o ) ( x ) ←− D y y y y = ( x ) ←− D y y ←− D y y = ( x ) ←− D y y = x , where ǫ is the sign such that w o x = ǫx (as in the statement of Item (4) – which exists byCorollary 9.13(2)). (cid:3) Corollary 10.3. Let x be a nonzero integral in B W . Let w be an element in the centralizerof w o . Let y = wx w o . Then, we have h ( x ) ←− D y , ( x ) ←− D y i 6 = 0 .Proof. Let the notation be as in the statement. By Lemma 10.2(4), we have h ( x ) ←− D y , ( x ) ←− D y i =( − ℓ ( w o ) ǫ h x, x i where ǫ is the sign such that w o x = ǫx (which exists by Corollary 9.13(2)).The result follows from this and Corollary 9.9(1). (cid:3) Disjoint systems and integrals In this section, we explain the relation between complete disjoint systems and integrals in B W . More specifically, we explain in Lemma 11.2: (1) ⇒ (3) how the existence of certainintegrals in B W implies commutativity relations up to scalar multiple. Lemma 11.1. Let D be a disjoint system of order two. Let w , w be some ordering of theelements of D . Let y = w x w o and let y = w x w o . If y y and y y are linearly dependent,then we necessarily have y y = ( − ℓ ( w o ) y y .Proof. Let the notation be as in the statement. Suppose that y y and y y are linearlydependent. By Corollary 8.5, we know that both y y and y y are nonzero, hence lineardependence implies the existence of a nonzero scalar λ such that y y = λy y . If we apply N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 29 ←− D y y to both sides of this equality, we find in view of Corollary 8.5 that λ = ( − ℓ ( w o ) . Thiscompletes the proof. (cid:3) Lemma 11.2. Suppose that B W is finite dimensional. Let D be a complete disjoint systemof order r . Let w , . . . , w r be some ordering of the elements of D . Let y i = w i x w o for all ≤ i ≤ r . The following items are equivalent.(1) The element y σ (1) · · · y σ ( r ) is a nonzero integral in B W for some σ ∈ S r .(2) The element y σ (1) · · · y σ ( r ) is a nonzero integral in B W for all σ ∈ S r .(3) We have y i y j = ( − ℓ ( w o ) y j y i for all ≤ i, j ≤ r .Proof. The Implication (2) ⇒ (1) is obvious. The Implication (1) ⇒ (2) follows from Corol-lary 8.5 and Proposition 9.5(1).Let us prove the Implication (2) ⇒ (3) . The equality claimed in Item (3) is obvious forall ≤ i = j ≤ r . Because of Lemma 6.5 and Corollary 6.6 both sides are zero. Let ≤ i = j ≤ r be arbitrary but fixed. By Proposition 9.5(1) and Item (2), there exists anonzero scalar λ such that y i y j Q = λy j y i Q where Q = y · · · ˆ y i · · · ˆ y j · · · y r . If we now apply ←− D Q to y i y j Q = λy j y i Q , we find in view of Lemma 5.9(6),(7) and Corollary 8.5that y i y j = λy j y i . In view of Lemma 11.1, we find that λ = ( − ℓ ( w o ) . This proves Item (3).Let us prove the Implication (3) ⇒ (1) . Let x = y · · · y r . By Lemma 6.5, Corollary 8.5and Item (3), the element x is a nonzero monomial which starts with γ for all γ ∈ R + . ByLemma 5.9(1) and Corollary 9.13(5), it follows that x is a nonzero integral in B W . (cid:3) Conjecture 11.3. Suppose that B W is finite dimensional. Let D be a complete disjointsystem of order r . Let w , . . . , w r be some ordering of the elements of D . Let y i = w i x w o forall ≤ i ≤ r . Then, the element y · · · y r is a nonzero integral in B W . Remark 11.4 (Motivational remark) . In type A and A , the assumptions and the conclusionof Conjecture 11.3 are satisfied for trivial reasons and by [32, Example 6.4]. This simple ob-servation makes part of our motivation for the notion of disjiont systems, for Conjecture 11.3and for the proceeding in this paper in general.12. Coproducts of integrals In this section, we explain how finite dimensionality of B W in general implies commutativityrelations up to multiplication with a nonzero element in B W (cf. Theorem 12.3). Definition 12.1. Let V be a Z ≥ -graded vector space. Let Y be a Z ≥ -graded vectorsubspace of V . We call U a Z ≥ -graded complement of Y in V if U is a Z ≥ -graded vectorsubspace of V such that V ∼ = Y ⊕ U . It is clear that a Z ≥ -graded complement of Y in V exists for every V and Y as above. Lemma 12.2 (Key lemma on concrete commutativity) . Let Θ , Θ ⊆ R + be two subsets.Let y be a monomial which does only involve Θ , and let y be a monomial which does onlyinvolve Θ . Let y, ¯ y be monomials which do only involve Θ ∪ Θ such that yy y and ¯ yy y arelinearly independent. Let b, ¯ b ∈ B W be homogeneous elements with respect to the Z ≥ -gradingand the W -grading such that byy y and ¯ b ¯ yy y are nonzero integrals in B W . Then, thereexist monomials y ′ , ¯ y ′ which do only involve Θ ∪ Θ and homogeneous elements b ′ , ¯ b ′ ∈ B W with respect to the Z ≥ -grading and the W -grading such that deg Z ≥ y + deg Z ≥ ¯ y < deg Z ≥ y ′ + deg Z ≥ ¯ y ′ , where deg Z ≥ denotes the Z ≥ -degree, and such that b ′ y ′ y y and ¯ b ′ ¯ y ′ y y are nonzero integralsin B W .Proof. Let the notation be as in the statement. Without loss of generality, we may assumethat the Z ≥ -degree of b is larger or equal than the Z ≥ -degree of ¯ b . By assumption andProposition 9.5(1), there exists a nonzero scalar λ such that byy y = λ ¯ b ¯ yy y . Let ∆( b ) = 1 ⊗ b + X i b i (1) ⊗ b i (2) , ∆(¯ b ) = 1 ⊗ ¯ b + X i ¯ b i (1) ⊗ ¯ b i (2) be decompositions such that b i (2) is homogeneous with respect to the W -grading and homo-geneous of Z ≥ -degree less than the Z ≥ -degree of b and such that ¯ b i (2) is homogeneous withrespect to the W -grading and homogeneous of Z ≥ -degree less than the Z ≥ -degree of ¯ b . Let U be a Z ≥ -graded complement of k yy y ⊕ k ¯ yy y in B W . Let P be the natural projection P : B W ∼ = k yy y ⊕ k ¯ yy y ⊕ U → k yy y ∼ = k onto k yy y ∼ = k . If we apply now the coproduct to both sides of the equation byy y = λ ¯ b ¯ yy y , apply further P ⊗ to the result and identify it along the isomorphism k ⊗ B W ∼ = B W ,we find that b + X i,j λ i,j b i (2) ( yy y ) j = λ X j ¯ λ j ¯ b (¯ yy y ) j + λ X i,j ¯ λ i,j ¯ b i (2) (¯ yy y ) j where the λ i,j , ¯ λ j , ¯ λ i,j are scalars which result from the application of P to the first tensorfactors of the decompositions of ∆( byy y ) and ∆(¯ b ¯ yy y ) and where the ( yy y ) j and (¯ yy y ) j are sub-monomials of yy y and ¯ yy y which are nonconstant whenever the correspondingscalar is nonzero. If we multiply the previous equation with yy y from the right, we canfind a monomial y ′ which does only involve Θ ∪ Θ such that deg Z ≥ y < deg Z ≥ y ′ and suchthat b ′ y ′ y y is a nonzero integral in B W where b ′ is either b i (2) , ¯ b or ¯ b i (2) . If we set ¯ y ′ = ¯ y and ¯ b ′ = ¯ b , we have found y ′ , ¯ y ′ , b ′ , ¯ b ′ with all the properties required in the statement. Thiscompletes the proof. (cid:3) Theorem 12.3 (Concrete commutativity) . Let Θ , Θ ⊆ R + . Let y be a monomial whichdoes only involve Θ , and let y be a monomial which does only involve Θ such that y y and y y are both nonzero. If B W is finite dimensional, then there exist monomials y, ¯ y which doonly involve Θ ∪ Θ such that k yy y = k ¯ yy y = 0 . Proof. Let the notation be as in the statement. Suppose that B W is finite dimensional.Suppose that y y and y y are linearly independent. Otherwise, we can set y = ¯ y = 1 andare done by assumption. Let y, ¯ y be monomials which do only involve Θ ∪ Θ such that yy y and ¯ yy y are linearly independent and such that deg Z ≥ y + deg Z ≥ ¯ y is maximal. Such achoice of monomials clearly exists by assumption and because the Z ≥ -degree of any nonzerohomogeneous element is bounded above by the Z ≥ -degree of any nonzero integral in B W byProposition 9.5(1). If we now apply Lemma 12.2 to this situation, we can find monomials y ′ , ¯ y ′ which do only involve Θ ∪ Θ such that deg Z ≥ y + deg Z ≥ ¯ y < deg Z ≥ y ′ + deg Z ≥ ¯ y ′ , N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 31 and such that y ′ y y and ¯ y ′ y y are linearly dependent and nonzero – as claimed. (cid:3) Remark 12.4. Note that Lemma 12.2 and Theorem 12.3 have right analogues which followby application of ρ and Remark 5.6.13. Reduction of monomials Under the title of “Reduction of monomials” , we discuss in this section the reductionof arbitrary monomials modulo a suitable ideal (or what amounts more or less to the samemodulo multiplication with a hypothetical element – a notion which will be introduced shortlyafter in Section 14) to a monomial which does only involve ∆ , i.e. to a basis element of thestandard basis of N W . For the moment, we achieve this reduction procedure only in type A using the known relations of B S m (i.e. the Fomin-Kirillov relation, the nilpotent relationand the commutation relation as in [8, Example 4.4]). The corresponding result in type A isstated in Corollary 13.7 based on Lemma 13.4. Definition 13.1. We denote by J l W and J r W the Z ≥ -graded and W -graded left ideal andright ideal in B W defined by the equalities J l W = P α ∈ R + \ ∆ B W x α and J r W = P α ∈ R + \ ∆ x α B W . Remark 13.2. Note that we have the trivial relations J l W = ρ ( J r W ) and J r W = ρ ( J l W ) bydefinition of ρ . Lemma 13.3. We have N W ∩ J l W = N W ∩ J r W = 0 .Proof. The first claimed equality follows by application of ρ and Remark 13.2 and Fact 6.1(4).We prove the second claimed equality now. To this end, it suffices to prove that x w / ∈ J r W forall w ∈ W because the ideal J r W is W -graded by definition. Suppose for a contradiction that x w = P α ∈ R + \ ∆ x α M α for some w ∈ W and some monomials M α of Z ≥ -degree ℓ ( w ) − . If we apply ←− D w − to theprevious displayed equation, we find in view of Fact 6.1(3) and Lemma 5.9(6),(7) that P α ∈ R + \ ∆ x α ( M α ) ←− D w − . But the right side of the last equation must be zero for Z ≥ -degree reasons as stipulated in[8, Remark 2.16] which is clearly a contradiction. (cid:3) From now on and for the rest of this section, we assume that R is of type A .Lemma 13.4 (Key lemma on reduction of monomials) . Let λ ∈ k and α , . . . , α m ∈ R + besuch that not all α , . . . , α m are simple roots and such that the monomial M = λx α . . . x α m satisfies M + J r W = 0 . Let ≤ j < m be the maximal index such that α , . . . , α j ∈ ∆ andsuch that α j +1 / ∈ ∆ (which exists by the assumptions in the previous sentence). Then, thereexist λ i ∈ k and α i , . . . , α ij +1 ∈ R + such that M + J r W = P i λ i x α i · · · x α ij +1 x α j +2 · · · x α m + J r W ,α i , . . . , α ij ∈ ∆ and ht( α ij +1 ) < ht( α j +1 ) . Proof. Let the notation be as in the statement. We prove the lemma by induction on j .Assume first that j = 1 . It is clear that ( α , α ) > since otherwise M + J r W = 0 by [8,Example 4.4]. It follows that γ = α − α is a positive root. If we set α = α , we find that M + J r W = x γ x α x α · · · x α m + J r W = 0 by [8, Example 4.4]. Consequently, the root γ is simple,and we are done because α is simple by assumption.We prove the induction step. Suppose that j > and that the assertion is true for allmonomials and all integers < j . Let w = s α · · · s α j . We distinguish three cases now. In thefirst case, we assume that there exists β ∈ ∆ such that w ( β ) < and such that ( α j +1 , β ) > .Suppose that a β as in the previous sentence is given. Then, we know that γ = α j +1 − β is apositive root. By repeated application of the induction hypothesis and by [8, Example 4.4],we compute that M + J r W = x ws β x γ x β x α j +2 · · · x α m − x ws β x β + γ x γ x α j +2 · · · x α m + J r W = λx ws β s γ x β x α j +2 · · · x α m + µx ws β s αj +1 x γ x α j +2 · · · x α m + J r W for some λ, µ ∈ k (which might possibly be zero). This completes the proof of the first casebecause the height of γ is strictly less than the height of α j +1 .We consider the second case. In the second case, we assume that there exists β ∈ ∆ suchthat w ( β ) < and such that ( α j +1 , β ) = 0 . Suppose that a β as in the previous sentenceis given. By repeated application of the induction hypothesis and by [8, Example 4.4], wecompute that M + J r W = x ws β x α j +1 x β x α j +2 · · · x α m + J r W = λx ws β s αj +1 x β x α j +2 · · · x α m + J r W for some λ ∈ k (which is unique and necessarily nonzero). This clearly completes the proofof the second case.We consider the third case. In the third case, we assume that for all β ∈ ∆ such that w ( β ) < we necessarily have ( α j +1 , β ) < . Let us fix some arbitrary γ ∈ ∆ such that w ( γ ) < . Let α = α j +1 for short. By repeated application of the induction hypothesis andby [8, Example 4.4], we compute that M + J r W = x ws γ x α x α + γ x α j +2 · · · x α m + x ws γ x α + γ x γ x α j +2 · · · x α m + J r W = λx ws γ s α x α + γ x α j +2 · · · x α m + µx ws γ s α + γ x γ x α j +2 · · · x α m + J r W for some λ, µ ∈ k (which might possibly be zero). To complete the proof of the third case,we only have to analyse further the first summand in the previous equation. Let β ∈ ∆ bethe unique simple root such that ( α, β ) > and ( γ, β ) < . It follows that α ′ = α − β is apositive root. (The definition of the roots α, α ′ , β, γ is illustrated in Figure 2.) With thesedefinitions we compute that ws γ s α ( β ) = − w ( α ′ ) < where the first equality follows because α ′ is orthogonal to γ and where the second inequality follows because all simple roots β ′ inthe support of α (and hence of α ′ ) satisfy ( α, β ′ ) ≥ and consequently w ( β ′ ) > by theassumption in the third case under consideration. Using this insight, the orthogonality of β and α + γ , repeated application of the induction hypothesis and [8, Example 4.4], we computethat x ws γ s α x α + γ x α j +2 · · · x α m + J r W = x ws γ s α s β x α + γ x β x α j +2 · · · x α m + J r W = λx ws γ s α s β s α + γ x β x α j +2 · · · x α m + J r W for some λ ∈ k (which might possibly be zero). This completes the proof of the third case,of the induction step and hence of the lemma. (cid:3) Corollary 13.5. Let λ ∈ k and α , . . . , α m ∈ R + be such that the monomial M = λx α . . . x α m N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 33 β ⊥ α + γ γα ′ α / ∈ ∆ Figure 2. Illustration of the situation in the third case of the induction step in the proof ofLemma 13.4. We want to thank Black Mild for providing the code for this figure in an answer toour question on TEX StackExchange [38]. satisfies M + J r W = 0 . Let ≤ j ≤ m be an arbitrary index. Then, there exists a unique µ ∈ k (which is necessarily nonzero) such that M + J r W = µx s α ··· s αj x α j +1 · · · x α m + J r W . Proof. Let the notation be as in the statement. By repeated application of Lemma 13.4,there exist λ i ∈ k and α i , . . . , α ij ∈ ∆ such that M + J r W = P i λ i x α i · · · x α ij x α j +1 · · · x α m + J r W . Because J r W is W -graded by definition, and by possibly discarding summands in the abovesum, we may assume that each summand in the above sum has the same W -degree equal tothe W -degree of M , and that x s α ··· s αj = x α i · · · x α ij for all i . If we set µ = P i λ i , we havefound the desired expression for M + J r W with a scalar µ which is unique (and necessarilynonzero) because M + J r W is nonzero by assumption. (cid:3) Remark 13.6. Note that Lemma 13.4 and Corollary 13.5 have obvious left analogues (i.e.analogues for the left ideal J l W instead of the right ideal J r W ) which follow by application of ρ and Remark 13.2. Corollary 13.7 (Reduction of monomials) . Let M ∈ B W be a monomial of W -degree w .Then, there exist a unique λ ∈ k such that M + J l W = λx w + J l W and M + J r W = λx w + J r W . In particular, it follows that M + J l W = 0 if and only if M + J r W = 0 .Proof. The “In particular” is clear from the second sentence in the statement of the lemma.The first claimed equality follows from the second by application of ρ and Remark 13.2. Weprove the second equality now. The uniqueness of λ is clear because x w + J r W is nonzero byLemma 13.3. The existence of λ follows directly from Corollary 13.5 applied to j = m incase M + J r W = 0 , and is trivial in case M + J r W = 0 (we simply set λ = 0 and this is theonly λ which fits). (cid:3) Corollary 13.8 (Isomorphism theorems) . We have natural isomorphisms of Z ≥ -graded and W -graded vector spaces B W / J l W ∼ = N W and B W / J r W ∼ = N W . Proof. This corollary is immediate from Lemma 13.3 and Corollary 13.7. (cid:3) Hypothetical elements In this section, we use the results in Section 13, to show in Theorem 14.5 how nonzeroleft or right hypothetical elements (cf. Definition 14.1) in type A can be lifted to nonzerointegrals in B W under the assumption that B W is finite dimensional. As a consequence ofthis, we can derive Theorem 14.8 which corresponds to Theorem 1.4 in the introduction. Definition 14.1. Let P ∈ B W be such that ( P ) ←− D β = 0 for all β ∈ ∆ . • We say that P is a left hypothetical element if x α P = 0 for all α ∈ R + \ ∆ . Wesay that P is a nonzero left hypothetical element if P is nonzero and if P is a lefthypothetical element. • We say that P is a right hypothetical element if P x α = 0 for all α ∈ R + \ ∆ . Wesay that P is a nonzero right hypothetical element if P is nonzero and if P is a righthypothetical element. • We say that P is a hypothetical element if P is a left and right hypothetical ele-ment. We say that P is a nonzero hypothetical element if P is nonzero and if P is ahypothetical element. Remark 14.2. If B W is finite dimensional, it is clear that there exist monomials which doonly involve R + \ ∆ and which are nonzero hypothetical elements. Indeed, similar as inthe proof of Corollary 9.13(6), one considers nonzero monomials in the subalgebra of B W generated by x α where α ∈ R + \ ∆ of maximal Z ≥ -degree (cf. Lemma 5.9(6)). Lemma 14.3. An element P ∈ B W is a left hypothetical element if and only if ρ ( P ) is aright hypothetical element. An element P ∈ B W is a right hypothetical element if and only if ρ ( P ) is a left hypothetical element.Proof. This result follows immediately from Corollary 7.2, [8, Proposition 3.7(6)] and thedefinition of ρ . (cid:3) Lemma 14.4. Let P ∈ B W be a nonzero element such that ( P ) ←− D β = 0 for all β ∈ ∆ . Then,we have x w o P = 0 and P x w o = 0 .Proof. Let the notation be as in the statement. In order to proof the lemma, it suffices toproof that ( x w o P ) ←− D w o = 0 and ( P x w o ) ←− D w o = 0 . But, by Fact 6.1(3), Lemma 5.9(7) andCorollary 8.3, those expressions evaluate as w o P and P , respectively, which are both nonzeroby assumption. (cid:3) From now on and for the rest of this section, we assume that R is of type A .Theorem 14.5 (Main theorem on hypothetical elements (I)) . (1) Let P ∈ B W be a nonzero left hypothetical element. If B W is finite dimensional, then x w o P is a nonzero integral in B W (2) Let P ∈ B W be a nonzero right hypothetical element. If B W is finite dimensional,then P x w o is a nonzero integral in B W Proof. Item (1) follows from Item (2), Fact 6.1(4), Lemma 14.3 and [8, Proposition 3.7(6)].We prove Item (2) now. Let P ∈ B W be a nonzero right hypothetical element, and supposethat B W is finite dimensional. Let M be a monomial chosen as in the proof of Proposi-tion 9.5(3) and Corollary 9.13(6) with respect to the nonzero component of P of smallest Z ≥ -degree such that P M is a nonzero integral in B W . Let m be the Z ≥ -degree of M , and N THE DIMENSION OF THE FOMIN-KIRILLOV ALGEBRA AND RELATED ALGEBRAS 35 let w be the W -degree of M . Then, we necessarily have ℓ ( w o ) ≤ m by Lemma 14.4, andfurther P M = λP x w for some unique λ ∈ k (which is necessarily nonzero) by Corollary 13.7.It follows that w = w o and hence that P x w o is a nonzero integral in B W – as desired. (cid:3) Theorem 14.6 (Main theorem on hypothetical elements (II)) . Suppose that B W is finitedimensional.(1) Every left or right hypothetical element is a hypothetical element. Following andconsistent with the convention in Remark 9.7, we speak from now on, except in theproof of this theorem, only about bona fide hypothetical elements.(2) Every hypothetical element P satisfies P = ( x ) ←− D w o for some uniquely determinedintegral x , namely x = P x w o , in B W . In particular, the vector space of all hypotheticalelements is one-dimensional.(3) We have h P ∗ , P i 6 = 0 for any nonzero hypothetical elements P and P ∗ .Proof. We start to prove the first sentence of Item (2) for right hypothetical elements. Therest of the content of Item (2) will follow from Remark 14.2 and Proposition 9.5(1) onceItem (1) is established. Note first that the uniqueness of x , once its existence is established,is clear from Proposition 9.10(1). Note further that we may assume that P = 0 since oth-erwise we have to set x = 0 and the claimed assertion is obvious. In this sense, let P bea nonzero right hypothetical element. We know from Theorem 14.5(2) that x = P x w o isa nonzero integral in B W which satisfies P = ( x ) ←− D w o by Fact 6.1(3) and Lemma 5.9(7).In order to prove Item (1), we only have to note, in view of Lemma 14.3 and [8, Proposi-tion 3.7(6)], that for the P under consideration, we have P ∼ ρ ( P ) . But this follows fromFact 6.1(2), Proposition 7.1, Corollary 9.13(2), the portion of Item (2) we already proved and[8, Remark 3.16, Proposition 6.5]. To understand the last item, we just have to note thatfor two nonzero hypothetical elements P and P ∗ the bracket h P ∗ , P i is a nonzero multiple of h ( x ) ←− D w o , ( x ) ←− D w o i for some nonzero integral x in B W – by Proposition 9.5(1) and Item (2) –which is indeed nonzero by Corollary 10.3. (cid:3) Definition 14.7. Let B ′ W be the subalgebra of B W generated by x α where α ∈ R + \ ∆ . 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