Open Access This article is
- freely available
Axioms 2017, 6(4), 32; doi:10.3390/axioms6040032
Factorization of Graded Traces on Nichols Algebras
Department of Mathematics, University Hamburg, Bundesstraße 55, D-20146 Hamburg, Germany
Faculty of Mathematics and Informatics, Philipps-University Marburg, Hans-Meerwein-Straße, D-35032 Marburg, Germany
Author to whom correspondence should be addressed.
Received: 27 October 2017 / Accepted: 25 November 2017 / Published: 4 December 2017
A ubiquitous observation for finite-dimensional Nichols algebras is that as a graded algebra the Hilbert series factorizes into cyclotomic polynomials. For Nichols algebras of diagonal type (e.g., Borel parts of quantum groups), this is a consequence of the existence of a root system and a Poincare-Birkhoff-Witt (PBW) basis basis, but, for nondiagonal examples (e.g., Fomin–Kirillov algebras), this is an ongoing surprise. In this article, we discuss this phenomenon and observe that it continues to hold for the graded character of the involved group and for automorphisms. First, we discuss thoroughly the diagonal case. Then, we prove factorization for a large class of nondiagonal Nichols algebras obtained by the folding construction. We conclude empirically by listing all remaining examples, which were in size accessible to the computer algebra system GAP and find that again all graded characters factorize.
Keywords:Nichols algebra; Hilbert series; graded traces
A Nichols algebra over a finite group is a certain graded algebra associated with a given Yetter–Drinfel’d module over this group. For example, the Borel subalgebras of the Frobenius–Lusztig kernels are finite-dimensional Nichols algebras over quotients of the root lattice of as an abelian group. Nichols algebras are in fact braided Hopf algebras enjoying several universal properties.
In this article, we discuss the following curious phenomenon that is apparent throughout the ongoing classification of finite-dimensional Nichols algebras: the graded dimension or Hilbert series of any finite-dimensional Nichols algebra known so far factorizes as a polynomial in one variable into the product of cyclotomic polynomials. The root system theory and PBW-basis of Nichols algebras developed by [1,2] precisely explains a factorization of as graded vector space (even Yetter–Drinfel’d module) into Nichols subalgebras of rank 1 associated with each root. This explains the complete factorization for Nichols algebras over abelian groups. However, for Nichols algebras over nonabelian groups, the Nichols subalgebras of rank 1 may still be large complicated algebras, so the root system cannot explain the complete factorization of the Hilbert series that we observe.
A very bold assumption could be that the complete factorization points to the existence of a somehow finer root system. For a large family of examples, this is literally the case, as we will prove (see below). This should be the most important message of the present article.
Another key interest of the present article is to moreover consider the entire graded character of the group acting on the Nichols algebra (the Hilbert series is the graded character at the identity). As empirical data, Section 4 contains a list of all known finite-dimensional Nichols algebras of rank 1 that were accessible to us in size by GAP.
We calculate in each case the graded characters and verify that these characters again factorize completely into cyclotomic polynomials. We also point to examples, where there is no associated factorization as graded G-representations.
A very interesting class of examples is the Fomin–Kirillov algebra  associated with a Coxeter group. For the Coxeter groups , the Fomin–Kirillov algebras are of finite dimension 2, 12, 576, 8,294,400, 64. These quadratic algebras independently appeared as the first Nichols algebras over nonabelian groups , associated with the conjugacy class(es) of reflections in the Coxeter group. Since they are built on a single conjugacy class (resp. 2), their generalized root system is trivial (resp. ).
For , it is an important open problem whether the respective Nichols algebras are finite-dimensional and coincide with the Fomin–Kirillov algebras. Already, Fomin and Kirillov pointed out that, in this case, the graded dimension does not factorize into cylcotomic factors and suspect this to rule out finite dimension altogether.
The main goal of this article is to study systematically factorization mechanisms beyond the root system. It has been shown in a series of joint papers of the second author [5,6,7] that the existence of such a complete factorization into cyclotomic polynomial implies strong bounds on the number of relations in low degree in the Nichols algebra. On the other hand, the first author has obtained in  new families of Nichols algebras over nonabelian groups, where we can now indeed prove the factorization of graded characters: they are constructed from diagonal Nichols algebras by a folding technique and, from this, they retain a finer root system (e.g., inside ). This includes the Fomin–Kirillov algebra over , which has a finer root system inside . A recent classification  for rank >1 shows that the folding construction already exhausts all finite-dimensional Nichols algebras of rank >3 over finite groups.
We do not know whether a similar construction produces the remaining known examples of Nichols algebras of small rank over nonabelian groups, let alone Nichols algebras over non-semisimple Hopf algebras. The first author has recently studied a construction of such Nichols algebras in , and, again, the factorization follows from a by-construction finer root system.
The content of this article is as follows:
In Section 2, we review graded traces and basic facts about graded traces on finite-dimensional Nichols algebras, including additivity and multiplicativity with respect to the representations, rationality, and especially Poincaré duality. All of these facts have appeared in literature, and we gather them here for convenience.
In Section 3, we study factorization mechanisms for Nichols algebras, and hence for their graded traces and especially their Hilbert series. The root system of a Nichols algebra in the sense of [1,2] directly presents a factorization of as a graded Yetter–Drinfel’d module: this completely explains the factorization of the graded trace of an endomorphism Q that respects the root system grading.
For Nichols algebras over abelian groups, we use the theory of Lyndon words to significantly weaken the assumption on Q. On the other hand, we give an example of an endomorphism (the outer automorphism of containing a loop) where this mechanism fails; a factorization of the graded trace is nevertheless observed and can be tracked to the surprising existence of an alternative “symmetrized” PBW-basis. The formulae appearing involve the orbits of the roots under Q and are in resemblance to the formulae given in  Section 13.7 for finite Lie groups.
For Nichols algebras over nonabelian groups, the root system factorization discussed above still applies, but is too crude in general to explain the full observed factorization into cyclotomic polynomials. Most extremely, the large Nichols algebra of rank 1 have a trivial root system. A large family of examples is constructed by the folding construction of the first author in . Here, the complete factorization can again be tracked back to a finer PBW-basis.
Finally, we give results on the divisibility of the Hilbert series derived by the second author in . By the freeness of the Nichols algebra over a sub-Nichols algebra [14,15], one can derive a divisibility of the graded trace by that of a sub-Nichols algebra. Moreover, in many cases, there is a shift-operator for some , which can be used to prove that there is an additional cyclotomic divisor of the graded trace. This holds in particular for commuting with g.
Section 4 finally displays a table of graded characters for all known examples of finite-dimensional Nichols algebras of nonabelian group type and rank 1, which were computationally accessible to us. We observe that, again, all graded characters factorize in these examples.
2. Hilbert Series and Graded Traces
2.1. The Graded Trace
In the following, we suppose to be a graded vector space with finite-dimensional layers . We frequently call a linear map an operator Q. We denote the identity operator by and the projectors to each by . An operator Q is called graded if one of the following equivalent conditions is fulfilled:
- Q is commuting with all projections ;
- Q preserves all layers .
We denote the restriction of a graded operator to each by . An operator Q is called algebra operator resp. Hopf algebra operator, if V is a graded algebra resp. Hopf algebra and Q is an algebra resp. Hopf algebra morphism.
For a graded vector space V with finite-dimensional layers, we define
- the grading operator as
- the exponentiated grading operator as the -valued formal power series , so .
If V is finite-dimensional, then is a polynomial with well-defined value at the total dimension of V.
For V a graded vector space with finite-dimensional layers and Q a graded operator, define the graded trace of Q as the power series
Obviously, the graded trace is linear in Q and the graded trace of the identity is again the Hilbert series.
For V, the graded algebrasthe graded dimensions areFor V, the free algebra in K variables, the graded dimension isIn particular, for a finitely generated algebra, the graded trace of any algebra operator converges in some neighbourhood of .
If A is an infinite-dimensional algebra, then is known to be a rational function if one of the following holds (see e.g., ):
- A is commutative and finitely generated;
- A is right noetherian with finite global dimension;
- A is regular.
For graded vector spaces with finite-dimensional layers, the sum and product again are graded vector spaces with finite-dimensional layers. The codiagonal grading by definition impliesThen, the following properties for the graded trace hold immediately from the respective properties of the trace:
We conclude by two important examples of algebras, where the graded dimensions factorize nicely into the cyclotomic polynomials above:
Let W be a Weyl group acting on having positive roots . Then, a standard fact in invariant theory states  Section 2.4:where the homogeneous polynomials have fundamental degrees , fulfilling and . This isomorphism clearly implies that the Hilbert series factorizes as follows:
This implies the following formula, which is important, e.g., for the order of Lie groups over finite fields  Section 2.9: Let be the group ring of a Weyl group graded by the length of reduced expressions. Then,
As an example for , we have and this yields
The cohomology ring of a Lie group has the addition structure of a skew-commutative Hopf algebra (this observation by Hopf in 1941 was actually the beginning of the subject) and it factorizes accordingly into terms and . For example, the Hilbert series of the compact unitary groups isObserve the symmetry called Poincaré duality.
For affine Lie algebras (and other vertex algebras), the graded traces and characters of representations (via Weyl character formula) are of utmost importance and they typically can be easily shown to factorize.
Finally, we briefly discuss examples arising from physics:
Interprete V as a space of states, graded by the energy, with grading operator E the Hamilton operator. Then, the Boltzmann factor is the (unnormalized) probability measure on V at inverse temperature β, and a similar role is played by in quantum field theory. In particular, the Hilbert series is the partition function and the normalized graded trace of an operator Q is the expectation value
For example, a free boson in dimension 1 has a state space V consisting of all differential polynomials in a function ϕ, so is the number of partitions . Taking the differentials in degree k as a set of algebra generators, then the graded dimension factorizes
2.2. Poincaré Duality
A finite-dimensional Nichols algebra (in fact, any finite-dimensional graded Hopf algebra) exhibits a remarkable Poincaré Duality where is the subspace of degree k and L is the largest degree (see, e.g.,  Rem. 2.2.4). This is visible at the level of Hilbert series by
We easily generalize this argument to the calculation of graded traces of algebra operators in Nichols algebras:
Let be a finite-dimensional Nichols algebra with top degree L and integral Λ. Let Y be an arbitrary algebra automorphism of and the scalar such that . Then, .
Let and be two bases of with for all . Then, it holds that☐
Let be a finite-dimensional Nichols algebra with top degree L and integral Λ. Let Q be an arbitrary algebra automorphism of and the scalar such that . Then,
We apply Lemma 2 to and :☐
The special case recovers the Poincaré duality of the Hilbert series, and therefore for all l.
2.3. An Example for Factorization Only in the Trace
As a toy example, we want to present a type of graded representations that exhibits a seemingly paradoxical property: their graded characters factor nicely, whereas the representations themselves do not. We will encounter this property in the case of Nichols algebras of non-abelian group type of rank 1.
Let G be the dihedral group . It acts on its group algebra by conjugation (the action factors through . It also preserves the following -graduation:
The action then has graded characters , , and . Denote by the trivial G-representation, and with the other one-dimensional representations factorizing through . Then, is isomorphic toas a graded G-representation. While the graded characters factor nicely, the representation itself does not.
3. Graded Traces and Hilbert Series over Nichols Algebras
In this section, we study factorization mechanisms for Nichols algebras, and hence for their graded traces and especially their Hilbert series.
The root system of a Nichols algebra , introduced in Section 3.1 Theorem 1, directly presents a factorization of as graded vectorspace:
Note, however, that, over nonabelian groups, the root system factorization is too crude in general to explain the full observed factorization into cyclotomic polynomials; especially for the rank 1 cases in the next section, the factorization obtained this way is trivial.
Nevertheless, we will start in Section 3.2 by demonstrating a factorization of the graded trace of an endomorphism Q that stabilizes a given axiomatized Nichols algebra factorization, such as the root system above:
Let be a Nichols algebra with factorization and Q an algebra operator that stabilizes this factorization. Then,
In Section 3.3, we focus on Nichols algebras over abelian groups. The preceding corollary immediately gives an explicit trace product formula for endomorphisms Q stabilizing the root system, in terms of cylcotomic polynomials. In particular, it shows the complete factorization of their Hilbert series.
Using the theory of Lyndon words, we are able to weaken the assumptions on Q to only normalize the root system, i.e., acting on it by permutation. We give such examples where Q interchanges two disconnected subalgebras in the Nichols algebra, as well as the outer automorphism of a Nichols algebra of type . The authors expect that a more systematic treatment via root vectors will carry over to endomorphisms normalizing the root system of a non-abelian Nichols algebra as well.
In Section 3.4, we present an example of a Nichols algebra of type and an endomorphism Q induced by its outer automorphism that fails the normalizing condition on the non-simple root. Note that, in contrast to the -example above, there is an edge flipped by the automorphism, which is called a “loop” in literature (e.g., , p. 47ff). Nevertheless, one observes a factorization of the graded trace of Q, and, in this example, this can be traced back to a surprising and apparently new “symmetrized” PBW-basis.
In Section 3.5, we start approaching Nichols algebras over nonabelian groups, where one observes astonishingly also factorization of graded traces into cyclotomic polynomials. This cannot be explained by the root system alone and might indicate the existence of a finer root system, which is not at the level of Yetter–Drinfel’d modules.
We can indeed give a family of examples constructed as covering Nichols algebras by the first author . By construction, these Nichols algebras possess indeed such a finer root system of different type (e.g., ). In these examples, the root systems lead to a complete factorization, but this mechanism does not seem to easily carry over to the general case.
3.1. Nichols Algebras over Groups
The following notions are standard. We summarize them to fix notation and refer to  for a detailed account.
A Yetter–Drinfel’d module M over a group G is a G-graded vector space over denoted by layers with a G-action on M such that . To exclude trivial cases, we call M indecomposable iff the support generates all G and faithful iff the action is.
Note that, for abelian groups, the compatibility condition is just the stability of the layers under the action of G.
The notion of a Yetter–Drinfel’d module automatically brings with it a braiding on M—in fact, each group G defines an entire braided category of G-Yetter–Drinfel’d modules with graded module homomorphisms as morphisms (e.g., , Def. 1.1.15).
Consider , for all and . Then, τ fulfills the Yang–Baxter-equationturning M into a braided vector space.
In the non-modular case, the structure of Yetter–Drinfel’d modules is well understood ( Section 3.1) and can be summarized in the following three lemmata:
Let G be a finite group and let be an algebraically closed field whose characteristic does not divide . Then, any finite-dimensional Yetter–Drinfel’d module M over G is semisimple, i.e., decomposes into simple Yetter–Drinfel’d modules (the number is called rank of M):
Let G be a finite group, arbitrary and the character of an irreducible representation V of the centralizer subgroup . Define the Yetter–Drinfel’d module to be the induced G-representation . It can be constructed more explicitly as follows:
- Define the G-graduated vector space by
- Choose a set of representatives for the left -cosets . Then, for any g-conjugate there is precisely one with .
- For the action of any on any for determine the unique , such that and . Then, and using the given -action on V, we may define .
Then, is simple as a Yetter–Drinfel’d module and and are isomorphic if and only if g and are conjugate and χ and are isomorphic.
If is the character of a one-dimensional representation , we will identify with the action . In positive characteristics, we will restrict to , where the character determines its representation.
Let G be a finite group and let be an algebraically closed field whose characteristic does not divide . Then, any simple Yetter–Drinfel’d module M over G is isomorphic to some for some and a character of an irreducible representation V of the centralizer subgroup .
For finite and abelian G over algebraically closed with , we have one-dimensional simple Yetter–Drinfel’d modules and hence the braiding is diagonal (i.e., ) with braiding matrix .
Consider the tensor algebra , i.e., for any homogeneous basis , the algebra of words in all . We may uniquely define skew derivations on this algebra, i.e., maps by , , and .
The Nichols algebra is the quotient of by the largest homogeneous ideal invariant under all , such that .
In specific instances, the Nichols algebra may be finite-dimensional. This is a remarkable phenomenon (and the direct reason for the finite-dimensional truncations of for q a root of unity):
Take and the Yetter–Drinfel’d module with dimensions i.e., , then and hence the Nichols algebra has dimension 2.
More generally, a one-dimensional Yetter–Drinfel’d module with a primitive n-th root of unity has Nichols algebra .
Take and the Yetter–Drinfel’d module with dimensions i.e., then
In the abelian case, Heckenberger (e.g., ) introduced q-decorated diagrams, with each node corresponding to a simple Yetter–Drinfel’d module decorated by , and each edge decorated by and edges are drawn if the decoration is ; it turns out that this data is all that is needed to determine the respective Nichols algebra.
Let be a Nichols algebra of finite dimension over an arbitrary group G; then, there exists a root system with positive roots and a truncated basis of monomials in . Namely, the multiplication in is an isomorphism of graded vector spaces .
Let M be the diagonally braided vector space with basis and braiding fulfilling . The associated diagram is:
Direct calculations of the quantum symmetrizer (or general results of Kharchenko) directly show that the following relations hold:
Define the nonzero element in degree 2; then, we consider the three one-dimensional braided subspaces of the Nichols algebra
The corresponding Nichols algebras are subalgebras of . It turns out that the commutator and truncation relations are defining and the Nichols algebra is eight-dimensional with a PBW-Basis with . This can be formulated as: multiplication in yields a bijection of -graded vector spaces:
This Nichols algebra appears as the positive part of a quantum group and has a root system of type with three roots .
In the same sense, over abelian G for , any proper Cartan matrix of a semisimple Lie algebra is realized for braiding matrix .
However, several additional exotic examples of finite-dimensional Nichols algebras exist that possess unfamiliar Dynkin diagrams, such as a multiply-laced triangle, and where Weyl reflections may connect different diagrams (yielding a Weyl groupoid). Heckenberger completely classified all Nichols algebras over abelian G in .
Also over nonabelian groups, much progress has been made:
- Andruskiewitsch, Heckenberger, and Schneider studied the Weyl groupoid in this setting as well and established a root system and a PBW-basis for finite-dimensional Nichols algebras in .
- On the other hand, finite-dimensional indecomposable examples over nonabelian groups were discovered—first, Nichols algebras of type over the group and of rank 1 over (Schneider et al. ), some examples of rank 1 over various groups , as well as infinite families of large-rank Nichols algebras of Lie type constructed in  via diagram folding over central extensions of abelian groups.
- Recently, in , Heckenberger and Vendramin have classified all finite-dimensional indecomposable Nichols algebras of rank >1, thereby discovering several new Nichols algebras in rank 2 and 3. The case of rank 1 remains open. All examples in rank >3 turn out to be the foldings in the previous bullet.
3.2. A First Trace Product Formula
In the following we want to use the root system of a Nichols algebra to derive a trace product formula.
A factorization of a Nichols algebra of a braided vectorspace is a collection of braided vector spaces with some arbitrary index set , such that:
- All are braided subspaces and are homogeneous with respect to the -grading of .
- The multiplication in induces a graded isomorphism of braided vector spaces : .
An operator Q on is said to stabilize the factorization iff for all . In this case, we denote by the restriction. Q is said to normalize the factorization iff for each there is a with . In this case, Q acts on by permutations and we denote the shifting restriction with image in .
The root system of a Nichols algebra of a Yetter–Drinfel’d module M (see Section 3.1) is the leading example of a factorization.
A factorization of a Nichols algebra can be used to derive a product formula of the trace and graded trace of some operator Q. A first application is:
Let be a Nichols with factorization and Q an algebra operator that stabilizes this factorization. Then, the trace of Q is .
We evaluate the trace in the provided factorization: let Q act diagonally on the tensor product by acting on each factor via the restriction , which is possible because Q was assumed to stabilize this factorization. The action of an algebra operator Q commutes with the multiplication so the traces of Q acting on each sides coincide. The trace on a tensor product is the product of the respective traces and hence we get☐
To calculate the graded trace with the preceding lemma, first note that is a graded algebra automorphism, so if Q fulfills the conditions of the lemma, so does . Hence, we find :
Let be a Nichols algebra with factorization and Q an algebra operator that stabilizes this factorization. Then,
Let the braided vector space be defined by . Then, the diagonal Nichols algebra is of standard Cartan type and possesses a factorization with
All braidings are , hence . This implies that the multiplication in is an isomorphism of graded vector spaces:
This shows that the Hilbert series is
In the previous example of a Nichols algebra , let be defined by and . This map preserves the braiding and hence extends uniquely to an algebra automorphism on , in particular, holds. A direct calculation yields:
The product formula returns for the same trace:
In the next section, we study the special case of a diagonal Nichols algebra, and we will also study examples of operators Q, which neither stabilize nor normalize the root system. However, their graded trace is still factorizing, which indicates the existence of alternative PBW-basis. We will construct such for the case .
3.3. Nichols Algebra over Abelian Groups
We now restrict our attention to the Nichols algebra of a Yetter–Drinfel’d module M of rank n over an abelian group and . This means M is diagonal i.e., the sum of 1-dimensional braided vector spaces . According to , possesses an arithmetic root system that can be identified with a set of Lyndon words in n letters, with word length corresponding to the grading in the Nichols algebra. Such a Lyndon word corresponds to iterated q-commutators in the letters according to iterated Shirshov decomposition of the word.
For any positive root , we denote by the order of the self-braiding . It is known that this determines and we denote by the length of the Lyndon word resp. the degree of the root vector in the Nichols algebra grading.
We further denote by the G-grading of , extending the G-grading of M on simple roots. Moreover, we denote the scalar action of any on by , extending the G-action of M on simple roots. We frequently denote .
Let Q be an algebra operator on a diagonal Nichols algebra that stabilizes the root system, i.e., for all the root vector is an eigenvector to Q with eigenvalue . Then, the product formula of Corollary 3 reads as follows:
Especially for the action of a group element , we get
The factorization follows from Corollary 3. We yet have to verify the formula on each factor . has a basis for with degrees . By assumption, Q acts on via the scalar and by multiplicativity on by . Altogether:☐
Let the braided vector space be defined by with q a primitive -th root of unity. Then, the diagonal Nichols algebra is of standard Cartan type and possesses a factorization with , , and , where . This implies that the multiplication in is an isomorphism of graded vector spaces:
This obviously agrees with the Hilbert series in Lemma 8:
Let us now apply the formula of Lemma 8 to calculate the graded trace of the action of group elements (which stabilize the root system): we realize the braided vector space M as a Yetter–Drinfel’d module over , such that is -graded and is graded, with suitable actions:
Then, we get for the action of each group element :
We now look at graded traces of automorphisms Q, where Q does not stabilize the root system. We restrict ourselves to diagonal Nichols algebras , so we may use the theory of Lyndon words.
The PBW-basis consists of monotonic monomials
The PBW-basis carries the lexicographic order < and, by , (Remark after Thm 3.5) this is the same as the lexicographic order of the composed words . For a sequence of Lyndon words , not necessarily monotonically sorted, we define . In particular, sorted sequences correspond to the PBW-basis. For any sequence of Lyndon words , denote by its monotonic sorting.
- For any sequence of Lyndon words , not necessarily monotonically sorted, we have , where and “smaller” denotes linear combinations of PBW-elements lexicographically smaller than the PBW-element .
- Let and ; then,
Claim (a): We perform induction on the multiplicity of the highest appearing Lyndon word: thus, for a fixed suppose that the claim has been proven for all sequences with and strictly less then N indices i with .
Consider then a sequence with and precisely N indices i with . We perform a second induction on the index i of the leftmost appearing :
- If we may consider the sequence having strictly less w-multiplicity. By induction hypothesis, . Since the assertion then also holds for .
- Otherwise, let be the leftmost appearance of w, especially . By  Prop. 3.9, we then have where and “smaller” means products of Lyndon words with . Thus, all products contain w with a multiplicity less then ; by induction hypothesis, these are a linear combination of PBW-elements lexicographically strictly smaller then . The remaining summand has in a leftmore position and the claim follows by the second induction hypothesis.
Claim (b): We proceed by induction on the length of , which is the length of any reduced expression for . For , we are done, so assume for some i that ; hence, with shorter.
Again, by  Prop. 3.9, we have:
Moreover, for any sequences of Lyndon words , claim (a) proves that is a linear-combination of PBW-elements lexicographically smaller than . Hence, inductively,
By the same argument (again using claim (a)), we may multiply both sides with the remaining factors:
We may now use the induction hypothesis on , which is shorter. ☐
Let be a finite-dimensional Nichols algebra over a Yetter–Drinfel’d module M over an abelian group G. Let Q be an automorphism of the graded algebra permuting the roots and denote the action on root vectors by
On any orbit all orders of coincide for and we denote this value by . Similarly, all degrees α coincide and we denote the sum over the orbit in slight abuse of notation . Then,with the q-symbol and the scalar braiding factor of Q acting as an element of on A, as in the preceding lemma.
We start with the factorization along the root systemwith as assumed. The action of Q on monomials can be calculated using the previous lemma:where denotes the scalar braiding factor of Q acting as an element of on A. The trace over may be evaluated in this monomial basis and the only contributions come from monomials with all equal:
Thus, we can calculate the trace in terms of q-symbols: We sum up the scalar action factors on all Q-fixed basis elements for and multiply by the level:for and as in the assertion independent of . Therefore,☐
We now give examples where this formula can be applied. Note that the normalizing-condition seems to be very restrictive and the examples below crucially rely on exceptional behaviour for . Nevertheless, we obtain nontrivial examples, such as , and will use the previous formula systematically in the last subsection in conjunction with the finer root system presented in the examples there.
Let q be a primitive -th root of unity and M a Yetter–Drinfel’d module with braiding matrix corresponding to and . Extend to an algebra automorphism of . Note that any other off-diagonal entries would let Q fail to preserve the braiding matrix. We have and explicitly calculate . Hence, all contributions to the graded trace are balanced monomials with , yielding
Consider again the example with and q a primitive -th root of unity. Consider in this case the diagram automorphism again. In the theory of Lie algebra foldings, the flipped edge is called a loop . We easily calculate that, in such cases, the standard root system is never normalized by Q, since : . We will discuss this example and its factorization in the next subsection.
Consider the braiding matrixwhich gives rise to a Nichols algebra of type and hence a root system(these are choices), with
Notice that, for the specific choice of , by chance, we also have
Consider the diagram automorphism that preserves the braiding matrix and hence gives rise to an algebra automorphism of . We show that it normalizes the chosen factorization:
Hence, our product formula yields for the graded trace of Q:
3.4. A Non-Normalizing Example with Alternative PBW-Basis
Consider Example 15 in the previous subsection, which is not normalized, for . We first calculate the graded trace directly on the basis with :
We observe that, in this case, we have the following symmetric analog to a PBW-basis, which explains this graded trace: Denote and . Then, these elements have a common power:
Moreover, we have the relation and up to r the elements form a basis. More precisely, we have an alternative presentation for the Hilbert seriesthat could be reformulated on the level of graded vector spaces:
This factorization with relation is stabilized by the action of Q ( even and odd), from which we conclude with the formula of Lemma 8:
3.5. Factorization Mechanism for Large Rank over Nonabelian Groups
We shall finally consider a family of examples over nonabelian groups obtained by the first author in : for a finite-dimensional semisimple simply-laced Lie algebra with a diagram automorphism , consider the diagonal Nichols algebra of type . We define a covering Nichols algebra over a nonabelian group G (an extraspecial 2-group) and with folded Dynkin diagram . The covering Nichols algebra is isomorphic to as an algebra; however, there exist nondiagonal Doi twists, which leave the Hilbert series invariant.
The root system of is , but because the root spaces are mostly two-dimensional, this cannot explain the full factorization of the Hilbert series. The old factorization along the -root system is not a factorization into sub-Yetter–Drinfel’d modules, but nevertheless still shows the full factorization.
Let the dihedral group and consider the Nichols algebra where the centralizer characters are , , and (respectively, for the nondiagonal Doi twist). This was the first known example for a finite-dimensional Nichols algebra in  and it is known to be of type . We have with respective Hilbert series . From the root system, we can only explain the factorization into three factors. However, this Nichols algebra is the covering Nichols algebra of a diagonal Nichols algebra of type and, from this presentation, we may read off the full factorization in an inhomogeneous PBW-basis:
3.6. Factorization by Sub-Nichols-Algebras
During this subsection, let be an arbitrary field, G a finite group and consider a rank-1-Yetter–Drinfel’d module for some and a one-dimensional representation of . Denote the conjugacy class of g in G by X. Define the enveloping group
The Nichols algebra is naturally graded by . Now, , for all establishes as a canonical quotient of for all .
The original group G is another quotient of , and this induces an action of on .
Denote the generators of the Nichols algebra by and by the dual base.
Define for by and minimal such that for all diagonal elements . As we only consider rank one, does not depend on x, and we call the order of q. Throughout this section, assume that each coefficient is an (not necessarily primitive) m-th root of unity.
In , the second author derived divisibility relations for the Hilbert series of Nichols algebras by an analysis of the modified shiftand similar maps, where is arbitrary and is the opposite braided derivation.
For each , is a linear isomorphism, leaving invariant for all , where is the braided derivation. If is the group generated by all , , the orbit of under linearly spans . Finally, let be some group epimorphism, such that . Then, maps the -layer of degree h to the layer of degree for all (see , Proposition 9). For us, the relevant quotient H of will not be G, but as chosen above.
In addition, for each , satisfies , where we identify with the action of on :for all (the second equality is due to ).
Let λ be a k-th root of unity (not necessarily primitive). Let M be some finite-dimensional graded vector space and . Consider as a quotient of , then M has a -grading. With respect to this grading, for all if and only if the -graded character is divisible by .
This is a straightforward generalization of Lemma 6 of : given a polynomial p, denote with the coefficient of in (or zero if ). Set .
“⇒”: Choose for each and , hence . Let be such that is larger than the top degree of M. Summation of the previous equation then yields for each the telescoping sum
The two sums on the left-hand side sum to and , respectively, so by assumption, the left hand side is zero. is zero by definition (), hence is zero. This proves that p is a polynomial, and from follows .
“⇐”: Let for some polynomial p. We have and therefore for eachfrom which follows . ☐
Let G be a finite group, a proper subgroup, and be arbitrary. Let χ be a one-dimensional representation of , and let be its restriction to . Set and . Set X and to be the conjugacy classes of h in G and , respectively. Let be arbitrary and identify g with its actions on and .
(1) Then, is divisible by .
(2) Assume there is some , such that for some m-th root of unity λ, where m is the order of q. Then, is divisible by .
(1) is free as a -module, so there is a linear isomorphism mediated by multiplication (e.g., [14,15]). K and are both closed under the action of (K is closed because is closed under conjugation). Therefore, as -representations and .
(2) We show that is divisible by . Set (layer j of with ). The modified shift operator establishes a linear isomorphism between and for each . Let B be a basis for and , and denote with the dual basis element corresponding to for the basis B and for the basis , respectively. Then,holds. Apply Lemma 10. ☐
The condition of part (2) of Theorem 3 is fulfilled for and with , .
Choose and a transposition, so X and are the conjugacy classes of transpositions. Choose χ and to be the alternating representations of G and . Their Nichols algebras will appear again in Section 4.6 and Section 4.1, respectively. Choose and . Then, g and x commute, and Theorem 3 explains why contains the factor .
4. Calculations for Small Rank-1 Nichols Algebras
The following results have been calculated with the help of GAP  in a straightforward way: first, calculate a linear basis for the given Nichols algebra; then, generate the representing matrix of the action of each element of the conjugacy class X, which also generates G, and then calculate the graded traces of all conjugacy classes.
The Nichols algebra of Section 4.4 admits a large dimension of 5184. For this size, it was not possible for us to calculate all matrices we needed. In this special case, we made use of Corollary 1, so we could restrict our matrix calculations to the lower half of grades and compute the full graded trace by Poincaré duality.
The Nichols algebras of dimensions 326,592 and 8,294,400 are computationally not yet accessible with this method.
Let and representing the conjugacy class of transpositions. The centralizer of g is isomorphic to , and we choose to be its alternating representation (otherwise the Nichols algebra is infinite-dimensional). Then, the Nichols algebra is generated by with relations and some higher relations. This Nichols algebra was studied in  and is also the Fomin–Kirillov algebra for the Coxeter group . It was shown to have finite dimension and a basis in each grade is
G acts by conjugation with signs. The graded traces can be calculated and factorized by hand: obviously, the graded dimension is
The element g fixes (up to sign) only elements in degrees , so the only contributions to the trace come from , Λ. Thereby,
Similarly (and using the relations in degree 2), one calculates
From this (or directly), we can calculate the decomposition into irreducible G-representations in each degree. If we denote the trivial irreducible, alternating, and standard representations of by T, A, and S of dimension , respectively, we findas G-representation. The factorization in line 2 results from a certain sub-Nichols-algebra (see Section 3.6) and implies the factorizations
To understand the factorization of the remaining terms , , and , this line of argument, however, fails, because does not factor into a tensor product of G-representations. For and , we may apply Theorem 3. (2) to explain the additional factors and , respectively, but this neither helps in the case , nor to understand the origin of the factors for .
If does not factor as a G-representation, one might think that it may still factor as an -representation for each , which would explain the factorization of the graded characters just as well. This, however, is wrong: take , which is of order 3. Let T be the trivial irreducible representation, B one of the non-trivial irreducible representations, and set . Then, isas an -representation and does not factor further.
Assume and admits a primitive third root of unity . Chooseand . The centralizer of g is . Choose . Then, is a faithful G-representation of dimension 432. G has nine conjugacy classes, and we choose as their representatives. Then, the graded characters of (not Brauer characters, but with values in ) are:
4.3. , or 72
Consider , and . The centralizer of g is isomorphic to , and choose to be the trivial irreducible representation. Then, is 36-dimensional with graded characters (not Brauer characters):
In characteristic , there is a very similar Nichols algebra of dimension 72: Assume and
Choose , then the centralizer is , to which we choose the representation . Then, the graded characters of are:
Consider the subgroup of G. The graded characters of the H-action on are exactly those of the left column in the above list. If considered in characteristic 2, these polynomials are divisible by the corresponding graded characters of the 36-dimensional Nichols algebra, with as common quotient.
Assume and that admits a primitive third root of unity . Chooseand . The centralizer of g is . Choose the representation . This leads to the following graded characters of :
In characteristic 2, yields a Nichols algebra with the same Hilbert series. Some of the above conjugacy classes merge in this case, because is a -extension of , but apart from that, the resulting graded characters are the same as above.
Chooseand . G is isomorphic to the GAP’s small group number 3 of size 20 , a semi-direct product of and . The centralizer of g is . Choose the representation . Then, is a faithful G-representation of dimension 1280 with the following graded characters:
There appears a second, non-isomorphic (but dual) Nichols algebra if one chooses , instead (see Example 2.1 in ). It features the same graded characters as above.
There are three pairwise non-isomorphic cases to consider with and .
First, choose and . The centralizer of g is . Choose the representation with and . The graded characters of are:
Now, choose the representation , instead. Then, the graded characters of are:
Third, choose and . The centralizer of g is . Choose the representation . Then, has the following graded characters:
Note how the graded characters differ pairwise for these three cases, a simple way to see that the three Nichols algebras obtained are non-isomorphic, not even as -representations, although the first and the second case are twist-equivalent to each other .
From the examples of the previous sections, we derive the following observations, which may help us in understanding the factorization of the Hilbert series and graded characters of any Nichols algebra. A theory of the representations coming from a Nichols algebra should be able to explain all of them.
- The zeros of the graded characters of all examples above are n-th roots of unity, where n most of the time is a divisor of , but not always: In Section 4.4, ninth roots of unity appear though ; in Section 4.5, we have , but has an eighth root of unity. A deeper understanding of why there are only roots of unity and which roots appear how often is eligible.
- Each of the characters with includes a factor (or in characteristic 2); therefore, the non-graded character vanishes. From this follows that all of the above Nichols algebras are (seen as their respective G-representations) multiples of the regular G-representation. The only exceptions to this are the 432-dimensional and the 72-dimensional Nichols algebras of Section 4.2 and Section 4.3, each of which admits a single non-trivial conjugacy class with non-vanishing character.
- The smallest common multiple p of the graded characters of a single Nichols algebra has a surprisingly small degree. We want to point out that the quotient typically is a polynomial whose roots have the same order as g has in G.
- Although all of the characters factor nicely (see point (1)), there is no corresponding factorization of the respective representations; we showed this in Section 4.1.
Quite often the factorization of the graded trace of a group acting on a finite-dimensional Nichols algebra can be explained by finer root systems. However in the most interesting cases of rank 1 we have no satisfying answer. We give these graded traces as empirical data and observe in particular that the G-representations do not factorize. We view our preliminary results on one hand as motivation to search for more systematic constructions of some of these Nichols algebras (even the Fomin Kirillov algebra of dimension 12 over ) in terms of folding, on the other hand some factorizations follow in these cases from subalgebras, which also could be the general picture.
The authors wish to thank I. Heckenberger for stimulating discussions.
Both authors contributed equally to research and writing of the manuscript.
Conflicts of Interest
The authors declare no conflicts of interest.
- Andruskiewitsch, N.; Heckenberger, I.; Schneider, H.J. The Nichols algebra of a semisimple Yetter-Drinfeld module. Am. J. Math. 2010, 132, 1493–1547. [Google Scholar]
- Heckenberger, I. Classification of arithmetic root systems. Adv. Math. 2009, 220, 59–124. [Google Scholar] [CrossRef]
- Fomin, S.; Kirillov, A.N. Quadratic algebras, dunkl elements, and schubert calculus. In Progress in Mathematics; Birkhäuser Boston: Cambridge, MA, USA, 1997; Volume 172, pp. 147–182. [Google Scholar]
- Milinski, A.; Schneider, H.J. Pointed indecomposable Hopf algebras over Coxeter groups. In New Trends in Hopf Algebra Theory, Proceedings of the Colloquium on Quantum Groups and Hopf Algebras, La Falda, Argentina, 9–13 August 1999; Number 267 in Contemporary Mathematics; AMS: Providence, RI, USA, 2000; pp. 215–236. [Google Scholar]
- Graña, M.; Heckenberger, I.; Vendramin, L. Nichols algebras of group type with many quadratic relations. Adv. Math. 2011, 227, 1956–1989. [Google Scholar] [CrossRef]
- Heckenberger, I.; Lochmann, A.; Vendramin, L. Braided racks, Hurwitz actions and Nichols algebras with many cubic relations. Transform. Groups 2012, 17, 157–194. [Google Scholar] [CrossRef]
- Heckenberger, I.; Lochmann, A.; Vendramin, L. Nichols algebras with many cubic relations. Trans. Am. Math. Soc. 2013, 367, 6315–6356. [Google Scholar] [CrossRef]
- Lentner, S. Orbifoldizing Hopf- and Nichols Algebras. Ph.D. Thesis, Ludwig-Maximilian-University Munich, Munich, Germany, 2012. [Google Scholar]
- Heckenberger, I.; Vendramin, L. A classification of Nichols algebras of semi-simple Yetter-Drinfeld modules over non-abelian groups. arXiv, 2014. [Google Scholar]
- Cuntz, M.; Lentner, S. A simplicial complex of Nichols algebras. Math. Z. 2015, 3–4, 647–683. [Google Scholar] [CrossRef]
- Carter, R.W. Finite Groups of Lie Type—Conjugacy Classes and Complex Characters; Wiley-Interscience: Hoboken, NJ, USA, 1985. [Google Scholar]
- Lentner, S. New large-rank Nichols algebras over nonabelian groups with commutator subgroup Z_2. J. Algebra 2014, 419, 1–33. [Google Scholar] [CrossRef]
- Lochmann, A. Divisibility relations for the dimensions and Hilbert series of Nichols algebras of non-abelian group type. J. Algebra 2013, 381, 21–36. [Google Scholar] [CrossRef]
- Graña, M. A freeness theorem for Nichols algebras. J. Algebra 2000, 231, 235–257. [Google Scholar] [CrossRef]
- Skryabin, S. Projectivity and freeness over comodule algebras. Trans. Am. Soc. 2007, 359, 2597–2623. [Google Scholar] [CrossRef]
- Jing, N.; Zhang, J.J. On the trace of graded automorphisms. J. Algebra 1997, 189, 353–376. [Google Scholar] [CrossRef]
- Ginzburg, V. (Ed.) Algebraic geometry and number theory. In Progress in Mathematics; Birkhäuser Boston: Cambridge, MA, USA, 2006.
- Heckenberger, I. Nichols Algebras. Lecture Notes at ECNU, Shanghai. Available online: http://www.mi.uni-koeln.de/~iheckenb/na.pdf (accessed on 4 December 2017).
- Andruskiewitsch, N.; Grana, M. Braided Hopf algebras over non-abelian groups. Boletin de la Academia Nacional de Ciencias en Córdoba 1999, 63, 45–78. [Google Scholar]
- Heckenberger, I.; Schneider, H.J. Root systems and Weyl groupoids for Nichols algebras. Proc. Lond. Math. Soc. 2010, 101, 623–654. [Google Scholar] [CrossRef]
- Andruskiewitsch, N.; Fantino, F.; Graña, M.; Vendramin, L. Pointed hopf algebras over some sporadic simple groups. Comptes Rendus Math. 2010, 348, 605–608. [Google Scholar] [CrossRef]
- Andruskiewitsch, N.; Zhang, S. On pointed Hopf algebras associated to some conjugacy classes in Sn. Proc. Am. Math. Soc. 2007, 135, 2723–2731. [Google Scholar] [CrossRef]
- Graña, M. Nichols Algebras of Nonabelian Group Type, Zoo of Examples, 2000. Available online: http://mate.dm.uba.ar/~matiasg/zoo.html (accessed on 4 December 2017).
- The GAP Group. Gap—Groups, Algorithms, and Programming, 2006. Available online: http://www.gap-system.org (accessed on 4 December 2017).
- Andruskiewitsch, N.; Fantino, F.; Garcia, G.A.; Vendramin, L. On Nichols algebras associated to simple racks. Contemp. Math. 2011, 537, 31–56. [Google Scholar]
- Vendramin, L. Nichols algebras associated to the transpositions of the symmetric group are twist-equivalent. Proc. Am. Math. Soc. 2012, 140, 3715–3723. [Google Scholar] [CrossRef]
© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).