1. Introduction

The main purpose of this article is to give a computer-aided proof of a surprising failure of the Koszul property for a family of algebras of interest in rational homotopy theory.

The study of formal spaces provides a natural setting for the question of whether $H^* (X,{{\mathbb Q}})$ is a Koszul algebra for a given space $X$. A space $X$ is called formal if its Sullivan model $A_{PL}(X)$ of polynomial differential forms is a formal graded-commutative differential graded algebra (cdga), that is, if $A_{PL}(X)$ is quasi-isomorphic as a cdga to $H^* (A_{PL}(X))$ with trivial differential. The ${{\mathbb Q}}$-completion of a formal space $X$, and hence the rational homotopy type of $X$, are completely determined by $H^* (X,{{\mathbb Q}})$. There are many examples of such spaces in topology, including compact Kähler manifolds and complements of complex hyperplane arrangements. The latter implies Eilenberg–Mac Lane spaces of type $K(G,1),$ where $G$ is a pure braid group are formal spaces.

Sullivan [24] and Morgan [17] showed that the fundamental group of a formal space is a 1-formal group. There exist non-formal spaces with 1-formal fundamental groups, but Sullivan and Morgan's theorem does have a partial converse. Papadima and Suciu [18] proved that for a connected, finite-type CW complex $X$, if $H^* (X,{{\mathbb Q}})$ is a Koszul algebra and $\pi _1(X)$ is a 1-formal group, then $X$ is a formal space.

In [4], Berceanu and Papadima prove certain motion groups, denoted $P\Sigma _n$, are 1-formal groups. The groups $P\Sigma _n$ have a long history. David Dahm, in his unpublished PhD thesis [10], considered arrangements of $n$ unknotted, unlinked circles in 3-space and the corresponding group $P\Sigma _n$ of motions of the arrangement in which each circle ends at its original position. In [12], Goldsmith explained Dahm's work and realized $P\Sigma _n$ as a subgroup of the automorphism group of the free group of rank $n$; she also determined specific generators of $P\Sigma _n$.

In [16], McCool found a presentation of $P\Sigma _n$ in terms of relations of the generators found by Goldsmith. In the paper [5], Brownstein and Lee were interested in the representation theory of $P\Sigma _n$ and successfully computed the second integral cohomology $H^2(P\Sigma _n, {{\mathbb Z}})$; they also conjectured a presentation for the entire cohomology algebra $H^* (P\Sigma _n, {{\mathbb Z}})$. Over 10 years later, Jensen, McCammond and Meier in [13] proved the conjecture of Brownstein and Lee. In particular, they showed $H^* (P\Sigma _n,{{\mathbb Q}})$ is a quadratic algebra, a necessary condition for Koszulity.

Berceanu and Papadima [4] also established 1-formality for the related group $P\Sigma _n^+ $ called the upper triangular McCool group. Cohen and Pruidze [8] proved $H^* (P\Sigma _n^+ ,{{\mathbb Q}})$ is a Koszul algebra, hence the Eilenberg–Mac Lane space of type $K(P\Sigma ^+_n, 1)$ is a formal space. They asked in that paper whether $H^* (P\Sigma _n,{{\mathbb Q}})$ is Koszul.

Our main theorem is the following.

This raises the possibility that $K(P\Sigma _n, 1)$ is not a formal space, but we do not know if this is the case. Papadima and Yuzvinsky showed in [19] that for $X$, a formal, connected, topological space with finite Betti numbers, the Koszul property of $H^* (X, {{\mathbb Q}})$ is equivalent to $X$ being a rational $K[\pi , 1]$ space, that is, the ${{\mathbb Q}}$-completion of $X$ is aspheric. The proof of Theorem 1.1 is given in Section 3. One should contrast Theorem 1.1 with the fact that several families of algebras naturally related to $H^* (P\Sigma _n, {{\mathbb Q}})$ are Koszul. Besides $H^* (P\Sigma _n^+ ,{{\mathbb Q}})$, the cohomology of the pure braid group is well known to be Koszul, as are the algebras studied in [3].

In Section 2, we show that the quadratic dual of $H^* (P \Sigma _n, k)$ is the enveloping algebra of a graded Lie algebra. We denote this enveloping algebra by $U({\mathfrak g}_n)$. As such, $U({\mathfrak g}_n)$ has a natural bialgebra structure. In Section 4, we study the algebra structure of $U({\mathfrak g}_n)$ and prove the following.

This theorem parallels results of [2, 9, 20].

2. Definitions of $U({\mathfrak g}_n)$ and $U({\mathfrak g}_n)^!$

Let $F_n$ denote the free group on $\{x_1,\ldots ,x_n\}$. For $n\ge 2$, the group $P\Sigma _n$ is the subgroup of ${\hbox {Aut}}(F_n)$ generated by the automorphisms \[\alpha _{ij}(x_k) = \left \{\begin {array}{ll}x_jx_ix_j^{-1} & \mbox {if } k=i{,} \\ x_k & \mbox {if } k\neq i,\\ \end {array}\right .\]

for all $1 \leq i \ne j \leq n$.

Elements of $P\Sigma _n$ have been called pure symmetric automorphisms and basis-conjugating automorphisms. McCool proved that the following relations determine a presentation of $P\Sigma _n$ [16]: \[\begin {align} &[\alpha _{ij},\alpha _{ik}\alpha _{jk}] \quad i, j, k {\text { distinct{,} }}\\ &[\alpha _{ij},\alpha _{kj}] \quad i, j, k {\text { distinct{,} }}\\ &[\alpha _{ij},\alpha _{kl}] \quad i, j, k, l {\text { distinct.}}\\ \end {align}\]

A presentation for the integral cohomology of $P\Sigma _n$ was conjectured by Brownstein and Lee [5] and proved by Jensen, McCammond, and Meier [13]. In this section and the next, we work with coefficients in ${{\mathbb Q}}$, though all results hold over a field ${\mathbb {k}}$ of characteristic 0.

The cohomology algebra $H^* (P\Sigma _n,{{\mathbb Q}})$ is a quadratic algebra, so it is natural to consider its quadratic dual algebra. We briefly recall this standard construction. The book [22] is an excellent reference on the theory of quadratic algebras.

Let ${\mathbb {k}}$ be a field. A ${\mathbb {k}}$-algebra $A$ is called quadratic if there exists a finite-dimensional ${\mathbb {k}}$-vector space $V$ and a subspace $R\subset V \otimes V$ such that $A\cong T(V)/ \langle R \rangle$, where $T(V)$ denotes the tensor algebra on $V$. The tensor algebra $T(V)$ is graded by tensor degree, and if $R\subset V \otimes V$, the factor algebra $T(V)/ \langle R \rangle$ inherits the tensor grading. Quadratic algebras are therefore ${{\mathbb N}}$-graded, have finite-dimensional graded components, and are connected, that is, $A_0={\mathbb {k}}$. The quadratic dual algebra $A^!$ is defined to be $T(V^* )/ \langle R^{\perp } \rangle ,$ where $V^* $ is the ${\mathbb {k}}$-linear dual of $V$ and $R^{\perp }\subset V^* \otimes V^* $ is the orthogonal complement of $R$ with respect to the natural pairing.

Throughout the paper, if $A$ is a quadratic algebra, then we use the term $A$-module to mean a ${{\mathbb Z}}$-graded $A$-module $M$ such that $\dim _{{\mathbb {k}}} M_i\lt \infty$ for all $i\in {{\mathbb Z}}$. If $M$ is an $A$-module and $j\in {{\mathbb Z}}$, then $M[j]$ denotes the shifted module with $M[j]_i = M_{i+j}$ for all $i\in {{\mathbb Z}}$. A homomorphism of $A$-modules $\varphi :M\rightarrow N$ is assumed to preserve degrees: $\varphi (M_i)\subset N_i$ for all $i\in {{\mathbb Z}}$. The graded Hom functor is \[\underline {{\hbox {Hom}}}_A(M,N):=\bigoplus _{j\in {{\mathbb Z}}} {\hbox {Hom}}_A^j(M,N),\] where ${\hbox {Hom}}_A^j(M,N)={\hbox {Hom}}_A(M,N[-j])$. The right-derived functors of the graded Hom functor are the graded Ext functors \[{{\rm Ext}}^i_A(M,N)=\bigoplus _{j\in {{\mathbb Z}}} {{\rm Ext}}_A^{i,j}(M,N).\] The Yoneda algebra of a quadratic algebra $A$ is the ${{\mathbb N}}\times {{\mathbb N}}$-graded ${\mathbb {k}}$-vector space ${\hbox {Ext}}_A({\mathbb {k}},{\mathbb {k}})=\bigoplus _{i,j} {\hbox {Ext}}_A^{i,j}({\mathbb {k}},{\mathbb {k}})$ endowed with the Yoneda composition product, which preserves the bi-grading. A quadratic algebra $A$ is called Koszul if ${\hbox {Ext}}_A^{i,j}({\mathbb {k}},{\mathbb {k}})=0$ whenever $i\neq j$, in which case ${\hbox {Ext}}_A({\mathbb {k}},{\mathbb {k}})\cong A^!$ as graded algebras. An algebra $A$ is Koszul if and only if $A^!$ is (see [22]).

In this section, and throughout most of this paper, we study the algebra $(E/I)^!$.

The similarity to the relations of $P\Sigma _n$ is not a coincidence. Lemma 2.2 shows that $(E/I)^!$ is the universal enveloping algebra of a quadratic Lie algebra which we denote by ${\mathfrak g}_n$. The Lie algebra ${\mathfrak g}_n$ is the associated graded Lie algebra of the Mal’cev Lie algebra of $P\Sigma _n$ (see [4]). For more on the construction of ${\mathfrak g}_n$, see Section 4. Henceforth, we denote the quadratic dual algebra by $U({\mathfrak g}_n)$ rather than $(E/I)^!$.

As a consequence of Lemma 2.2, it is easy to see that $U({\mathfrak g}_2)$ and $U({\mathfrak g}_3)$ are Koszul algebras; in fact, after a suitable change of variables, they are Poincaré-Birkhoff-Witt (PBW)-algebras in the sense of Priddy [23].

The analogous change of variables for $U({\mathfrak g}_4)$ does not produce a set of PBW generators, but many Koszul algebras are not PBW. The fact that the commutators $[x_{ij},x_{kl}]$ do not appear as relations of $U({\mathfrak g}_n)$ until $n=4$ underlies the failure of the PBW property. And as we will show, $U({\mathfrak g}_n)$ is not Koszul for $n=4$.

3. The factor algebras $U({\mathfrak g}_n/{\mathfrak h}_n)$

In this section, we describe a useful reduction from $U({\mathfrak g}_n)$ to a factor algebra whose failure of Koszulity implies $U({\mathfrak g}_n)$ is also not Koszul. We use the reduction to determine that $U({\mathfrak g}_4)$ is not Koszul. The next lemma shows this is sufficient to conclude $U({\mathfrak g}_n)$ is not Koszul for $n\ge 4$.

For each $1\le j\le n$, let $X_j = \sum _{i\neq j} x_{ij}$ denote the sum of the ‘$j$th column’ generators of the Lie algebra ${\mathfrak g}_n$. The following is based on an observation of Graham Denham.

Each of $U({\mathfrak g}_n)$, $U({\mathfrak h}_n)$, and $U({\mathfrak g}_n/{\mathfrak h}_n)$ is a quadratic algebra, so their Yoneda Ext-algebras are bigraded (see Section 2). We denote $H^{i,j}({\mathfrak g}_n,{{\mathbb Q}})={\hbox {Ext}}^{i,j}_{U({\mathfrak g}_n)}({{\mathbb Q}},{{\mathbb Q}})$ and similarly define $H^{i,j}({\mathfrak h}_n,{{\mathbb Q}})$ and $H^{i,j}({\mathfrak g}_n/{\mathfrak h}_n,{{\mathbb Q}})$. When only one grading component is listed, it is assumed to be the homological degree.

We now consider the case $n=4$ and, for the remainder of this section, we abbreviate $U({\mathfrak g}/{\mathfrak h})=U({\mathfrak g}_4/{\mathfrak h}_4)$. Eliminating generators $x_{41}, x_{42}, x_{43}$, and $x_{34}$ of $U({\mathfrak g}_4)$, we obtain the following presentation for $U({\mathfrak g}/{\mathfrak h})$.

Having passed to $U({\mathfrak g}/{\mathfrak h})$, the quadratic dual algebra also becomes smaller. The proof of the following lemma is a straightforward calculation analogous to that of Lemma 2.2.

If a quadratic algebra $A$ is Koszul, then the Hilbert series of $A$ and its quadratic dual algebra $A^!$ satisfy the relation $h_A(t)h_{A^{!}}(-t)=1$ (see [22]). The converse is false, there exist non-Koszul quadratic algebras for which the Hilbert series relation holds (see, for example [21]). The fact that $U({\mathfrak g}/{\mathfrak h})^!_3=0$ is noteworthy since the Hilbert series relation does imply Koszulity in this case [22, Corollary 2.2.4].

Thus the algebra $U({\mathfrak g}/{\mathfrak h})$ is Koszul if and only if its Hilbert series is $1/(1-4t)^2$. Indeed, this was our motivation for considering the quotient $U({\mathfrak g}/{\mathfrak h})$. The first nine terms of the Maclaurin series of $1/(1-4t)^2$ are \[1+8t+48t^2+256t^3+1280t^4+6144t^5+28\,672t^6+131\,072t^7+589\,824t^8+ \cdots .\] As discussed in more detail below, we used the computer algebra system bergman [1] (specifically the ncpbhgroebner method) to compute a Gröbner basis and the Hilbert series for $U({\mathfrak g}/{\mathfrak h})$ in degrees $\le 8$. We found the first eight terms of the Hilbert series to be \[1+8t+48t^2+256t^3+1280t^4+6144t^5+28\,672t^6+131\,072t^7+589\,8\mathbf {34}t^8+ \cdots .\] This computation demonstrates our result.

We first performed this calculation several years ago using the 32-bit Windows installation of bergman on an IBM ThinkPad T410 series with a 2.4 GHz Intel Core i5 M520 processor and 3 GB DDR3 RAM. The result has since been duplicated several times on various systems and architectures as well as under GAP [25] and other software packages, both by the authors and independently by Graham Denham, Frédéric Chapoton, Bérénice Oger, and Jan-Erik Roos. We wish to thank them for their interest in verifying Theorem 3.6.

Several additional remarks are in order. First, one can also use bergman to show the Hilbert series of $U({\mathfrak g}_4)$ agrees with $1/(1-4t)^3$ until $t^8$, at which point the coefficients again differ by 10, consistent with the collapse of the spectral sequence in Proposition 3.3. Since $U({\mathfrak g}_4)$ is much bigger than $U({\mathfrak g}_4/{\mathfrak h}_4)$, this calculation takes much longer and consumes much more memory. The calculation for $U({\mathfrak g}_4/{\mathfrak h}_4)$ takes less than 12  h on most machines we have tried. Secondly, the results above show the algebras $U({\mathfrak g}_4/{\mathfrak h}_4)$ and $U({\mathfrak g}_4)$ are 7-Koszul but not Koszul, meaning $H^{i,j}({\mathfrak g}_4)=0$ for $i\lt j\lt 8$. The delayed failure of Koszulity together with the exponential growth of $U({\mathfrak g}_4/{\mathfrak h}_4)$ make this calculation extremely difficult to perform by hand.

The Hilbert series calculation shows $\dim H^{3,8}({\mathfrak g}_4/{\mathfrak h}_4)=10$. Using bergman, we have also computed $\dim H^{3,9}({\mathfrak g}_4/{\mathfrak h}_4)=40$. This raises the following question.

4. $U({\mathfrak g}_n)$ as a smash product

In this section, we show that $U({\mathfrak g}_n)$ can be decomposed as a smash product of algebras. We briefly recall the smash product construction following [15]. Let ${\mathbb {k}}$ denote a field. Let ${\mathfrak g}$ be a ${\mathbb {k}}$-Lie algebra acting on a ${\mathbb {k}}$-algebra $R$ by ${\mathbb {k}}$-derivations. For $x \in {\mathfrak g}$ and $r \in R$, let $x(r)$ denote the action of $x$ on $r$. Let $\{x_i \, | \, i \in I\}$ be a basis for ${\mathfrak g}$. The smash product $R \# U({\mathfrak g})$ is defined to be the ${\mathbb {k}}$-algebra generated by $R$ and $\{x_i \, | \, i \in I\}$ with the relations $x_ir - rx_i = x_i(r)$ and $x_ix_j - x_jx_i = [x_i, x_j]$. As a vector space, $R \# U({\mathfrak g}) \cong R \otimes U({\mathfrak g})$.

Let $G$ be a group. The lower central series of $G$ is the sequence of normal subgroups of $G$ defined by $G_1 = G$, $G_{n+1} = [G_n, G]$. Let $G(n) = G_n/G_{n+1}$ denote the $n$th lower central series quotient group. Then ${\mathfrak g}= \bigoplus _n G(n) \otimes {{\mathbb Q}}$ is a Lie algebra over ${{\mathbb Q}}$ where the Lie bracket is induced by the group commutator. In the case $G=P\Sigma _n$, the resulting Lie algebra is ${\mathfrak g}_n$, see Theorem 5.4(2) in [4].

Let $1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$ be a split short exact sequence of groups. Assume $[A,C]\subset [A,A]$ or, equivalently, that $C$ acts trivially on $H_1(A)$. Then $0\rightarrow A(n)\rightarrow B(n)\rightarrow C(n)\rightarrow 0$ is a split exact sequence of abelian groups for all $n$, [11, 14]. And since ${{\mathbb Q}}$ is a flat ${{\mathbb Z}}$-module, the sequence \[0\rightarrow \mathfrak a\rightarrow \mathfrak b\rightarrow \mathfrak c\rightarrow 0\] is a split exact sequence of graded ${{\mathbb Q}}$-Lie algebras. It then follows (see 1.7.11 in [15]) that $U(\mathfrak b)\cong U(\mathfrak a)\#U(\mathfrak c)$.

Bardakov [2] has given two semidirect product decompositions of $P\Sigma _n$. Recall that $P\Sigma _n$ has a set of generators $\{\alpha _{ij} \, | \, 1 \leq i \ne j \leq n \}$. Suppose that $n \geq 3$, and set $m = \lfloor {n}/{2} \rfloor$. We define subgroups of $P\Sigma _n$ as follows. Let $K_n$ be the subgroup generated by $\{\alpha _{in}, \alpha _{ni} \, | \, 1 \leq i \lt n\}$; let $H_n$ be the subgroup generated by $\{\alpha _{2i-1, 2i}, \alpha _{2i, 2i-1} \, | \, i = 1, 2, \ldots , m \}$; let $G_n$ be the subgroup generated by the complement of $\{\alpha _{2i-1, 2i}, \alpha _{2i, 2i-1} \, | \, i = 1, 2, \ldots , m \}$ in the set of generators of $P\Sigma _n$. Then there are split short exact sequences of groups \[\begin {align} 1 &\longrightarrow K_n \longrightarrow P\Sigma _n \longrightarrow P\Sigma _{n-1} \longrightarrow 1,\\ 1 &\longrightarrow G_n \longrightarrow P\Sigma _n \longrightarrow H_n \longrightarrow 1. \end {align}\]

It is proved in [9] that the natural conjugation action of $P\Sigma _{n-1}$ on $H_1(K_n)$ is trivial. Similarly, the formulas in the proof of Theorem 2 of [2] show that the action of $H_n$ on $H_1(G_n)$ is also trivial. Let $R_n$, $S_n$, $T_n$ denote the enveloping algebras of the graded ${{\mathbb Q}}$-Lie algebras associated to the lower central series of the groups $G_n$, $H_n$, $K_n$, respectively. The following theorem is immediate.

It is worthwhile to note that [2] shows that $H_n$ is isomorphic to the direct product of $m$ copies of $F_2$, the free group on two generators. Hence $S_n$ is a tensor product of $m$ copies of a free associative algebra on two generators.

We conclude by proving that $T_n$ and $R_n$ are not finitely presented algebras.

This result should be compared with Pettet's results in [20].

Although these algebras are not finitely presented it is straightforward to compute minimal defining relations in any given degree. We have checked that the quadratic closures (the algebras on the same sets of generators but with only the defining quadratic relations) of $T_n$ and $R_4$ are Koszul. Further study of these algebras would be interesting.