# Solving spin systems: the Babylonian way.

Nicola Kistler  Nicola Kistler
J.W. Goethe-Universität Frankfurt, Germany<EMAIL_ADDRESS>
###### Abstract.

We show that spin systems with generic (ferro- or paramagnetic, or random)
interactions are ”completely integrable”. The approach is worked out, by way
of example, for the Sherrington-Kirkpatrick model: we derive an exact, closed
formula for the quenched free energy in finite volume which involves an
integral over a Gaussian field with correlation structure given by the
interaction matrix (with a twist).

This work has been supported by a DFG research grant, contract number
2337/1-1. It is my pleasure to thank Stephan Gufler, Adrien Schertzer, Marius
A. Schmidt, and Giulia Sebastiani for the endless discussions on the topic of
these notes.

The Sherrington-Kirkpatrick (SK) model [10] for mean field spin glasses is
constructed as follows: for $N\in\mathbb{N}$, consider centered Gaussians (the
disorder) $(g_{ij})_{1\leq i<j\leq N}$ issued on some probability space
$(\Omega,\mathcal{F},\mathbb{P})$, all independent and with variance $1/N$.
The Ising configuration space $\Sigma_{N}=\\{\pm 1\\}^{N}$ is endowed with
coin tossing measure, $P_{o}(\sigma)=2^{-N}$ for $\sigma\in\Sigma_{N}$. We
denote by $E_{o}$ expectation under $P_{o}$. The SK-Hamiltonian is then

$H_{N}(\sigma)=\sum_{1\leq i<j\leq
N}g_{ij}\sigma_{i}\sigma_{j}=\frac{1}{2}\sum_{i,j=1}^{N}g_{ij}\sigma_{i}\sigma_{j}\,,$
(1)

with (symmetrised) random interaction matrix $g_{ij}=g_{ji}$, and $g_{ii}=0$.
The quenched, finite volume SK-free energy to inverse temperature $\beta>0$
and external field $h\in\mathbb{R}$ is

$Nf_{N}(\beta,h)\equiv\log E_{o}\left[\exp\left(\beta
H_{N}(\sigma)+h\sum_{i=1}^{N}\sigma_{i}\right)\right]\,.$ (2)

Theorem. Let ${\sf X}=({\sf X}_{i})_{i=1}^{N}$ be a centered Gaussian random
field with covariance

${\sf{E}}{\sf X}_{i}{\sf
X}_{j}=\begin{cases}\sum_{k=1}^{N}\left|g_{ik}\right|\,,&i=j,\\\
g_{ij}\,,&i\neq j.\end{cases}$ (3)

Then the exact formula for the quenched SK-free energy holds:

$Nf_{N}(\beta,h)=\log{\sf{E}}\exp\left(\sum_{i=1}^{N}\log\cosh\left(h+\sqrt{\beta}{\sf
X}_{i}\right)\right)-\frac{\beta}{2}\sum_{i,j=1}^{N}|g_{ij}|\,.$ (4)

The formula (4) is quite fascinating: a minor yet crucial modification of the
diagonal turns the Gaussian disorder matrix $(g_{ij})$ into the covariance of
a Gaussian field.

It will become clear below that the treatment also yields exact formulas for
spin magnetizations, correlations etc. under the quenched Gibbs measure.

Before giving the (truly elementary) proof of the Theorem, some comments are
in order.

First, we emphasize that neither the Ising nature of the spins, nor the
Gaussian character of the disorder play any structural role: the treatment
allows, mutatis mutandis, to derive exact formulas for the Hopfield111In the
Hopfield model one obtains a closed formula for the free energy which differs
from the classical one obtained via plain Hubbard-Stratonovich transformation
[7]. model [5], the independent set problem [8], the Edwards-Anderson (EA)
model [2] or, incidentally, also classical systems such as the Ising model on
$\mathbb{Z}_{d}$ for $d\geq 1$, or, for that matter, on any (random) graph
etc. The approach relies on the Babylonian trick222as Res Jost referred to the
method of completing the squares. to decouple the two body interaction, and
thus applies in vast generality. As a matter of fact, the procedure is
flexible enough to cover also the case of generic $p$-body interactions,
$p\geq 2$: how to do this is sketched in Remark 1 below.

The case of the EA-model is particularly interesting. The model itself is to
these days deemed intractable, also from the standpoint of theoretical
physics. Indeed, even numerical simulations have been frighteningly
inconclusive so far. Now, a minor modification of formula (4) holds true for
this model as well (at the risk of being opaque: one simply replaces the
external field with the boundary conditions, and sums over nearest neighbors).
It thus follows that all physically relevant quantities can be efficiently
simulated via polynomially many Gaussian random variables only. There is good
reason to believe that this will help to settle the long-standing debate
between the droplet picture of Fisher and Huse [3] in low dimensions and the
replica symmetry breaking scenario predicted by the Parisi theory [7].

It should be mentioned that the decoupling is straightforward also in models
where the underlying measure ($E_{o}$) itself is interacting, such as the
Heisenberg model [4] for magnetism ($E_{o}\to$ spherical measure) or, say, the
Domb-Joyce model [1] for weakly self avoiding walks ($E_{o}\to$ law of simple
random walk), but in such cases the ensuing formulas are more involved.

Concerning the aforementioned disordered mean field models: the Theorem
reduces the computation of the limiting free energy to the asymptotic
evaluation of Gaussian integrals akin to (4), which - albeit quenched - are
reminiscent of the Parisi solution [7]. This, in turn, brings into the game
random matrices of Wigner-type (remark that the entries of the covariance (4)
are not independent under $\mathbb{P}$: terms on and above the diagonal are
correlated). We will report on this in future work.

Finally, the Babylonian trick stems from an effort to tame the issue of
frustration, ever so present in disordered systems. How to overcome this
problem, see (5)-(8) below, is perhaps the main observation of these notes.

###### Proof of the Theorem.

Some conventions: we will mostly omit the parameter $N$ from our notations. In
particular we write $\sum_{i}a_{i}\equiv\sum_{i=1}^{N}a_{i}$ and
$\sum_{ij}a_{ij}\equiv\sum_{i,j=1}^{N}a_{ij}$ for double sums. We will use the
classical notation

$g_{ij}^{+}\equiv\max\left\\{g_{ij},0\right\\},\qquad
g_{ij}^{-}\equiv\max\left\\{-g_{ij},0\right\\}\,,$ (5)

in which case

  * i)

both $g^{+}$ and $g^{-}$ are positive: $g_{ij}^{+}\geq 0$ and $g_{ij}^{-}\geq
0$;

  * ii)

the decomposition holds true: $g_{ij}=g_{ij}^{+}-g_{ij}^{-}$;

  * iii)

similarly, it holds: $\left|g_{ij}\right|=g_{ij}^{+}+g_{ij}^{-}$.

We first split the Hamiltonian

$\displaystyle\sum_{i,j}g_{ij}\sigma_{i}\sigma_{j}$
$\displaystyle\stackrel{{\scriptstyle\text{ii)}}}{{=}}\sum_{i,j}g_{ij}^{+}\sigma_{i}\sigma_{j}+\sum_{i,j}\left(-g_{ij}^{-}\right)\sigma_{i}\sigma_{j}$
(6)
$\displaystyle=\sum_{i,j}g_{ij}^{+}\sigma_{i}\sigma_{j}+\sum_{i,j}g_{ij}^{-}\left(-\sigma_{i}\sigma_{j}\right)\,.$

The Babylonian trick amounts to writing

$\sigma_{i}\sigma_{j}=\frac{1}{2}(\sigma_{i}+\sigma_{j})^{2}-1,\qquad-\sigma_{i}\sigma_{j}=\frac{1}{2}(\sigma_{i}-\sigma_{j})^{2}-1,$
(7)

(also using the simplification $\sigma_{i}^{2}=1$, valid for Ising spins).
This implies

$\eqref{split}=\frac{1}{2}\sum_{i,j}g_{ij}^{+}(\sigma_{i}+\sigma_{j})^{2}+\frac{1}{2}\sum_{i,j}g_{ij}^{-}(\sigma_{i}-\sigma_{j})^{2}+G\,,$
(8)

where the $\sigma$-independent term reads

$G\equiv-\sum_{i,j}g_{ij}^{+}-\sum_{i,j}g_{ij}^{-}=-\sum_{i,j}\left(g_{ij}^{+}+g_{ij}^{-}\right)\stackrel{{\scriptstyle\text{iii)}}}{{=}}-\sum_{i,j}|g_{ij}|\,.$
(9)

The quenched free energy is thus

$\displaystyle Nf_{N}(\beta,h)=\log
E_{o}\left[\exp\left(\frac{\beta}{4}\sum_{i,j}g_{ij}^{+}(\sigma_{i}+\sigma_{j})^{2}+\frac{\beta}{4}\sum_{i,j}g_{ij}^{-}(\sigma_{i}-\sigma_{j})^{2}+h\sum_{i}\sigma_{i}\right)\right]-\frac{\beta}{2}G\,.$
(10)

We henceforth focus on the partition function

$Z_{N}(\beta,h)\equiv
E_{o}\left[\exp\left(\frac{\beta}{4}\sum_{i,j}g_{ij}^{+}(\sigma_{i}+\sigma_{j})^{2}+\frac{\beta}{4}\sum_{i,j}g_{ij}^{-}(\sigma_{i}-\sigma_{j})^{2}+h\sum_{i}\sigma_{i}\right)\right]\,.$
(11)

The double sums in the exponential consist of positive terms: we can thus
linearise via Hubbard-Stratonovich transformation [6]. To do so we introduce
$(\tilde{X}_{ij}),(\tilde{Y}_{ij})$ standard Gaussians, all independent,
issued on some probability space $(\Omega^{\prime},\mathcal{F}^{\prime},{\sf
P})$. We denote by ${\sf{E}}$ their (joint) expectation, and shorten

${\sf X}_{ij}\equiv\tilde{X}_{ij}\sqrt{g_{ij}^{+}},\qquad{\sf
Y}_{ij}\equiv\tilde{Y}_{ij}\sqrt{g_{ij}^{-}}\,.$ (12)

We stress that the disorder matrix $(g_{ij})$ is quenched: $g_{ij}^{+}$ and
$g_{ij}^{-}$ are constants. Furthermore, the $g^{+},g^{-}$ matrices are by
construction symmetric, but the ${\sf X},{\sf Y}$ are not. By Hubbard-
Stratonovich transformation, and interchanging the order of integration, we
get

$Z_{N}(\beta,h)\equiv{\sf{E}}E_{o}\left[\exp\left(\sqrt{\frac{\beta}{2}}\sum_{i,j}{\sf
X}_{ij}(\sigma_{i}+\sigma_{j})+\sqrt{\frac{\beta}{2}}\sum_{i,j}{\sf
Y}_{ij}(\sigma_{i}-\sigma_{j})+h\sum_{i}\sigma_{i}\right)\right]\,.$ (13)

We write the double sums in (13) as

$\displaystyle\sum_{i,j}{\sf X}_{ij}(\sigma_{i}+\sigma_{j})+\sum_{i,j}{\sf
Y}_{ij}(\sigma_{i}-\sigma_{j})=\sum_{i}\left({\sf X}_{i\bullet}+{\sf
X}_{\bullet i}+{\sf Y}_{i\bullet}-{\sf Y}_{\bullet i}\right)\sigma_{i}\,,$
(14)

with the shorthand notation $A_{i\bullet}=\sum_{j}a_{ij}$ and $A_{\bullet
j}\equiv\sum_{i}a_{ij}$, which we use for any $N\times N$ matrix $A=(a_{ij})$.
Using (LABEL:double_sum) in (13), and integrating out the Ising spins yields

$\displaystyle
Z_{N}(\beta,h)={\sf{E}}\exp\left(\sum_{i=1}^{N}\log\cosh\left(h+\sqrt{\frac{\beta}{2}}\left({\sf
X}_{i\bullet}+{\sf X}_{\bullet i}+{\sf Y}_{i\bullet}-{\sf Y}_{\bullet
i}\right)\right)\right)\,.$ (15)

This is, in essence, the claim of the Theorem, the rest is only a covariance
check. Clearly:

${\sf{E}}\left[{\sf X}_{i\bullet}+{\sf X}_{\bullet
i}\right]=0\,,\,\qquad{\sf{E}}\left[{\sf Y}_{i\bullet}-{\sf Y}_{\bullet
i}\right]=0\,.$ (16)

Furthermore ${\sf X}=({\sf X}_{ij})$ and ${\sf Y}=({\sf Y}_{ij})$ are
independent: together with (16) this yields

${\sf var}\left[\left({\sf X}_{i\bullet}+{\sf X}_{\bullet i}\right)+\left({\sf
Y}_{i\bullet}-{\sf Y}_{\bullet i}\right)\right]={\sf var}\left[{\sf
X}_{i\bullet}+{\sf X}_{\bullet i}\right]+{\sf var}\left[{\sf
Y}_{i\bullet}-{\sf Y}_{\bullet i}\right]\,,$ (17)

Again by (16), the variance equals the second moment and thus

$\displaystyle{\sf var}\left[{\sf X}_{i\bullet}+{\sf X}_{\bullet
i}\right]=\sum_{j,k}\left({\sf{E}}{\sf X}_{ij}{\sf X}_{ik}+{\sf{E}}{\sf
X}_{ij}{\sf X}_{ki}+{\sf{E}}{\sf X}_{ji}{\sf X}_{ik}+{\sf{E}}{\sf X}_{ji}{\sf
X}_{ki}\right)\,.$ (18)

The middle terms vanish (recall also that $g_{ii}=0$, hence
$\tilde{X}_{ii}=0$), whereas the first and last are $\neq 0$ only if $k=j$.
Using furthermore that the $g$-matrix is symmetric, we get

${\sf var}\left[{\sf X}_{i\bullet}+{\sf X}_{\bullet
i}\right]=2\sum_{j}{\sf{E}}\left[{\sf
X}_{ij}^{2}\right]\stackrel{{\scriptstyle\eqref{sho}}}{{=}}2\sum_{j}{\sf{E}}\left[\left(\sqrt{g_{ij}^{+}}\tilde{X}_{ij}\right)^{2}\right]=2\sum_{j}g_{ij}^{+}=2g_{i\bullet}^{+}\,.$
(19)

The computation of the second term on the r.h.s. of (17) is just as
straightforward: observing also that ${\sf{E}}{\sf
Y}_{ij}^{2}=g_{ij}^{-}={\sf{E}}{\sf Y}_{ji}^{2}$ (symmetry of $g^{-}$), one
steadily checks that

${\sf var}\left[{\sf Y}_{i\bullet}-{\sf Y}_{\bullet
i}\right]={\sf{E}}\left[\left({\sf Y}_{i\bullet}-{\sf Y}_{\bullet
i}\right)^{2}\right]=2\sum_{j}g_{ij}^{-}=2g_{i\bullet}^{-}\,.$ (20)

Plugging (19) and (20) in (17) therefore yields

${\sf var}\left[\left({\sf X}_{i\bullet}+{\sf X}_{\bullet i}\right)+\left({\sf
Y}_{i\bullet}-{\sf Y}_{\bullet
i}\right)\right]=2\left(g_{i\bullet}^{+}+g_{i\bullet}^{-}\right)\stackrel{{\scriptstyle
iii)}}{{=}}2\sum_{k}\left|g_{ik}\right|\,.$ (21)

We next compute, for $i\neq j$, the covariance:

$\displaystyle{\sf C}_{ij}$ $\displaystyle\equiv{\sf{E}}\left\\{\left({\sf
X}_{i\bullet}+{\sf X}_{\bullet i}\right)+\left({\sf Y}_{i\bullet}-{\sf
Y}_{\bullet i}\right)\right\\}\left\\{\left({\sf X}_{j\bullet}+{\sf
X}_{\bullet j}\right)+\left({\sf Y}_{j\bullet}-{\sf Y}_{\bullet
j}\right)\right\\}$ (22) $\displaystyle={{\sf{E}}\left[\left({\sf
X}_{i\bullet}+{\sf X}_{\bullet i}\right)\left({\sf X}_{j\bullet}+{\sf
X}_{\bullet j}\right)\right]}+{{\sf{E}}\left[\left({\sf Y}_{i\bullet}-{\sf
Y}_{\bullet i}\right)\left({\sf Y}_{j\bullet}-{\sf Y}_{\bullet
j}\right)\right]}\,,$

again since $\sf X$ and $\sf Y$ are independent (and centered). The $\sf
X$-contribution reads

$\displaystyle{\sf{E}}\left[\left({\sf X}_{i\bullet}+{\sf X}_{\bullet
i}\right)\left({\sf X}_{j\bullet}+{\sf X}_{\bullet
j}\right)\right]={\sf{E}}\left[{\sf X}_{i\bullet}{\sf
X}_{j\bullet}\right]+{\sf{E}}\left[{\sf X}_{i\bullet}{\sf X}_{\bullet
j}\right]+{\sf{E}}\left[{\sf X}_{\bullet i}{\sf
X}_{j\bullet}\right]+{\sf{E}}\left[{\sf X}_{\bullet i}{\sf X}_{\bullet
j}\right].$ (23)

First and last term on the r.h.s. of (LABEL:first_x) vanish since for $i\neq
j$ the involved random variables are independent (and centered), whereas
second and third terms read, respectively,

$\displaystyle{\sf{E}}\left[{\sf X}_{i\bullet}{\sf X}_{\bullet
j}\right]=\sum_{k,l}{\sf{E}}\left[{\sf X}_{ik}{\sf
X}_{lj}\right]=\sum_{k,l}1_{k=j,l=i}{\sf{E}}\left[{\sf X}_{ik}{\sf
X}_{lj}\right]={\sf{E}}\left[{\sf X}_{ij}^{2}\right]=g_{ij}^{+},$ (24)
$\displaystyle{\sf{E}}\left[{\sf X}_{\bullet i}{\sf
X}_{j\bullet}\right]=\sum_{k,l}{\sf{E}}\left[{\sf X}_{ki}{\sf
X}_{jl}\right]=\sum_{k,l}1_{k=j,l=i}{\sf{E}}\left[{\sf X}_{ki}{\sf
X}_{jl}\right]=g_{ji}^{+}=g_{ij}^{+},$

again by independence (and symmetry in the last step). Using all this in
(LABEL:first_x) yields

${\sf{E}}\left[\left({\sf X}_{i\bullet}+{\sf X}_{\bullet i}\right)\left({\sf
X}_{j\bullet}+{\sf X}_{\bullet j}\right)\right]=2g_{ij}^{+}\,.$ (25)

The computation of the ${\sf Y}$-contribution in (22) is fully analogous: the
upshot reads

${\sf{E}}\left[\left({\sf Y}_{i\bullet}-{\sf Y}_{\bullet i}\right)\left({\sf
Y}_{j\bullet}-{\sf Y}_{\bullet j}\right)\right]=-2g_{ij}^{-}\,.$ (26)

Combining (25) and (26) yields, for $i\neq j$,

$\displaystyle{\sf C}_{ij}=2g_{ij}^{+}-2g_{ij}^{-}\stackrel{{\scriptstyle
ii)}}{{=}}2g_{ij}.$ (27)

Using (21) and (LABEL:ultimate) in (15) settles the claim.

∎

######  Remark 1.

The treatment works also for generic $p$-spin interactions. Consider e.g. the
disordered mean field 3-spin (Ising) Hamiltonian

$H_{N,3}(\sigma)\equiv\sum_{1\leq i<j<k\leq
N}g_{ijk}\sigma_{i}\sigma_{j}\sigma_{k}.$ (28)

with $(g_{ijk})$ independent centered Gaussians. The idea is to first use the
splitting _ii)_ and then to apply the following Babylonian trick :

$\sigma_{i}\sigma_{j}\sigma_{k}=\frac{1}{2}\left(\sigma_{i}+\sigma_{j}\sigma_{k}\right)^{2}-1,\qquad-\sigma_{i}\sigma_{j}\sigma_{k}=\frac{1}{2}\left(\sigma_{i}-\sigma_{j}\sigma_{k}\right)^{2}-1\,.$
(29)

followed by Hubbard-Stratonovich transformation. This leads to a non-
interacting Hamiltonian (”$\sigma_{i}$”), and a two body interaction
(”$\sigma_{j}\sigma_{k}$”), but one may simply repeat the procedure on the
latter, iterating until full decoupling is reached.

######  Remark 2.

It is possible to derive exact, spin-independent expressions also for
nonlinear models such as the perceptron [9]: one can Taylor expand the
activation function, and apply the procedure to the Taylor terms, one by one.

## References

  * [1] Domb, C., A. J. Barrett, and M. Lax. Self-avoiding walks and real polymer chains. Journal of Physics A: Mathematical, Nuclear and General 6.7 (1973): L82.
  * [2] Edwards, Samuel Frederick, and Phil W. Anderson. Theory of spin glasses. Journal of Physics F: Metal Physics 5.5 (1975): 965.
  * [3] D. S. Fisher and D. A. Huse. Absence of many states in realistic spin glasses. J. Phys. A, 20(15):L1005–10, 1987.
  * [4] Heisenberg, Werner. Zur theorie des ferromagnetismus. Original Scientific Papers Wissenschaftliche Originalarbeiten. Springer, Berlin, Heidelberg, 1985. 580-597.
  * [5] Hopfield, J. J. Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences. 79 (8): 2554–2558. (1982)
  * [6] Hubbard, John. Calculation of partition functions. Physical Review Letters 3.2 (1959): 77.
  * [7] Mézard, Marc, Giorgio Parisi, and Miguel A. Virasoro. Spin glass theory and beyond. World Scientific, Singapore (1987).
  * [8] Moon, J. W., and L. Moser. On cliques in graphs. Israel Journal of Mathematics 3 : 23-28 (1965)
  * [9] Rosenblatt, Frank (1957). The Perceptron—a perceiving and recognizing automaton. Report 85-460-1. Cornell Aeronautical Laboratory.
  * [10] Sherrington, David, and Scott Kirkpatrick. Solvable model of a spin-glass. Physical review letters 35.26: 1792 (1975).


###### Abstract

The objective of this research paper is to study the relationship between a
block of a finite group and a defect group of such block. We define a new
notion which is called a strongly $k(D)$-block and give a necessary and
sufficient condition of a block with a cyclic defect group to be a
$k(D)$-block in term of its inertial index.

ON k(D)-BLOCKS

Ahmad M. Alghamdi

Department of Mathematical Sciences

Faculty of applied Sciences

Umm Alqura University

P.O. Box 14035, Makkah 21955, Saudi Arabia.

e-mail<EMAIL_ADDRESS>
Mathematics Subject Classification 2010: 20C20, 20C15.

Keywords: Brauer’s k(B) Problem, Blocks with cyclic defect groups.

##  1 Introduction

Let $p$ be a prime number and $B$ a $p$-block of a finite group $G$ with a
defect group $D$ of order $p^{d}$. In [1], we proved that $k(B)\equiv k(D)\
Mod(p)$, where $k(B)$ is the number of the ordinary irreducible characters
belonging to $B$ and $k(D)$ is the number of the ordinary irreducible
characters of the defect group $D$, which is an extra special $p$-group of
order $p^{3}$ and exponent $p$, for an odd prime number $p$. In other words,
we relate the number of the ordinary irreducible characters of a finite group
which belong to a certain block and the number of the conjugacy classes of the
defect group of that block under consideration.

This result led us to think about numerical relationships between a $p$-block
and its defect group. In the present work, we are free from any condition
about the prime number $p$. A question about the existence of a function on
the natural numbers which relates some block-invariants under consideration is
known as Brauer’s $21$-problem (see [2, 13]). In fact, Brauer asked whether it
is the case that $k(B)\leq|D|$ in general. This question is known as Brauer
$k(B)$-conjecture.

For a bound of $k(B)$, it is well known that $k(B)\leq p^{2d-2}$ for $d\geq 3$
and $k(B)\leq p^{d}$ for $d\in\\{0,1,2\\}$. See [2], [13, 14], [6] and [7, 8]
for more details and discussions in this direction.

However, we have arisen a question about which blocks and which conditions
ensure the equality $k(B)=k(D)$ as well as ensure the congruency $k(B)\equiv
k(D)\ Mod(p)$. We have studied some general cases as well as some examples for
small $G$. Then we try to characterize such blocks which have cyclic defect
groups in the term of the order of the inertial subgroups.

However, as far as we know, we have not seen a similar relation in the
literature. In fact, most of the examples which we have already considered
satisfy the equality $k(B)=k(D)$. However, for $G=A_{5}$ with $p=5$, we find
that $k(B_{0}(G))=4$, where $B_{0}(G)$ is the principal $5$-block of $G$. But
$k(D)=5$, where $D\in Syl_{5}(G)$. Since $A_{5}$ is a simple group and some
unusual properties arise from such group, we think that there are some classes
in group theory in such a way, $p$-blocks satisfy either the equality or the
congruency relation. We call those groups which do not satisfy our congruency
relation exotic groups.

######  Definition 1.1

Let $p$ be a prime number, $G$ a finite group and $B$ a $p$-block of $G$ with
defect group $D$. Write $k(B)$ to mean the number of ordinary irreducible
characters of $B$ and write $k(D)$ to mean the number of ordinary irreducible
characters of $D$. We call $B$ a strongly $k(D)$-block if $k(B)=k(D)$.

We can consider an equality mod the prime number $p$ as in the following
definition.

######  Definition 1.2

Let $p$ be a prime number, $G$ a finite group and $B$ a $p$-block of $G$ with
defect group $D$. Write $k(B)$ to mean the number of ordinary irreducible
characters of $B$ and write $k(D)$ to mean the number of ordinary irreducible
characters of $D$. We call $B$ a $k(D)$-block if $k(B)\equiv k(D)Mod(p)$.

It is clear that a strongly $k(D)$-block is a $k(D)$-block. However, we shall
see in Example 1.3 some $k(D)$-blocks which are not strongly $k(D)$-blocks.

Our main concern is to study finite groups and their blocks which satisfy
Definitions 1.1 and 1.2. Note that it is well known that $k(D)$ is the number
of the conjugacy classes of $D$. It is well known that blocks with cyclic
defect groups are well understood. This theory is rich and has many
applications. So, we shall start by doing some sort of characterization of
strongly $k(D)$-blocks with cyclic defect groups. Our main tool is Dade
theorem for the number of irreducible characters of a block with a cyclic
defect group; ( see [3] and [5, Page 420]).

In the end of the paper, we use the computations and the results in [9, 10] to
see that such phenomena do occur quite often in block theory.

###  1.1 Examples of strongly $k(D)$-blocks, $k(D)$-blocks and non
$k(D)$-blocks:

We shall start with some examples which illustrate the phenomenon of
$k(D)$-blocks.

######  Example 1.3

For $G=S_{n}$; the symmetric group of $n$ letters and
$n\in\\{2,3,4,5,6,7,8\\}$, it happens that $k(B_{0}(G))=k(D)$, for $D\in
Syl_{p}(G)$ and $p\in\\{2,3,5,7\\}$. However, for any prime number $p\geq 5$,
the defect group of the principal $p$-block of the symmetric group $S_{2p}$ is
an abelain $p$-group of order $p^{2}$ and $k(B_{0})=\frac{p^{2}+3p}{2}$. In
this situation, we obtain $k(D)$-blocks which are not strongly $k(D)$-blocks.
A similar conclusion holds for the principal $p$-block of the symmetric group
$S_{3p}$, with $p\geq 7$. However, when $p=3$ and $G=S_{9}$, then
$22=k(B_{0}(S_{9}))\neq k(D)=17$, where $D\in Syl_{3}(S_{9})$.

######  Example 1.4

Let $G$ be the dihedral group of order $8$. It has a unique $2$-block with
five ordinary irreducible characters which obviously coincide with the number
of the conjugacy classes of $G$.

######  Example 1.5

Let $G$ be the alternating group $A_{4}$. Then for $p=2$, $G$ has a unique
$2$-block with four ordinary irreducible characters which is the same as the
number of the conjugacy classes of the defect group. For $p=3$, we see that
$G$ has one $3$-block of defect zero and the principal $3$-block with three
ordinary irreducible characters with the same number as the number of the
conjugacy classes of a Sylow $3$-subgroup of $G$.

######  Example 1.6

For $G=SL(2,3)$; the special linear group, we have for $p=2$, the principal
$2$-block has the quaternion group $Q_{8}$ as a defect group and indeed,
$k(B_{0}(SL(2,3)))=7$ and $5=k(Q_{8})$. Then $k(B_{0}(SL(2,3)))$ is a
$k(D)$-block.

######  Example 1.7

Now, we have faced the first example which does not obey our speculation. It
is the first non abelian simple group; $A_{5}$. Although, for $p\in\\{2,3\\}$,
$k(B_{0}(A_{5}))=k(D)$, where $D\in Syl_{p}(A_{5})$, we observed that for
$p=5$, $k(B_{0}(A_{5}))=4\neq k(D)=5$, where, $D\in Syl_{5}(A_{5})$. Hence,
$A_{5}$ is the first exotic group in this sense. The same obstacle we have
faced for the group $G=GL(3,2)$, since $k(B_{0}(G))\neq k(D),$ when $D\in
Syl_{7}(G)$

######  Example 1.8

The principal $3$-block for $G=PSL(3,3)$; the projective special linear group,
satisfies $11=k(B_{0}(G))=k(D)=11,$ where $D$ is an extra special $3$-group of
order $27$ and exponent $3$.

###  1.2 General cases for the notion of $k(D)$-blocks

  1. 1.

Let $p$ be a prime number and $G$ a finite group. Assume that the prime number
$p$ does not divide the order of the group $G$. Then each block of $G$ has
defect zero, (see Theorem 6.29 Page 247 in [11] ). Hence such block is a
strongly $k(D)$-block.

  2. 2.

It is well known, see Problem 13 in Chapter 5 page 389 in [11] that if
$G=DO_{p^{{}^{\prime}}}(G)$ is a $p$-nilpotent group with a Sylow $p$-subgroup
$D$ and the maximal normal $p^{{}^{\prime}}$-subgroup of $G$ then $k(B)=k(D)$.
Certainly, a nilpotent $p$\- block is a strongly $k(D)$-block.

  3. 3.

We know that if $G$ is a $p$-group then it has a unique $p$-block, namely, the
principal block and such block is an strongly $k(D)$-block.

  4. 4.

For $p$-blocks with dihedral defect groups, we have $k(B)=k(D)$. Hence such
blocks are examples of strongly $k(D)$-blocks.

##  2  $k(D)$-blocks with cyclic defect groups

In this section, we discuss $p$-blocks with cyclic defect groups. Recall that
a root of a $p$-block $B$ of a finite group $G$ with defect group $D$ is a
$p$-block $b$ of the subgroup $DC_{G}(D)$ such that $b^{G}=B$ (see Chapter 5
Page 348 in [11]), where $C_{G}(D)$ is the centralizer subgroup of $D$ in $G$.
Now for the root $b$, we define the inertial index of $B$ to be the natural
number $e:=e(B)=[I_{G}(b):DC_{G}(D)]$, where $I_{G}(b)=\\{g\in G:b^{g}=b\\}$.
It is clear that $I_{G}(b)$ is a subgroup of $G$ which contains $DC_{G}(D)$
and the index $[I_{G}(b):DC_{G}(D)]$ is well-defined. The number $e$ above is
crucial to investigate some fundamental results in block theory.

Let us restate the following well known result which was established by Dade
regarding the number of irreducible characters in a block with a cyclic defect
group. For more detail, the reader can see the proof and other constructions
in [3, 12].

######  Lemma 2.1

Let $B$ be a $p$-block of a finite group $G$ with a cyclic defect group $D$ of
order $p^{d}$. Then $B$ has $e+\frac{p^{d}-1}{e}$ ordinary irreducible
characters, where $|D|=p^{d}$.

With the above notation, we characterize strongly $k(D)$-blocks in the term of
the inertial index for blocks with cyclic defect groups. Also, we believe that
it is worth looking for some positive theorems regarding the notion of
$k(D)$-blocks.

######  Theorem 2.2

Let $B$ be a $p$-block of a finite group $G$ with a cyclic defect group $D$ of
order $p^{d}$. Then $B$ is a strongly $k(D)$-block if and only if $e=1$ or
($e=p-1$ and $d=1$).

Proof: Assuming that $B$ is a strongly $k(D)$-block and using Lemma 2.1, we
can write $p^{d}=e+\frac{p^{d}-1}{e}.$ Then we have
$e^{2}-p^{d}e+(p^{d}-1)=0.$ Letting $e$ be the variable, we see that the only
solution we have is that $e=1$ or $e=p^{d}-1$. The result follows as $e$
divides $p-1$. The converse is clear and the main result follows.

######  Remark 2.3

We get an analogue result of Theorem 2.2 for $k(D)$-blocks with cyclic defect
groups, by solving the congruency equation
$p^{d}\equiv_{p}e+\frac{p^{d}-1}{e}.$

##  3 The interplay with fundamental results

There are fundamental progress in solving Brauer problems. We recast the
following result which is due to R. Kessar and G. Malle [10, HZC1]. This
result can be used to see an strongly block $B$ with abelian defect group $D$
of order $p^{d}$ as such that $k_{0}(B)=k(D)$, where $k_{0}(B)$ is the number
of ordinary irreducible characters of height zero belonging to $B$.

######  Lemma 3.1

Let $G$ be a finite group, $B$ be a $p$-block of $G$ with defect group $D$. If
$D$ is abelian then every ordinary irreducible character of $B$ has height
zero.

Let us conclude this paper by mentioning the following lemma in such a way
that we rely on the computation in [9, Proposition 2.1] by B. Kulshammer and
D. Sambale. These computations guarantee that the phenomena of strongly
$k(D)$-block occur quite often in the theory of blocks.

######  Lemma 3.2

Let $G$ be a finite group, $B$ be a $p$-block of $G$ with a defect group $D$.
If $D$ is an elementary abelian of order 16 then $B$ is a $k(D)$-block.

In fact, Lemma 3.2 can be replaced by the following much stronger result.

######  Theorem 3.3

Let $G$ be a finite group and $p=2$ or $3$. Then each $p$-block of $G$ with
abelian defect group is a $k(D)$-block.

## References

  * [1] A. Alghamdi, The ordinary weight conjecture and Dade’s projective conjecture for $p$-blocks with an extra special defect group, Ph.D. Thesis, University of Birmingham, 2004.
  * [2] R. Brauer and W. Feit, _On the number of irreducible characters of finite groups in a given block,_ Proc. Nat. Acad. Sci. U. S. A. 45 (1959), 361-365. 127–131.
  * [3] E. Dade, _Counting characters in blocks with cyclic defects groups,_ J. Algebra 186 (3), 934-969, (1996).
  * [4] E. Dade, _Blocks With Cyclic Defect Groups,_ Annals of Mathematics, Second Series, Vol. 84, No. 1 (Jul., 1966), pp. 20-48.
  * [5] L. Dornhoff, _Group Representation Theory,_ Part B: Modular representation theory. Marcel Dekker Inc., New York, (1972).
  * [6] R. Knorr, _On the number of characters in a $p$-block of a $p$-solvable group,_ Illinois J. Math. 28, 181-210 (1984).
  * [7] B. Kulshammer and G. Robinson, _Alperin-Makey Implies Brauer’s Problem 21,_ J. Algebra 180, 208–210 (1996).
  * [8] B. Kulshammer, _Modular representations of finite groups: conjectures and examples,_ Jena, (1996).
  * [9] B. Kulshammer and B. Sambale, _The 2-blocks of defect 4,_ Journal: Represent. Theory 17 (2013), 226-236, Published electronically: May 2, 2013.
  * [10] R. Kessar and G. Malle, _Quasi-isoloated blocks and height zero Conjecture,_ J. Algebra 147, 450–455 (1992).
  * [11] H. Nagao and Y. Tsushima, _Representation of finite groups,_ Academic Press Inc., Boston, MA, Translated from Japanese, (1989).
  * [12] G. Navarro, _Characters and blocks of finite groups,_ Volume 250 of London Mathematical Society Lecture Notes Series. Cambridge University Press, Cambridge 1998.
  * [13] G. R. Robinson, _Some open conjectures on representation theory,_ J. Ohio State Univ. Math. Res. Inst. Publ.,de Gruyter, Berlin,1997, Volume 6, Representation theory of finite groups (Columbus, OH, 1995), 127–131.
  * [14] G. R. Robinson, _On Brauer’s $k(B)$ Problem,_ J. Algebra 147, 450-455 (1992). 127–131.


# Brackets by any other name

<EMAIL_ADDRESS>
###### Abstract.

Brackets by another name - Whitehead or Samelson products - have a history
parallel to that in Kosmann-Schwarzbach’s “From Schouten to Mackenzie: notes
on brackets”. Here I _sketch_ the development of these and some of the other
brackets and products and braces within homotopy theory and homological
algebra and with applications to mathematical physics. In contrast to the
brackets of Schouten, Nijenhuis and of Gerstenhaber, which involve a relation
to another graded product, in homotopy theory many of the brackets are free
standing binary operations. My path takes me through many twists and turns;
unless particularized, _bracket_ will be the generic term including product
and brace. The path leads beyond binary to multi-linear $n$-ary operations,
either for a single $n$ or for whole coherent congeries of such assembled into
what is known now as an $\infty$-algebra, such as in homotopy Gerstenhaber
algebras. It also leads to more subtle invariants. Along the way, attention
will be called to interaction with ‘physics’; indeed, it has been a two-way
street.

###### Key words and phrases:

brackets, products, braces, deformation theory, Lie, Leibniz and Filippov
algebras, higher homotopy algebras and operations

###### 1991 Mathematics Subject Classification:

Primary: 01A65, 55Q15, 55S30, 16E40, 18N40; Secondary: 53D55.

Jim Stasheff

109 Holly Dr

Lansdale PA 19446, USA

_In Memory of Kirill Mackenzie (1951-2020)_

###### Contents

  1. 1 Introduction
  2. 2 Brackets known as products
  3. 3 The Gerstenhaber bracket revisited
  4. 4 $\mathbb{n}$-variations on a theme
  5. 5 Brace algebras
  6. 6 Secondary and higher products and brackets
  7. 7 Leibniz “brackets”
  8. 8 Evolution of notation
  9. 9 For Kirill
  10. 10 Acknowledgements
  11. 11 Appendix

##  1\. Introduction

Here is a complement to [35] in this volume, emphasizing a parallel
development of ‘brackets’ (and products and braces, etc.) from a
homotopy/homological point of view (cf. also [jh:kirill]). I will include
_higher structures_ 111 There is now the _Journal of Higher Structures_. in
two versions: $n$-ary operations for $n>2$ and in the sense of classic
secondary operations, defined only when a primary operation vanishes222 When
iterated operations are used, that will be made explicit.. Brackets by another
name - Whitehead or Samelson products - have a history parallel to that in
Kosmann-Schwarzbach’s “From Schouten to Mackenzie: notes on brackets”. Here I
_sketch_ their development as well as that of _some_ of the other brackets and
products and braces within homotopy theory and homological algebra and
applications to mathematical physics. In contrast to the brackets of Schouten,
Nijenhuis and of Gerstenhaber, which involve a relation to another graded
product, in homotopy theory many of the brackets are free standing binary
operations. My path takes me through many twists and turns; unless
particularized, _bracket_ will be the generic terms including products and
braces. The path leads beyond binary to multi-linear $n$-ary operations,
either for a single $n$ or for whole coherent congeries of such assembled into
what is known now as an $\infty$-algebra, such as in homotopy Gerstenhaber
algebras, as well as more subtle invariants; for example, secondary operations
which are defined only when a primary operation vanishes. It reveals missed
(or seriously delayed) opportunities for collaboration within mathematics and
between mathematics and physics; some leads have ‘withered on the vine’. As
Dyson remarked about missed opportunities [18]:

> occasions on which mathematicians and physicists lost chances of making
> discoveries by neglecting to talk to each other.

The situation has improved since Dyson’s lecture, but not completely; a common
vocabulary does not necessarily imply a common viewpoint.

##  2\. Brackets known as products

In 1941, J.H.C. Whitehead constructed his eponymous _Whitehead product_ 333
The name appears to have been used for the first time in 1953 by Hilton and
Whitehead [31]. [76]. It might better be called the _Whitehead bracket_ ;
indeed, it is denoted by $[\ ,\ ]$ and later was shown to satisfy a shifted
graded version of the Jacobi identity. For a space $X$ and base point $x$, the
homotopy classes of maps $f:S^{p}\to X$ (respecting base points) form a group
$\pi_{p}(X)$. For classes $\alpha\in\pi_{p}(X),\ \beta\in\pi_{q}(X)$, the
Whitehead product $[\alpha,\beta]\in\pi_{p+q-1}$ is represented as follows:
Let $f:S^{p}\to X$ and $g:S^{q}\to X$ represent $\alpha,\beta$ respectively.
The product $S^{p}\times S^{q}$ has a cellular decomposition as $S^{p}\vee
S^{q}\smile e^{p+q}$, where the cell $e^{p+q}$ is attached by a map
$h:S^{p+q-1}\to S^{p}\vee S^{q}$. The composition $(f\vee g)\circ h$
represents the homotopy class $[\alpha,\beta]\in\pi_{p+q-1}$. However, it was
not until the mid-1950s that several authors444 S. C. Chang, Hilton, Massey-
Uehara, Nakaoka-Toda,, G.W. Whitehead and others not published.,
independently, showed that the Whitehead product satisfied a graded version of
the Jacobi identity. It was around the same time that Schouten introduced his
bracket of multi-vector fields on a manifold, followed by the contributions of
his student Nijenhuis. These two visions of _graded Lie algebra_ were
contemporaneous, but no-one noticed, at least no-one remarked, until decades
later. Here in this section is a sketch of the ‘other’ development of the
theory of graded Lie brackets in early homotopy theory, based primarily on
historical research in 1995 by Kristen Haring: _On the Events Leading to the
Formulation of the Gerstenhaber Algebra:1945-1966_ 555 Including contributions
from many of the main researchers involved:. Gerstenhaber, Hochschild, Lian,
Massey, Nijenhuis, Samelson, Serre, Spencer, Sternberg, G.W. Whitehead and
Zuckerman. [30]. Nijenhuis’ title _Jacobi-type identities for bilinear
differential concomitants of certain tensor fields_ [53] indicates the pangs
of birthing the new concept of _graded Lie bracket_ , as did the many attempts
to recognize it in homotopy theory. The graded Jacobi identity  There was a
long delay from Whitehead’s 1941 paper until 1954 when many independent proofs
of the Jacobi identity for Whitehead products were published or acknowledged
elsewhere666 See Haring [30] for a chronological account of the submission of
proofs and the recognition within those articles of the existence of still
others.. The difficulty (beyond the issue of grading dependent signs) was that
$(f\vee g)\circ h$ defines the homotopy class $[\alpha,\beta]\in\pi_{p+q-1}$,
but the Jacobi identity at the map level will hold only up to homotopy. As
Massey remarked to Haring:

> this question was “in the air” among homotopy theorists in the early 1950’s,
> I don’t believe you can point to any one person and say he or she raised
> this question.

One of the issues that may have delayed recognition/formulation was the fact
that all three brackets (Nijenhuis-Schouten, Gerstenhaber, Whitehead) were
graded Lie only after a shift. Samelson’s development [63] of what is now
called the _Samelson product_ produced the shift topologically by passing to
the based loop space $\Omega X$ of a pointed space $X$ where $\pi_{p}(\Omega
X)$ is isomorphic to $\pi_{p+1}(X)$. Samelson’s product

$\pi_{p}\Omega X\otimes\pi_{q}\Omega X\to\pi_{p+q}\Omega X$

was realized in terms of the commutator with respect to loop multiplication,
though his emphasis was on the corresponding Pontryagin product after applying
the Hurewicz morphism. As a graded commutator at the level of homotopy
classes, the Jacobi identity follows as for a graded associative algebra.

##  3\. The Gerstenhaber bracket revisited

Gerstenhaber’s bracket was originally defined on the Hochschild cochain
complex $C^{\bullet}(A,A)$ of an associative algebra $A$ with
$C^{p}(A,A):=Hom(A^{\otimes p},A)$ and coboundary $\delta:C^{p}(A,A)\to
C^{p+1}(A,A)$ given, for $f\in C^{p}(A,A)$,by

$(\delta f)(a_{1}\cdots a_{p+1})=a_{1}f(a_{2}\cdots a_{p+1})+\sum_{i=1}^{p}\pm
f(a_{1}\cdots a_{i}a_{i+1}\cdots a_{p+1})\pm f(a_{1}\cdots a_{p})a_{p+1}$ (1)

This cochain complex is itself an associative algebra with respect to the _cup
product_ $f\smile g:=f(\cdots)g(\cdots)$. Notice, as in algebraic topology,
$\smile$ is not commutative, even up to sign. However, Gerstenhaber introduced
his bracket $[\ ,\ ]$ with a description that related it to the cup product
by777 The resemblance to Steenrod’s relation between $\smile$ and $\smile_{1}$
was first observed (as far as I know) by Kadeishvili in 1988 [33].

$\delta[f,g]=[\delta f,g]\pm[f,\delta g]\pm f\smile g$ (2)

On the Hochschild cochain complex above, Gerstenhaber defined his $\circ$
operation: $f\circ g\in C^{p+q-1}$ for $f\in C^{p},g\in C^{q}$ by summing
operations $\circ_{i}$ inserting (with signs) $g$ into $f$ in the i-th place.
On the cohomology, what is now known as the Gerstenhaber bracket $[f,g]$ is
represented by the commutator $f\circ g\pm g\circ f.$ The combination of the
cup product $f\smile g$ with the Gerstenhaber bracket $[f,g]$ and the
Hochschild differential satisfies many of the relations of a differential
graded Poisson algebra, but not all. For example, the operation $[h,\ \ ]$ is
up to sign a right inner derivation of degree 1 of $f\smile g$ of the graded
algebra $(C^{*}(A,A),\smile),$ but generally not a left derivation. Instead,
the deviation from being a left derivation of degree 1 is given by a
coboundary.

######  Remark 1.

The nomenclature evolved from ‘graded or odd Poisson algebra’ to G-algebra
around 1987 and then to Gerstenhaber algebra. According to Google Scholar, the
first published appearance of “Gerstenhaber algebra” was in 1992 [42, 55].

This brings us back to one of the subtleties of Gerstenhaber’s construction:
what is now called a _Gerstenhaber algebra_ exists on the cohomology of the
complex of Hochschild cochains. Gerstenhaber originally apologized888 “at the
risk of seeming old fashioned” [23]. for _not_ presenting his results on
Hochschild cohomology in terms of a more modern $Ext$-functor (at the
cohomology level). I beg to differ. He derived his structures from operations
at the cochain level where some of the defining relations held only up to
homotopy. The cochain homotopy structure as he developed it bore fruit in ways
the cohomology level $Ext$-functor might not have suggested. The Fröhlicher-
Nijenhuis bracket of vector-valued forms is the graded commutator of
derivations of the algebra of differential forms [35]. Gerstenhaber’s is also
a commutator, in fact, from two different points of view: first, in
Gerstenhaber, as a commutator with respect to the $\circ$-product, later as
the commutator of coderivations of $\bigoplus A^{p}$ as a coalgebra with the
de-concatenation diagonal. Indeed, the Hochschild chain complex can be
described in terms of the _bar construction_ on $A$ in one of its
incarnations. This is how I came to see Gerstenhaber’s structures, which led
me to the interpretation of $C(A,A)\equiv Hom(TA,A)$ as $Coder(TA)$, the space
of coderivations of $TA$, the tensor _coalgebra_ on $A$. There I saw, many
years later, the Gerstenhaber bracket as (up to sign) the commutator bracket
of coderivations [68].

######  Remark 1.

The bracket notation $[a,b]$ was often used to indicate a commutator in an
associative algebra and only gradually became standard for an abstract Lie
algebra. Even the appellation “Lie algebra”, due to Weyl, did not occur until
the 1930’s ! (Compare the alternate notations discussed in [35].)

Homotopy Gerstenhaber algebras In a letter to several of us [15], P. Deligne
pointed out that since the structure of what is now called a Gerstenhaber
algebra can be described as the structure of an algebra over the _homology_ of
the little discs operad [50], for this relationship which exists at the
homology level there must be more between the little disks operad and the
Hochschild cochains. Later this became quoted as (some variant of) the
following conjecture. Deligne’s Conjecture now a Theorem: The structure of an
algebra over the homology of the little disks operad on the Hochschild
cohomology may be naturally lifted to the cochain level. This gave rise to a
‘cottage industry’ producing a variety of proofs, first by by Tamarkin [71]
though unpublished, then A. Voronov [74], then McClure-Smith [51] and several
others. The essential point is that homotopy relations in Gerstenhaber’s
original construction are but a small part of a whole congery of higher
homotopy relations, similar to those appearing in the algebraic topology of
based loop spaces about the same time, beginning in 1963 [66], the same year
as [23]. It was a very good year! Like the Whitehead product, these homotopy
relations did not involve another basic operation such as the cup product or
exterior product. Still more higher homotopies were needed for iterated loop
spaces, giving rise to May’s introduction of _operads_ [50]. Intuition similar
to Deligne’s is a recurring theme: classical structure on the homology of a
chain complex indicates the possible presence of a corresponding
$\infty$-structure on the original chain complex. Conversely, there is the
process of _homotopy transfer_ of a strict graded algebra on the chain level
(e.g. a differential graded associative algebra) which will often give rise to
an $\infty$-structure on the homology so as to be suitably equivalent to the
original on the chain complex. A pioneering example is due to Kadeshvili,
giving rise in the associative setting to an $A_{\infty}$-structure [67]:

######  Theorem 3.1.

[34] Given a differential graded algebra $(A,d)$, there is an
$A_{\infty}$-algebra structure on $H(A,d)$ which is equivalent as
$A_{\infty}$-algebra to $(A,d)$.

$L_{\infty}$-algebras, deformation theory and physics Since the Schouten-
Nijenhuis bracket developed out of differential geometry, it is not surprising
it had physical relevance; what has been called _cohomological physics_ can be
traced at least as far back as to Gauss [22]. More surprising, perhaps, is
that Gerstenhaber’s bracket from homological algebra appeared in physics in
the development of _deformation quantization_ [8, 9]. Here the Gerstenhaber
bracket gave the first obstruction to the existence of a _star product_. If
that obstruction vanishes, then there are higher order obstructions as in
general deformation theory, including the original example for deformations of
complex structure [17]. Kosmann-Schwarzbach comments that in 1964, Nijenhuis
and Richardson remarked on the striking similarity of Gerstenhaber’s treatment
of deformations and the existing treatments of deformations of complex
structure. Indeed, Gerstenhaber confirms that deformations of complex
structure inspired his own work. For the Whitehead product or for the
Gerstenhaber bracket, the Jacobi identity holds only up to homotopy. A choice
of homotopy can be taken as part of the structure on a chain complex $(L,d)$
with a bracket $[x,y]$ and a ternary operation $[x,y,z]$ such that999
According to folklore, “there exists a set of signs” which are spelled out in
many of the relevant references.

$\begin{split}&[x,[y,z]]\pm[[x,y],z]\pm[y,[x,z]]\quad=\quad
d[x,y,z]\pm[dx,y,z]\pm[x,dy,z]\pm[x,y,dz]\end{split}$ (3)

or for cycles x, y, z:

$\begin{split}&[x,[y,z]]\pm[[x,y],z]\pm[y,[x,z]]\quad=\quad
d[x,y,z]\end{split}$ (4)

Asking if $[x,y,z]$ satisfies an appropriate relation leads to the concept of
an $L_{\infty}$-algebra. These algebras are a generalization of differential
graded Lie algebras in which the Jacobi identity again holds only up to
homotopy, but now included is a choice of such a homotopy and a relation
involving four elements, to be satisfied up to homotopy and on and on.
$L_{\infty}$-algebras were first introduced in the context of deformations of
rational homotopy types, in preliminary versions of [64], but other later
basic references are listed in the n-lab entry for _L-infinity-algebra_.

######  Definition 3.1.

An _$L_{\infty}$ -algebra_101010 There are alternate descriptions with
different notation, e.g. $[\ ,\cdots,\ ]_{n}$ , with shifted grading and with
$d$ of degree $-1$. consists of a differential graded vector space $(L,d)$
with differential $d=\ell_{1}$ of degree $1$ and graded skew-symmetric $n$-ary
brackets $\ell_{n}:L^{\otimes n}\to L$ of degree $2-n$ satisfying a coherent
set of differential relations, called _generalized Jacobi relations._

An $L_{\infty}$-algebra may arise from an $A_{\infty}$-algebra by suitable
skew-symmetrization; in return, there is an $A_{\infty}$-algebra given by a
_universal enveloping_ functor [40]. In 1989, the $L_{\infty}$ structure of
CSFT (Closed String Field Theory) was first recognized as such when Zwiebach
fortuitously gave a talk in Chapel Hill at the last GUT (Grand Unification
Theory) Workshop [77, 78]. This was the proper ‘birth certificate’ for
$L_{\infty}$-structures in physics, which gradually became a part of the
‘standard tool kit’ in gauge field theory (cf. [42]). Closed String Field
Theory is a field theory in the physics sense, starting with an algebra of
functions on the space of loops (closed strings) on a manifold. A convolution
product on this algebra produces a binary bracket and a full panoply of higher
homotopies. A little later, Lian and Zuckerman [45] were working on _conformal
field theory_ (CFT) in the BRST formalism, which includes a generalization of
a Chevalley-Eilenberg complex. The theories studied by Lian and Zuckerman led
to a BV-algebra (Batalin-Vilkovisky) on cohomology, which extends a
Gerstenhaber algebra. Just as Gerstenhaber presented some first level
homotopies which ultimately gave rise to _homotopy Gerstenhaber algebras_ ,
Lian and Zuckerman presented some first level homotopies which ultimately gave
rise to _homotopy BV-algebras_. (See the next section for Nambu Hamiltonian
dynamics.)

##  4\. $\mathbb{n}$-variations on a theme

So far, the emphasis has been on either binary operations or systems of
compatible multilinear operations: $\infty$-algebras. There is another
tradition of $n$-ary algebras: vector spaces (without grading) with only _one_
multilinear $n$-ary operation $V^{\otimes n}\to V$, $n\geq 3$. According to an
excellent survey of many $n$-ary operations by de Azcarraga and Izquierdo
[13]:

> ternary operations appeared for the first time associated with the cubic
> matrices that had been introduced by A. Cayley in the middle of the XIXth
> century and that were also considered by J.J. Sylvester some forty years
> later.

$\textdbend$ WARNING: There is a confusion of nomenclature, especially for
variations on the definition of Lie algebra, including both $n$-Lie algebras
and Lie $n$-algebras, as well as $Lie(n)$ and $Lie_{n}$. These are _not_ all
the same, not at all (see the Appendix for clarification). n-ary Lie algebras
Let us begin with $n$-ary Lie algebra, which has two major variants,
corresponding to two versions of a (generalized Jacobi) _characteristic
identity_ 111111 In older literature, these two different generalized Jacobi
relations are distinguished by respective _fundamental identities_ , but, as
suggested in [13], a better name is _characteristic identities_.. In a “Lie”
context, appropriate graded symmetries are always assumed. One version is the
defining relation for an $L_{\infty}$-algebra restricted to a vector space
with just one $n$-ary bracket,

$\sum\pm l_{n}(l_{n}(v_{\sigma(1)}\otimes\dots\otimes v_{\sigma(n})\otimes
v_{\sigma(n+1)}\otimes\dots\otimes v_{\sigma(2n-1)})=0;$ (5)

no grading is involved. Such algebras have been studied quite independently of
$\infty$-structures and of each other by Hanlon and Wachs [28] (combinatorial
algebraists) and by de Azcarraga and Bueno [14] (physicists). n-ary Filippov
algebra A major alternative _generalized Jacobi relation_ was introduced by
Filippov.

######  Definition 4.1.

An _$n$ -ary Filippov algebra_ (or $n$-Filippov algebra) [19] consists of a
vector space $V$ with an $n$-ary bracket $[\ \ ,\cdots,\ \ ]:V^{\otimes n}\to
V$ such that $[X_{1},X_{2},\dots,X_{n-1},\quad]$ acts as a left
derivation121212 That identity was known also to Sahoo and Valsakumar [62].:

$[X_{1},X_{2},\dots,X_{n-1},[Y_{1},Y_{2},\dots,Y_{n}]]=$
$[[X_{1},X_{2},\dots,X_{n-1},Y_{1}],Y_{2},\dots,Y_{n}]+\dots+[Y_{1},\dots,Y_{n-1},[X_{1},\dots,X_{n-1},Y_{n}]].$

Unfortunately, algebras with either characteristic identity are often called
$n$-Lie algebras; I urge use of _$n$ -ary Filippov_ to lessen the confusion of
terms (see the Appendix for clarification of these and other terms). As I
recall, I first learned of the Filippov identity from Alexander Vinogradov
when we met at the Conference on Secondary Calculus and Cohomological Physics,
Moscow, August 1997. (This identity arose independently by yet another name in
[20]) See A. and M. Vinogradov’s [73] for a comparison of these two distinct
generalizations of the ordinary Jacobi identity to $n$-ary brackets. Nambu
$n$-Hamiltonian mechanics Nambu mechanics [52, 70] is a generalization of
Hamiltonian mechanics proposed by Yoichiro Nambu in 1973131313 Gaetano Vilasi
has alerted me to a possible predecessor in 1887, Albeggiani’s $n$-Poisson
bracket [4].. In his formulation, a triple (or, more generally, n-tuple) of
“canonical variables” replaces a canonically conjugated pair in the
Hamiltonian formalism and a ternary (or, more generally, n-ary) operation the
_Nambu bracket_ , which generalizes the Poisson bracket to $n$-variables,
$n\geq 3$. Nambu dynamics is described by the flow given by Nambu-Hamilton
equations of motion—a system of ODE’s which involves $n-1$ “Hamiltonians”.
Nambu’s original work [52] was a generalization of the binary Poisson bracket
of Hamiltonian mechanics to a ternary bracket (6). In [7], Bayen and Flato
formalized the notion of a _Nambu algebra_ , including the $n$-ary version.
The defining relation was the characteristic identity for the Filippov bracket
above, though apparently not remarked at the time. Similarly, in [70] there is
no mention of Filippov; apparently the Siberian journal was not well read in
Moscow or Leningrad. Takhtajan emphasizes the role ternary and higher order
algebraic operations and mathematical structures related to them play in
passing from Hamilton’s to Nambu’s dynamical picture. He writes:

> We start by formulating the fundamental identity (FI) for the Nambu bracket
> as a consistency condition for Nambu’s dynamics. It yields the analog of the
> Poisson theorem that the Poisson bracket of integrals of motion is again an
> integral of motion. A ‘canonical’ Nambu bracket is defined for a triple of
> classical observables on the three-dimensional phase space $\mathbb{R}^{3}$
> with coordinates $x,y,z$ by the following beautiful formula
>
>
> $\\{f_{1},f_{2},f_{3}\\}=\frac{\partial(f_{1},f_{2},f_{3})}{\partial(x,y,z)},$
> (6)
>
> where the right-hand side stands for the Jacobian of the mapping
> $f=(f_{1},f_{2},f_{3}):\mathbb{R}^{3}\mapsto\mathbb{R}^{3}$. This formula
> naturally generalizes the usual Poisson bracket from binary to ternary
> operation on classical observables141414 M. Flato informed Takhtajan that,
> apparently, Nambu introduced this bracket in order to develop a “toy model”
> for quarks considered as triples and that, together with Fronsdal, Flato
> independently introduced such a relation..

The Vinogradovs [73] provide a thorough survey, both historically and cross-
culturally (analysis, combinatorics, homological algebra, deformation theory)
including mathematical physics. My understanding (though I’ve been unable to
verify it) is that Filippov had noticed that, for $n=3$, Nambu’s bracket was a
specific example [52, 70, 69].

As can be seen, there has been a rather spotty interaction among the various
points of view in both physics and mathematics, delaying opportunities for
interaction. All these algebras are important in geometry and in physics where
the corresponding structures are on vector bundles over a smooth manifold
(see, for example, [75] and references therein). For these, there are a
variety of terms, e.g. $n$-(ary) Lie algebroids. Mackenzie played a leading
role in clarifying the basic mathematical structure here [48].

##  5\. Brace algebras

_Products and brackets and braces -oh my! 151515 Lions and tigers, and bears,
oh my! - Dorothy in Wizard of Oz (1939)_ Recall that Gerstenhaber’s bracket on
the Hochschild cochain complex arose from summing operations $\circ_{i}$
inserting (with signs) $g$ into $f$ in the i-th place. To handle more and more
significantly coherent muti-linear operations, there are more elaborate
structures, _brace algebras_ and _symmetric brace algebras_ 161616 The name
_brace_ refers to the symbol $\\{\ ,\ \\}$, not to be confused with it’s use
for Poisson brackets. Parentheses, brackets, and braces are sometimes referred
to as ”round,” ”square,” and ”curly”. in which the insertions occur
simultaneously in multiple slots, not necessarily consecutive. Such braces
were originally introduced in 1988 (without the name and with different
notation) by Kadeishvili [33]; he recognized later that his $\smile_{1}$ on
the Hochschild complex is exactly Gerstenhaber’s circle product. For
uniqueness and functorality of the minimal $A_{\infty}$-model, he needed the
notion of $A_{\infty}$-_morphisms_ of $A_{\infty}$\- algebras, for which he
constructed the higher braces. Later in 1993, Getzler [25] used essentially
the above description, again without the name ‘brace’. Not until 1995 in work
of Gerstenhaber and A. Voronov [24] was it shown that the braces on the
Hochschild cochain complex satisfy certain identities; they called the
resulting algebric structure a _brace algebra_ with a homotopy Gerstenhaber
algebra structure as an applicaion.

######  Definition 5.1.

A (non-symmetric) _brace algebra_ is a graded vector space $\mathcal{B}$
together with a collection of degree $0$ multilinear braces
$x,x_{1},\ldots,x_{n}\longmapsto x\\{x_{1},\ldots,x_{n}\\}$ that satisfy the
identities

$\displaystyle x\\{x_{1},\cdots,x_{m}\\}\\{y_{1},\cdots,y_{n}\\}=$
$\displaystyle\sum\epsilon\
x\\{y_{1},\cdots,y_{i_{1}},x_{1}\\{y_{i_{1}+1},\cdots,y_{j_{1}}\\},y_{j_{1}+1},\cdots,y_{i_{m}},$
$\displaystyle\quad\quad\quad
x_{m}\\{y_{i_{m}+1},\cdots,y_{j_{m}}\\},y_{j_{m}+1},\cdots,y_{n}\\},$

where the sum is taken over all sequences $0\leq i_{1}\leq j_{1}\leq\cdots\leq
i_{m}\leq j_{m},\leq n$ where $x\\{\thickspace\\}=x$ and $\epsilon$ is an
appropriate sign.

There is a corresponding symmetric version of the brace algebra where the sum
is taken over unshuffle sequences with appropriate signs [41]. They provide an
important machinery for handling panoplies of higher operations, for example,
in the definition of homotopy Gerstenhaber algebras and homotopy BV-algebras
and applications to physics. A still more elaborate notion of _multi-braces_
is due to and applied by Akman [3, 2], again in physics.

##  6\. Secondary and higher products and brackets

There is an ancient and honorable world view of _secondary_ or _conditional_
invariants, ranging from classical algebra to algebraic topology and even
mathematical physics. The essential idea:  _When a known invariant vanishes,
it is often possible to define a secondary invariant, but only on objects
where the primary invariant vanishes._ More precisely [58], given a set of
transformations $T$, a _conditional invariant modulo R_ means invariance for
an object under only those transformation $T$ for which R
holds171717Physicists might say “on shell”. In cohomology, there are _Massey
products_ , defined on cohomology classes $u,v,w$ if the products $uv=0=vw$
[49]. Then suitable vanishing for Massey products of four classes leads to a
ternary invariant and so on. This works for general dg associative algebras
and for $A_{\infty}$-algebras with a little more effort.

For a dg Lie algebra, the analogous operations are named _Massey-Lie brackets_
, developed by Retakh [60]. After transferring the structure of a dg Lie
algebra $(L,d)$ to an $L_{\infty}$-algebra with respect to the differential
(the 1-bracket), there are again multi-linear operations on the cohomology
$H(L)$ which are all primary in the sense of $L_{\infty}$-algebras. Toda
(secondary) brackets The _Toda bracket_ is an operation on homotopy classes of
maps, in particular on homotopy groups of spheres, defined by Hiroshi Toda
[72]. Here the algebra in question concerns the composition of maps up to
homotopy, in particular maps from spheres to spheres and hence on the algebra
of homotopy groups of spheres. For spaces $A$ and $B$, denote by $[A,B]$ the
set of homotopy classes of maps $A\to B$. Suppose that

$W\overset{f}{\to}X\overset{g}{\to}Y\overset{h}{\to}Z$

is a sequence of maps between spaces such that the compositions $g\circ f$ and
$h\circ g$ are both nullhomotopic. Given a space $A$, let $CA$ denote the cone
on $A$. . Then we get a (non-unique) map

$F:CW\to Y$

induced by a homotopy from $g\circ f$ to a trivial map. Similarly we get a
non-unique map

$G:CX\to Z$

induced by a homotopy from $h\circ g$ to a trivial map. Appropriate
compositions give two maps $CW\to Z$ which agree on $W$. Joining them together
on the suspension $SW$. the union of these two cones, we get a map

$<f,g,h>:SW\to Z.$

The homotopy class of this map is called the _Toda bracket_ of (the classes
of) $f,g$ and $h$. It is not well defined since choices of homtopies were
used. The indeterminacy is given by maps $h[SW,Y]$ and $[SX,Z]f$ (compare
Massey products or Massey-Lie brackets.) There are also higher Toda brackets
of several elements, defined when suitable lower Toda brackets vanish. This
parallels the theory of Massey products in cohomology and, indeed, there is a
common framework in which these and many other classes of secondary operations
are situated [6], including _higher Whitehead products_ [29, 56, 1].

##  7\. Leibniz “brackets”

In 1993, Loday [47], for use in algebraic K-theory, formalized the notion of a
_(right) Leibniz algebra 181818 Some authors have called these _Loday
algebras_ , but Loday himself strongly urged calling them _Leibniz
algebras_.,_ using the $[\ ,\ ]$-notation to display the defining relation as
visibly equivalent to the Jacobi relation:

$[[x,y],z]=[x,[y,z]]+[[x,z],y].$

Such algebras with the left handed convention had been described earlier by
Bloh [11] with the name (in Russian) _left $\Delta$-algebra_, where $\Delta$
refers to _distributive_ , although it could just as well stand for
_derivation_. In [47], there is a hint of relation to (Hamiltonian) physics.
Earlier, in the late 1980’s, Dirac’s theory of constraints led to work of
Irene Dorfman in the context of Dirac structures in field theory [16]. She
developed a bracket that bears her name, that is a special case of what is now
known as a _Leibniz bracket (or product)_. In the context of Courant
algebroids, Liu, Weinstein and Xu [46] introduced a non-skewsymmetric bracket
they called ‘a twisted191919 A term much overworked, even in this intersection
of math and physics. bracket’. Later Ševera and Weinstein wrote [65]:

> _It was observed in 1998 by Kosmann-Schwarzbach, Xu, and Ševera (all
> unpublished !!) that the non-skewsymmetric version of the bracket satisfied
> the Jacobi identity written in Leibniz form._

Roytenberg in his thesis [61] pointed out that their formula agreed with
Dorfman’s. He also showed that this bracket can be expressed as the _derived
bracket_ of the bracket of a differential graded Lie algebra introduced in
[37]): In any graded differential Lie algebra, $(A,[\ ,\ ],D)$, with bracket
of degree $\pm 1$, one can define a bilinear map, called the _derived bracket_
of $[\ ,\ ]$ by $D$, as

$(a,b)\in A\times A\mapsto(-1)^{|a|}[Da,b]\in A\ ,$

where $|a|$ is the degree of $a$ (see [36]). In 1994, Loday invited Shavkat
Ayupov for two months to IRMA (the Institute in Strasbourg of which Loday was
director). Ayupov found discussions with Loday very stimulating. Although
Loday studied these algebras from a cohomological point of view, upon return
to Tashkent, Ayupov and his colleagues began to develop the structure theory
of Leibniz algebras as algebras in their own rite; indeed, they are are
principal developers of this theory. For a quite complete picture of these
algebraic developments, see their book [5]. In physics, Leibniz algebras have
recently found increased use, in particular for gauging procedures in
supergravity, replacing the more classical Lie algebras of symmetries, while
using $\circ$ in place of the bracket notation $[\ ,\ ].$

######  Definition 7.1.

A (left) _Leibniz algebra_ $(V,\circ)$ consists of a vector space $V$ with a
bi-linear operation $\circ:V\otimes V\to V$ such that

$x\circ(y\circ z)=((x\circ y)\circ z)+y\circ(x\circ z)).$

The need for a unified perspective on gauging procedures in supergravity, as
well as in Double and Exceptional field theories, has been salient in
theoretical physics for some years now. The first author to notice that
Leibniz algebras could be a crucial element in gauging procedures in
supergravity may have been Strobl [39]. Recent mathematical interpretations of
such ‘physical’ structures have led to the development of _tensor hierarchies_
[39, 44, 43] which are differential graded Lie algebras (differential Lie
crossed modules are a particular case). Crucial to the development of such
tensor hierarchies is an _embedding tensor_ : for a Lie algebra $\mathfrak{g}$
and a Leibniz algebra $(V,\circ)$ for which $V$ is a $\mathfrak{g}$-module, an
embedding tensor is a map $\Theta:V\to\mathfrak{g}$ satisfying some
compatibility conditions. From that data, a tensor hierarchy can be
constructed, much as is done for Sullivan models or Postnikov towers. Many
physical gauge ‘field’ theories, especially of Lagrangian type, involve fields
which can be recognized as differential forms with the _infinitesimal_
symmetries of the theory being differential forms with values in a Lie
algebra. Certain field theories where the gauge structure of the ‘free’ (think
non-interacting particles) theory is given in terms of a strict Lie algebra
often require an $L_{\infty}$algebra for the interacting theory, the algebra
of gauge symmetries being field dependent. The latter is an idea going back to
[10] (compare [21]). Recently there has been further progress using
differential forms with values in an $L_{\infty}$-algebra. In fact, Hohm and
Zwiebach [32] have described general gauge invariant perturbative field
theories in terms of an $L_{\infty}$-algebra $L=\\{L_{n}\\}$ in which the
fields are elements of $L_{-1}$202020 Their convention is that the
differential $d=\ell_{1}$ is of degree $-1$, gauge parameters encoding the
gauge symmetries are elements of $L_{0}$ while $L_{-2}$ contains the _equation
of motion_ and $L_{-3}$ contains the Noether identities of her second
variational theorem [38, 54]. A promising alternative [12] is to start with an
algebra of differential forms with values in a Leibniz algebra and a
Lagrangian that ‘misbehaves’ due to a lack of ‘covariance’. To achieve
covariance, the Leibniz algebra of values is extended step by step to kill the
obstructions to covariance, arriving ultimately at an $L_{\infty}$-algebra.

##  8\. Evolution of notation

Another historical accident that may have hindered recognition of the these
two parallel developments of the Jacobi relation is the notation. First, in
the differential geometry of Schouten, ‘tensors’ (in fact tensor fiields) were
objects represented by symbols with indices with elaborate rules for combining
them, as Ricci, Levi-Civita and Einstein would have used the word212121 In
1923, Rainich [57] wrote:  As to the method of the study it seemed to me
better to avoid, as far as possible, the introduction of things which have no
intrinsic meaning, such as coordinates, the g’s, the three-indices symbols,…
Rainich tried again in 1950 to re-emphasize ‘the idea of the tensor itself and
to consider the components as something secondary’ [59]. However, heavy use of
indices persists to this day in the physics literature.. Nijenhuis grew up in
that tradition, but during the 1950’s, differential geometers gradually moved
away from heavy dependence on indices and Nijenhuis updated his notation. As
he wrote to Haring:

> I saw how to state theorems with minimal use of indices, but not how to
> prove them. Once I caught on, the transformation went quickly.

##  9\. For Kirill

I was able to meet Kirill at a conference in Paris in January 2007 including
an evening hosted for a few of us by Yvette and Bertram. Plans for him to meet
and work with me in the US were delayed by bureaucratic restrictions in the
United Kingom. I had hoped to develop the higher structure version of his
major work [48], which he with his collaborators had begun to do [27]. Then,
all too soon, sadly we lost him quite unexpectedly.

##  10\. Acknowledgements

Along the path I’ve trod, I’ve benefited from input by many people. I’m
especially grateful to Yvette Kosmann-Schwarzbach for comments of significant
depth as well as meticulous proof-reading, as did Sasha Voronov. I also
appreciate helpful remarks (in alphabetical order) of: J. de Azcarraga, M.
Gerstenhaber, A. Giaquinto, J. Heubschmann and for the algebraic theory of
Leibniz algebras, S. Ayupov and E. Stitzinger. The items that have been
mentioned in the text are meant only to whet the reader’s appetite in the hope
they will follow up on one or more of these tidbits. In particular, even at
the binary level, there are other brackets attributed to e.g. Vinogradov,
Balavoine and others.

##  11\. Appendix

There are two major distinct meanings of $n$_-algebra_ :

  * •

$n$ indicating an algebra of $n$-ary operations

  * •

$n$ indicating an algebra structure on a graded vector space $V=\\{V_{i}\\}$
for $0\leq i\leq n-1$.

Unfortunately, algebras satisfying one or the other _characteristic identity_
are called $n$-Lie algebras by many authors; I urge use of _$n$ -ary Filippov_
to lessen the confusion of terms. To further confusion, in the existing
literature, in addition to Lie $n$-algebra in the above sense, there are
terms: $Lie(k)$, $Lie_{n}$ and $L(m)$. $Lie(k)$ in Hanlon and Wachs [28] has
maps $V^{k+1}\to V$ satisfy an ungraded version of the $L_{\infty}$-relations.
These are also considered by de Azcarraga and Bueno [14] (physicists), but
defined in terms of structure constants for use in physics. There is a totally
different use of $Lie(n)$; it is a representation of the symmetric group on n
letters called Lie(n) which has dimension (n-1)!. It can be realized as a
certain sub-algebra of the free Lie algebra on $n$ ‘letters’. In contrast to
truncating an $L_{\infty}$-algebra definition by limiting grading to $0\leq
i\leq n-1$, Lada and Markl [40] define $L(m)$ via bounds on the $l_{k}$:
$\\{l_{k}|\ 1\leq k\leq m,\ k<\infty\\}.$

Gnedbaye [26] treats, from an operadic point of view, $k$-ary algebras
satisfying various possible generalizations of associativity, commutativity
and Lie structure.

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