1. CJM Online first
 Elduque, Alberto

Order $3$ elements in $G_2$ and idempotents in symmetric composition algebras
Order three elements in the exceptional groups of type $G_2$
are classified up to conjugation over arbitrary fields. Their
centralizers are computed, and the associated classification
of idempotents in symmetric composition algebras is obtained.
Idempotents have played a key role in the study and classification
of these algebras.
Over an algebraically closed field, there are two conjugacy classes
of order three elements in $G_2$ in characteristic not $3$ and
four of them in characteristic $3$. The centralizers in characteristic
$3$ fail to be smooth for one of these classes.
Keywords:symmetric composition algebra, Okubo algebra, automorphism group, centralizer, idempotent Categories:17A75, 14L15, 17B25, 20G15 

2. CJM 2017 (vol 69 pp. 992)
 Bremner, Murray; Dotsenko, Vladimir

Classification of Regular Parametrized Onerelation Operads
JeanLouis Loday introduced a class of symmetric operads generated
by one bilinear operation subject to one
relation making each leftnormed product of three elements equal
to a linear combination
of rightnormed products:
\[
(a_1a_2)a_3=\sum_{\sigma\in S_3}x_\sigma\, a_{\sigma(1)}(a_{\sigma(2)}a_{\sigma(3)})\
;
\]
such an operad is called a parametrized onerelation operad.
For a particular choice of parameters $\{x_\sigma\}$,
this operad is said to be regular if each of its components is
the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space $V$ is, as a
graded vector space, isomorphic to the tensor
algebra of $V$. We classify, over an algebraically closed field
of characteristic zero, all regular parametrized onerelation
operads.
In fact, we prove that each such operad is isomorphic to one
of the following five operads: the leftnilpotent operad
defined by the relation $((a_1a_2)a_3)=0$, the associative operad,
the Leibniz operad, the dual Leibniz (Zinbiel) operad, and the
Poisson operad.
Our computational methods combine linear algebra over polynomial
rings, representation theory of the symmetric group, and
GrÃ¶bner bases for determinantal ideals and their radicals.
Keywords:parametrized onerelation algebra, algebraic operad, Koszul duality, representation theory of the symmetric group, determinantal ideal, GrÃ¶bner basis Categories:18D50, 13B25, 13P10, 13P15, 15A54, 1704, , , , , 17A30, 17A50, 20C30, 68W30 

3. CJM 2017 (vol 69 pp. 721)
 Allison, Bruce; Faulkner, John; Smirnov, Oleg

Weyl Images of Kantor Pairs
Kantor pairs arise naturally in the study of
$5$graded Lie algebras. In this article, we introduce
and study Kantor pairs with short Peirce gradings and relate
them to Lie algebras
graded by the root system of type
$\mathrm{BC}_2$.
This relationship
allows us to define so called Weyl images
of short Peirce graded Kantor pairs. We use Weyl images to construct
new examples of Kantor pairs, including a class of infinite
dimensional
central simple Kantor pairs over a field of characteristic $\ne
2$ or $3$, as well as a family of forms of a split
Kantor pair of type
$\mathrm{E}_6$.
Keywords:Kantor pair, graded Lie algebra, Jordan pair Categories:17B60, 17B70, 17C99, 17B65 

4. CJM 2016 (vol 68 pp. 1285)
 Ehrig, Michael; Stroppel, Catharina

2row Springer Fibres and Khovanov Diagram Algebras for Type D
We study in detail two row Springer fibres of even orthogonal
type from an algebraic as well as topological point of view.
We show that the irreducible components and their pairwise intersections
are iterated $\mathbb{P}^1$bundles. Using results of Kumar and Procesi
we compute the cohomology ring with its action of the Weyl group.
The main tool is a type $\operatorname D$ diagram calculus labelling the
irreducible components in a convenient way which relates to a
diagrammatical algebra describing the category of perverse sheaves
on isotropic Grassmannians based on work of Braden. The diagram
calculus generalizes Khovanov's arc algebra to the type
$\operatorname
D$ setting and should be seen as setting the framework for generalizing
wellknown connections of these algebras in type $\operatorname A$ to other
types.
Keywords:Springer fibers, Khovanov homology, Weyl group type D Category:1711 

5. CJM 2016 (vol 68 pp. 841)
 Gupta, Sanjiv Kumar; Hare, Kathryn

Characterizing the Absolute Continuity of the Convolution of Orbital Measures in a Classical Lie Algebra
Let $\mathfrak{g}$ be a compact, simple Lie algebra of dimension
$d$. It is
a classical result that the convolution of any $d$ nontrivial,
$G$invariant,
orbital measures is absolutely continuous with respect to
Lebesgue measure on $\mathfrak{g}$ and the sum of any $d$ nontrivial
orbits
has nonempty interior. The number $d$ was later reduced to the
rank of the
Lie algebra (or rank $+1$ in the case of type $A_{n}$). More
recently, the
minimal integer $k=k(X)$ such that the $k$fold convolution of
the orbital
measure supported on the orbit generated by $X$ is an absolutely
continuous
measure was calculated for each $X\in \mathfrak{g}$.
In this paper $\mathfrak{g}$ is any of the classical, compact,
simple Lie
algebras. We characterize the tuples $(X_{1},\dots,X_{L})$, with
$X_{i}\in
\mathfrak{g},$ which have the property that the convolution of
the $L$orbital
measures supported on the orbits generated by the $X_{i}$ is
absolutely continuous and, equivalently, the sum of their orbits
has
nonempty interior. The characterization depends on the Lie type
of
$\mathfrak{g}$ and the structure of the annihilating roots of
the $X_{i}$.
Such a characterization was previously known only for type $A_{n}$.
Keywords:compact Lie algebra, orbital measure, absolutely continuous measure Categories:43A80, 17B45, 58C35 

6. CJM 2016 (vol 69 pp. 107)
 Kamgarpour, Masoud

On the Notion of Conductor in the Local Geometric Langlands Correspondence
Under the local Langlands correspondence, the conductor of an
irreducible representation of $\operatorname{Gl}_n(F)$ is greater than the
Swan conductor of the corresponding Galois representation. In
this paper, we establish the geometric analogue of this statement
by showing that the conductor of a categorical representation
of the loop group is greater than the irregularity of the corresponding
meromorphic connection.
Keywords:local geometric Langlands, connections, cyclic vectors, opers, conductors, SegalSugawara operators, ChervovMolev operators, critical level, smooth representations, affine KacMoody algebra, categorical representations Categories:17B67, 17B69, 22E50, 20G25 

7. CJM 2016 (vol 69 pp. 453)
 Marquis, Timothée; Neeb, KarlHermann

Isomorphisms of Twisted Hilbert Loop Algebras
The closest infinite dimensional relatives of compact Lie algebras are HilbertLie algebras, i.e. real Hilbert spaces with a Lie
algebra
structure for which the scalar product is invariant.
Locally affine Lie algebras (LALAs)
correspond to double extensions of (twisted) loop algebras
over simple HilbertLie algebras $\mathfrak{k}$, also called
affinisations of $\mathfrak{k}$.
They possess a root space decomposition
whose corresponding root system is a locally affine root system
of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$,
$D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some
infinite set $J$. To each of these types corresponds a ``minimal"
affinisation of some simple HilbertLie algebra $\mathfrak{k}$,
which we call standard.
In this paper, we give for each affinisation $\mathfrak{g}$ of
a simple HilbertLie algebra $\mathfrak{k}$ an explicit isomorphism
from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from
the classification
of locally affine root systems, but
for representation theoretic purposes it is crucial to obtain
it explicitly
as a deformation between two twists which is compatible
with the root decompositions.
We illustrate this by applying our isomorphism theorem to the
study of positive energy highest weight representations of $\mathfrak{g}$.
In subsequent work, the present paper will be used to obtain
a complete classification
of the positive energy highest weight representations of affinisations
of $\mathfrak{k}$.
Keywords:locally affine Lie algebra, HilbertLie algebra, positive energy representation Categories:17B65, 17B70, 17B22, 17B10 

8. CJM 2016 (vol 68 pp. 280)
 da Silva, Genival; Kerr, Matt; Pearlstein, Gregory

Arithmetic of Degenerating Principal Variations of Hodge Structure: Examples Arising from Mirror Symmetry and Middle Convolution
We collect evidence in support of a conjecture of Griffiths,
Green
and Kerr
on the arithmetic of extension classes of
limiting
mixed Hodge structures arising from semistable degenerations
over
a number field. After briefly summarizing how a result of Iritani
implies this conjecture for a collection of hypergeometric
CalabiYau threefold examples studied by Doran and Morgan,
the authors investigate a sequence of (nonhypergeometric) examples
in dimensions $1\leq d\leq6$ arising from Katz's theory of the
middle
convolution.
A crucial role is played by the MumfordTate
group (which is $G_{2}$) of the family of 6folds, and the theory
of boundary components of MumfordTate domains.
Keywords:variation of Hodge structure, limiting mixed Hodge structure, CalabiYau variety, middle convolution, MumfordTate group Categories:14D07, 14M17, 17B45, 20G99, 32M10, 32G20 

9. CJM 2015 (vol 68 pp. 150)
 Stavrova, Anastasia

Nonstable $K_1$functors of Multiloop Groups
Let $k$ be a field of characteristic 0. Let $G$ be a reductive
group over the ring of Laurent polynomials
$R=k[x_1^{\pm 1},...,x_n^{\pm 1}]$. Assume that $G$ contains
a maximal $R$torus, and
that every semisimple normal subgroup of $G$ contains a twodimensional
split torus $\mathbf{G}_m^2$.
We show that the natural map of nonstable $K_1$functors, also
called Whitehead groups,
$K_1^G(R)\to K_1^G\bigl( k((x_1))...((x_n)) \bigr)$ is injective,
and an isomorphism if $G$ is semisimple.
As an application, we provide a way to compute the difference
between the
full automorphism group of a Lie torus (in the sense of YoshiiNeher)
and the subgroup generated by
exponential automorphisms.
Keywords:loop reductive group, nonstable $K_1$functor, Whitehead group, Laurent polynomials, Lie torus Categories:20G35, 19B99, 17B67 

10. CJM 2015 (vol 68 pp. 258)
 Calixto, Lucas; Moura, Adriano; Savage, Alistair

Equivariant Map Queer Lie Superalgebras
An equivariant map queer Lie superalgebra is the Lie superalgebra
of regular maps from an algebraic variety (or scheme) $X$ to
a queer Lie superalgebra $\mathfrak{q}$ that are equivariant with respect
to the action of a finite group $\Gamma$ acting on $X$ and $\mathfrak{q}$.
In this paper, we classify all irreducible finitedimensional
representations of the equivariant map queer Lie superalgebras
under the assumption that $\Gamma$ is abelian and acts freely
on $X$. We show that such representations are parameterized
by a certain set of $\Gamma$equivariant finitely supported maps
from $X$ to the set of isomorphism classes of irreducible finitedimensional
representations of $\mathfrak{q}$. In the special case where $X$ is the
torus, we obtain a classification of the irreducible finitedimensional
representations of the twisted loop queer superalgebra.
Keywords:Lie superalgebra, queer Lie superalgebra, loop superalgebra, equivariant map superalgebra, finitedimensional representation, finitedimensional module Categories:17B65, 17B10 

11. CJM 2014 (vol 67 pp. 573)
 Chen, Fulin; Gao, Yun; Jing, Naihuan; Tan, Shaobin

Twisted Vertex Operators and Unitary Lie Algebras
A representation of the central extension of the
unitary Lie algebra
coordinated with a skew Laurent polynomial ring
is constructed using vertex operators over an integral $\mathbb Z_2$lattice.
The irreducible decomposition of the representation is explicitly computed and described.
As a byproduct, some fundamental representations of affine
KacMoody Lie algebra of type $A_n^{(2)}$ are recovered
by the new method.
Keywords:Lie algebra, vertex operator, representation theory Categories:17B60, 17B69 

12. CJM 2014 (vol 67 pp. 55)
 Barron, Tatyana; Kerner, Dmitry; Tvalavadze, Marina

On Varieties of Lie Algebras of Maximal Class
We study complex projective varieties that parametrize
(finitedimensional) filiform Lie algebras over ${\mathbb C}$,
using equations derived by Millionshchikov. In the
infinitedimensional case we concentrate our attention on
${\mathbb N}$graded Lie algebras of maximal class. As shown by A.
Fialowski
there are only
three isomorphism types of $\mathbb{N}$graded Lie algebras
$L=\oplus^{\infty}_{i=1} L_i$ of maximal class generated by $L_1$
and $L_2$, $L=\langle L_1, L_2 \rangle$. Vergne described the
structure of these algebras with the property $L=\langle L_1
\rangle$. In this paper we study those generated by the first and
$q$th components where $q\gt 2$, $L=\langle L_1, L_q \rangle$. Under
some technical condition, there can only be one isomorphism type
of such algebras. For $q=3$ we fully classify them. This gives a
partial answer to a question posed by Millionshchikov.
Keywords:filiform Lie algebras, graded Lie algebras, projective varieties, topology, classification Categories:17B70, 14F45 

13. CJM 2013 (vol 65 pp. 1287)
 Reihani, Kamran

$K$theory of Furstenberg Transformation Group $C^*$algebras
The paper studies the $K$theoretic invariants of the crossed product
$C^{*}$algebras associated with an important family of homeomorphisms
of the tori $\mathbb{T}^{n}$ called Furstenberg transformations.
Using the PimsnerVoiculescu theorem, we prove that given $n$, the
$K$groups of those crossed products, whose corresponding $n\times n$
integer matrices are unipotent of maximal degree, always have the same
rank $a_{n}$. We show using the theory developed here that a claim
made in the literature about the torsion subgroups of these $K$groups
is false. Using the representation theory of the simple Lie algebra
$\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a
combinatorial significance. For example, every $a_{2n+1}$ is just the
number of ways that $0$ can be represented as a sum of integers
between $n$ and $n$ (with no repetitions). By adapting an argument
of van Lint (in which he answered a question of ErdÅs), a simple,
explicit formula for the asymptotic behavior of the sequence
$\{a_{n}\}$ is given. Finally, we describe the order structure of the
$K_{0}$groups of an important class of Furstenberg crossed products,
obtaining their complete Elliott invariant using classification
results of H. Lin and N. C. Phillips.
Keywords:$K$theory, transformation group $C^*$algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism Categories:19K14, 19K99, 46L35, 46L80, , 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20 

14. CJM 2013 (vol 66 pp. 323)
 Hohlweg, Christophe; Labbé, JeanPhilippe; Ripoll, Vivien

Asymptotical behaviour of roots of infinite Coxeter groups
Let $W$ be an infinite Coxeter group. We initiate the study of the set
$E$ of limit points of ``normalized'' roots (representing the
directions of the roots) of W. We show that $E$ is contained in the
isotropic cone $Q$ of the bilinear form $B$ associated to a geometric
representation, and illustrate this property with numerous examples
and pictures in rank $3$ and $4$. We also define a natural geometric
action of $W$ on $E$, and then we exhibit a countable subset of $E$,
formed by limit points for the dihedral reflection subgroups of
$W$. We explain how this subset is built from the intersection
with $Q$ of the lines passing through two positive roots, and finally we
establish that it is dense in $E$.
Keywords:Coxeter group, root system, roots, limit point, accumulation set Categories:17B22, 20F55 

15. CJM 2013 (vol 66 pp. 453)
 Vaz, Pedro; Wagner, Emmanuel

A Remark on BMW algebra, $q$Schur Algebras and Categorification
We prove that the 2variable BMW algebra
embeds into an algebra constructed from the HOMFLYPT polynomial.
We also prove that the $\mathfrak{so}_{2N}$BMW algebra embeds in the $q$Schur algebra
of type $A$.
We use these results
to suggest a schema providing categorifications of the $\mathfrak{so}_{2N}$BMW algebra.
Keywords:tangle algebras, BMW algebra, HOMFLYPT Skein algebra, qSchur algebra, categorification Categories:57M27, 81R50, 17B37, 16W99 

16. CJM 2013 (vol 65 pp. 783)
 Garcés, Jorge J.; Peralta, Antonio M.

Generalised Triple Homomorphisms and Derivations
We introduce generalised triple homomorphism between Jordan Banach
triple systems as a concept which extends the notion of generalised homomorphism between
Banach algebras given by K. Jarosz and B.E. Johnson in 1985 and 1987, respectively.
We prove that every generalised triple homomorphism between JB$^*$triples
is automatically continuous. When particularised to C$^*$algebras, we rediscover
one of the main theorems established by B.E. Johnson. We shall also consider generalised
triple derivations from a Jordan Banach triple $E$ into a Jordan Banach triple $E$module,
proving that every generalised triple derivation from a JB$^*$triple $E$ into itself or into $E^*$
is automatically continuous.
Keywords:generalised homomorphism, generalised triple homomorphism, generalised triple derivation, Banach algebra, Jordan Banach triple, C$^*$algebra, JB$^*$triple Categories:46L05, 46L70, 47B48, 17C65, 46K70, 46L40, 47B47, 47B49 

17. CJM 2012 (vol 65 pp. 82)
 Félix, Yves; Halperin, Steve; Thomas, JeanClaude

The Ranks of the Homotopy Groups of a Finite Dimensional Complex
Let $X$ be an
$n$dimensional, finite, simply connected CW complex and set
$\alpha_X =\limsup_i \frac{\log\mbox{ rank}\, \pi_i(X)}{i}$. When
$0\lt \alpha_X\lt \infty$, we give upper and lower bound for $
\sum_{i=k+2}^{k+n} \textrm{rank}\, \pi_i(X) $ for $k$ sufficiently
large. We show also for any $r$ that $\alpha_X$ can be estimated
from the integers rk$\,\pi_i(X)$, $i\leq nr$ with an error bound
depending explicitly on $r$.
Keywords:homotopy groups, graded Lie algebra, exponential growth, LS category Categories:55P35, 55P62, , , , 17B70 

18. CJM 2012 (vol 64 pp. 721)
 Achab, Dehbia; Faraut, Jacques

Analysis of the BrylinskiKostant Model for Spherical Minimal Representations
We revisit with another view point the construction by R. Brylinski
and B. Kostant of minimal representations of simple Lie groups. We
start from a pair $(V,Q)$, where $V$ is a complex vector space and $Q$
a homogeneous polynomial of degree 4 on $V$.
The manifold $\Xi $ is an orbit of a covering of ${\rm Conf}(V,Q)$,
the conformal group of the pair $(V,Q)$, in a finite dimensional
representation space.
By a generalized KantorKoecherTits construction we obtain a complex
simple Lie algebra $\mathfrak g$, and furthermore a real
form ${\mathfrak g}_{\mathbb R}$. The connected and simply connected Lie
group $G_{\mathbb R}$ with ${\rm Lie}(G_{\mathbb R})={\mathfrak
g}_{\mathbb R}$ acts unitarily on a Hilbert space of holomorphic
functions defined on the manifold $\Xi $.
Keywords:minimal representation, KantorKoecherTits construction, Jordan algebra, Bernstein identity, Meijer $G$function Categories:17C36, 22E46, 32M15, 33C80 

19. CJM 2011 (vol 63 pp. 1083)
 Kaletha, Tasho

Decomposition of Splitting Invariants in Split Real Groups
For a maximal torus in a quasisplit semisimple simplyconnected group over a local field of characteristic $0$,
Langlands and Shelstad constructed a
cohomological invariant called the splitting invariant, which is an important
component of their endoscopic transfer factors. We study this invariant in the
case of a split real group and prove a
decomposition theorem which expresses this invariant for a general torus as a product of the corresponding
invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants
between different tori in the given real group.
Keywords:endoscopy, real lie group, splitting invariant, transfer factor Categories:11F70, 22E47, 11S37, 11F72, 17B22 

20. CJM 2009 (vol 62 pp. 382)
 Lü, Rencai; Zhao, Kaiming

Verma Modules over Quantum Torus Lie Algebras
Representations of various onedimensional central
extensions of quantum tori (called quantum torus Lie algebras) were
studied by several authors. Now we define a central extension of
quantum tori so that all known representations can be regarded as
representations of the new quantum torus Lie algebras $\mathfrak{L}_q$. The
center of $\mathfrak{L}_q$ now is generally infinite dimensional.
In this paper, $\mathbb{Z}$graded Verma modules $\widetilde{V}(\varphi)$ over $\mathfrak{L}_q$
and their corresponding irreducible highest weight modules
$V(\varphi)$ are defined for some linear functions $\varphi$.
Necessary and sufficient conditions for $V(\varphi)$ to have all
finite dimensional weight spaces are given. Also necessary and
sufficient conditions for Verma modules $\widetilde{V}(\varphi)$ to
be irreducible are obtained.
Categories:17B10, 17B65, 17B68 

21. CJM 2008 (vol 60 pp. 892)
 Neeb, KarlHermann; Wagemann, Friedrich

The Second Cohomology of Current Algebras of General Lie Algebras
Let $A$ be a unital commutative associative algebra over a field of
characteristic zero, $\k$ a Lie algebra, and
$\zf$ a vector space, considered as a trivial module of the Lie algebra
$\gf := A \otimes \kf$. In this paper, we give a
description of the cohomology space $H^2(\gf,\zf)$
in terms of easily accessible data associated with $A$ and $\kf$.
We also discuss the topological situation, where
$A$ and $\kf$ are locally convex algebras.
Keywords:current algebra, Lie algebra cohomology, Lie algebra homology, invariant bilinear form, central extension Categories:17B56, 17B65 

22. CJM 2008 (vol 60 pp. 88)
 Diwadkar, Jyotsna Mainkar

Nilpotent Conjugacy Classes in $p$adic Lie Algebras: The Odd Orthogonal Case
We will study the following question: Are nilpotent conjugacy
classes of reductive Lie algebras over $p$adic fields
definable? By definable, we mean definable by a formula in Pas's
language. In this language, there are no field extensions and no
uniformisers. Using Waldspurger's parametrization, we answer in the
affirmative in the case of special orthogonal Lie algebras
$\mathfrak{so}(n)$ for $n$ odd, over $p$adic fields.
Categories:17B10, 03C60 

23. CJM 2007 (vol 59 pp. 1260)
 Deng, Bangming; Du, Jie; Xiao, Jie

Generic Extensions and Canonical Bases for Cyclic Quivers
We use the monomial basis theory developed by Deng and Du to
present an elementary algebraic construction of the canonical
bases for both the RingelHall algebra of a cyclic quiver and the
positive part $\bU^+$ of the quantum affine $\frak{sl}_n$. This
construction relies on analysis of quiver representations and the
introduction of a new integral PBWlike basis for the Lusztig
$\mathbb Z[v,v^{1}]$form of~$\bU^+$.
Categories:17B37, 16G20 

24. CJM 2007 (vol 59 pp. 696)
 Bangoura, Momo

AlgÃ¨bres de Lie d'homotopie associÃ©es Ã une protobigÃ¨bre de Lie
On associe \`a toute structure de protobig\`ebre de Lie sur un espace
vectoriel $F$ de dimension finie des structures d'alg\`ebre de Lie
d'homotopie d\'efinies respectivement sur la suspension de l'alg\`ebre
ext\'erieure de $F$ et celle de son dual $F^*$. Dans ces alg\`ebres,
tous les crochets $n$aires sont nuls pour $n \geq 4$ du fait qu'ils
proviennent d'une structure de protobig\`ebre de Lie. Plus
g\'en\'eralement, on associe \`a un \'el\'ement de degr\'e impair de
l'alg\`ebre ext\'erieure de la somme directe de $F$ et $F^*$, une
collection d'applications multilin\'eaires antisym\'etriques sur
l'alg\`ebre ext\'erieure de $F$ (resp.\ $F^*$), qui v\'erifient les
identit\'es de Jacobi g\'en\'eralis\'ees, d\'efinissant les alg\`ebres
de Lie d'homotopie, si l'\'el\'ement donn\'e est de carr\'e nul pour
le grand crochet de l'alg\`ebre ext\'erieure de la somme directe de
$F$ et de~$F^*$.
To any protoLie algebra structure on a finitedimensional vector
space~$F$, we associate homotopy Lie algebra structures defined on
the suspension of the exterior algebra of $F$ and that of its dual
$F^*$, respectively. In these algebras, all $n$ary brackets for $n
\geq 4$ vanish because the brackets are defined by the protoLie
algebra structure. More generally, to any element of odd degree in
the exterior algebra of the direct sum of $F$ and $F^*$, we associate
a set of multilinear skewsymmetric mappings on the suspension of the
exterior algebra of $F$ (resp.\ $F^*$), which satisfy the generalized
Jacobi identities, defining the homotopy Lie algebras, if the given
element is of square zero with respect to the big bracket of the
exterior algebra of the direct sum of $F$ and~$F^*$.
Keywords:algÃ¨bre de Lie d'homotopie, bigÃ¨bre de Lie, quasibigÃ¨bre de Lie, protobigÃ¨bre de Lie, crochet dÃ©rivÃ©, jacobiateur Categories:17B70, 17A30 

25. CJM 2007 (vol 59 pp. 712)
 Billig, Yuly

Jet Modules
In this paper we classify indecomposable modules for the Lie algebra
of vector fields on a torus that admit a compatible action of the algebra
of functions. An important family of such modules is given by spaces of jets
of tensor fields.
Categories:17B66, 58A20 
