1. 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 

2. 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 

3. 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 

4. 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 

5. 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 

6. 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 

7. 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 

8. 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 

9. 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 

10. 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 

11. 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 

12. 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 

13. 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 

14. 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 

15. 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 

16. CJM 2006 (vol 58 pp. 1291)
 WeimarWoods, Evelyn

The General Structure of $G$Graded Contractions of Lie Algebras I. The Classification
We give the general structure of complex (resp., real) $G$graded
contractions of Lie algebras where $G$ is an arbitrary finite Abelian
group. For this purpose, we introduce a number of concepts, such as
pseudobasis, higherorder identities, and sign invariants. We
characterize the equivalence classes of $G$graded contractions by
showing that our set of invariants (support, higherorder identities,
and sign invariants) is complete, which yields a classification.
Keywords:Lie algebras, graded contractions Categories:17B05, 17B70 

17. CJM 2006 (vol 58 pp. 225)
 Azam, Saeid

Generalized Reductive Lie Algebras: Connections With Extended Affine Lie Algebras and Lie Tori
We investigate a class of Lie algebras which we call {\it generalized reductive
Lie algebras}. These are generalizations of semisimple, reductive, and affine
KacMoody Lie algebras. A generalized reductive Lie algebra which has an irreducible
root system is said to be {\it irreducible\/} and we note that this class of algebras
has been under intensive investigation in recent years. They have also been called
{\it extended affine Lie algebras}. The larger class of generalized reductive Lie
algebras has not been so intensively investigated. We study them in this paper and note
that one way they arise is as fixed point subalgebras of finite order automorphisms. We
show that the core modulo the center of a generalized reductive Lie algebra is a direct
sum of centerless Lie tori. Therefore one can use the results known about the
classification of centerless Lie tori to classify the cores modulo centers of
generalized reductive Lie algebras.
Categories:17B65, 17B67, 17B40 

18. CJM 2006 (vol 58 pp. 3)
19. CJM 2004 (vol 56 pp. 871)
 Schocker, Manfred

Lie Elements and Knuth Relations
A coplactic class in the symmetric group $\Sym_n$ consists of all
permutations in $\Sym_n$ with a given Schensted $Q$symbol, and may
be described in terms of local relations introduced by Knuth. Any
Lie element in the group algebra of $\Sym_n$ which is constant on
coplactic classes is already constant on descent classes. As a
consequence, the intersection of the Lie convolution algebra
introduced by Patras and Reutenauer and the coplactic algebra
introduced by Poirier and Reutenauer is the direct sum of all
Solomon descent algebras.
Keywords:symmetric group, descent set, coplactic relation, Hopf algebra,, convolution product Categories:17B01, 05E10, 20C30, 16W30 

20. CJM 2004 (vol 56 pp. 293)
 Khomenko, Oleksandr; Mazorchuk, Volodymyr

Structure of modules induced from simple modules with minimal annihilator
We study the structure of generalized Verma modules over a
semisimple complex finitedimensional Lie algebra, which are
induced from simple modules over a parabolic subalgebra. We consider
the case when the annihilator of the starting simple module is a
minimal primitive ideal if we restrict this module to the Levi factor of
the parabolic subalgebra. We show that these modules correspond to
proper standard modules in some parabolic generalization of the
BernsteinGelfandGelfand category $\Oo$ and prove that the blocks of
this parabolic category are equivalent to certain blocks of the
category of HarishChandra bimodules. From this we derive, in
particular, an irreducibility criterion for generalized Verma modules.
We also compute the composition multiplicities of those simple
subquotients, which correspond to the induction from simple modules
whose annihilators are minimal primitive ideals.
Keywords:parabolic induction, generalized Verma module, simple module, Ha\rish\Chand\ra bimodule, equivalent categories Categories:17B10, 22E47 

21. CJM 2003 (vol 55 pp. 1155)
 Đoković, Dragomir Ž.; Litvinov, Michael

The Closure Ordering of Nilpotent Orbits of the Complex Symmetric Pair $(\SO_{p+q},\SO_p\times\SO_q)$
The main problem that is solved in this paper has the following simple
formulation (which is not used in its solution). The group $K =
\mathrm{O}_p ({\bf C}) \times \mathrm{O}_q ({\bf C})$ acts on the
space $M_{p,q}$ of $p\times q$ complex matrices by $(a,b) \cdot x =
axb^{1}$, and so does its identity component $K^0 = \SO_p ({\bf C})
\times \SO_q ({\bf C})$. A $K$orbit (or $K^0$orbit) in $M_{p,q}$ is said
to be nilpotent if its closure contains the zero matrix. The closure,
$\overline{\mathcal{O}}$, of a nilpotent $K$orbit (resp.\ $K^0$orbit)
${\mathcal{O}}$ in $M_{p,q}$ is a union of ${\mathcal{O}}$ and some
nilpotent $K$orbits (resp.\ $K^0$orbits) of smaller dimensions. The
description of the closure of nilpotent $K$orbits has been known for
some time, but not so for the nilpotent $K^0$orbits. A conjecture
describing the closure of nilpotent $K^0$orbits was proposed in
\cite{DLS} and verified when $\min(p,q) \le 7$. In this paper we
prove the conjecture. The proof is based on a study of two
prehomogeneous vector spaces attached to $\mathcal{O}$ and
determination of the basic relative invariants of these spaces.
The above problem is equivalent to the problem of describing the
closure of nilpotent orbits in the real Lie algebra $\mathfrak{so}
(p,q)$ under the adjoint action of the identity component of the real
orthogonal group $\mathrm{O}(p,q)$.
Keywords:orthogonal $ab$diagrams, prehomogeneous vector spaces, relative invariants Categories:17B20, 17B45, 22E47 

22. CJM 2003 (vol 55 pp. 856)
 Su, Yucai

Poisson Brackets and Structure of Nongraded Hamiltonian Lie Algebras Related to LocallyFinite Derivations
Xu introduced a class of nongraded Hamiltonian Lie algebras. These
Lie algebras have a Poisson bracket structure. In this paper, the
isomorphism classes of these Lie algebras are determined by employing
a ``sandwich'' method and by studying some features of these Lie
algebras. It is obtained that two Hamiltonian Lie algebras are
isomorphic if and only if their corresponding Poisson algebras are
isomorphic. Furthermore, the derivation algebras and the second
cohomology groups are determined.
Categories:17B40, 17B65 

23. CJM 2003 (vol 55 pp. 766)
 Kerler, Thomas

Homology TQFT's and the AlexanderReidemeister Invariant of 3Manifolds via Hopf Algebras and Skein Theory
We develop an explicit skeintheoretical algorithm to compute the
Alexander polynomial of a 3manifold from a surgery presentation
employing the methods used in the construction of quantum invariants
of 3manifolds. As a prerequisite we establish and prove a rather
unexpected equivalence between the topological quantum field theory
constructed by Frohman and Nicas using the homology of
$U(1)$representation varieties on the one side and the
combinatorially constructed Hennings TQFT based on the quasitriangular
Hopf algebra $\mathcal{N} = \mathbb{Z}/2 \ltimes \bigwedge^*
\mathbb{R}^2$ on the other side. We find that both TQFT's are $\SL
(2,\mathbb{R})$equivariant functors and, as such, are isomorphic.
The $\SL (2,\mathbb{R})$action in the Hennings construction comes
from the natural action on $\mathcal{N}$ and in the case of the
FrohmanNicas theory from the HardLefschetz decomposition of the
$U(1)$moduli spaces given that they are naturally K\"ahler. The
irreducible components of this TQFT, corresponding to simple
representations of $\SL(2,\mathbb{Z})$ and $\Sp(2g,\mathbb{Z})$, thus
yield a large family of homological TQFT's by taking sums and products.
We give several examples of TQFT's and invariants that appear to fit
into this family, such as Milnor and Reidemeister Torsion,
SeibergWitten theories, Casson type theories for homology circles
{\it \`a la} Donaldson, higher rank gauge theories following Frohman
and Nicas, and the $\mathbb{Z}/p\mathbb{Z}$ reductions of
ReshetikhinTuraev theories over the cyclotomic integers $\mathbb{Z}
[\zeta_p]$. We also conjecture that the Hennings TQFT for
quantum$\mathfrak{sl}_2$ is the product of the ReshetikhinTuraev
TQFT and such a homological TQFT.
Categories:57R56, 14D20, 16W30, 17B37, 18D35, 57M27 

24. CJM 2002 (vol 54 pp. 595)
 Nahlus, Nazih

Lie Algebras of ProAffine Algebraic Groups
We extend the basic theory of Lie algebras of affine algebraic groups
to the case of proaffine algebraic groups over an algebraically
closed field $K$ of characteristic 0. However, some modifications
are needed in some extensions. So we introduce the prodiscrete
topology on the Lie algebra $\mathcal{L}(G)$ of the proaffine
algebraic group $G$ over $K$, which is discrete in the
finitedimensional case and linearly compact in general. As an
example, if $L$ is any sub Lie algebra of $\mathcal{L}(G)$, we show
that the closure of $[L,L]$ in $\mathcal{L}(G)$ is algebraic in
$\mathcal{L}(G)$.
We also discuss the Hopf algebra of representative functions $H(L)$ of
a residually finite dimensional Lie algebra $L$. As an example, we
show that if $L$ is a sub Lie algebra of $\mathcal{L}(G)$ and $G$
is connected, then the canonical Hopf algebra morphism from $K[G]$
into $H(L)$ is injective if and only if $L$ is algebraically dense
in $\mathcal{L}(G)$.
Categories:14L, 16W, 17B45 

25. CJM 2001 (vol 53 pp. 225)
 Britten, D. J.; Lemire, F. W.

Tensor Product Realizations of Simple Torsion Free Modules
Let $\calG$ be a finite dimensional simple Lie algebra over the
complex numbers $C$. Fernando reduced the classification of infinite
dimensional simple $\calG$modules with a finite dimensional weight
space to determining the simple torsion free $\calG$modules for
$\calG$ of type $A$ or $C$. These modules were determined by Mathieu
and using his work we provide a more elementary construction realizing
each one as a submodule of an easily constructed tensor product module.
Category:17B10 
