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26. CJM 2008 (vol 60 pp. 88)

 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

27. 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 Ringel--Hall 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 PBW-like basis for the Lusztig $\mathbb Z[v,v^{-1}]$-form of~$\bU^+$. Categories:17B37, 16G20

28. CJM 2007 (vol 59 pp. 696)

Bangoura, Momo
 AlgÃ¨bres de Lie d'homotopie associÃ©es Ã  une proto-bigÃ¨bre de Lie On associe \a toute structure de proto-big\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 proto-big\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 proto-Lie algebra structure on a finite-dimensional 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 proto-Lie 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 skew-symmetric 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, quasi-bigÃ¨bre de Lie, proto-bigÃ¨bre de Lie, crochet dÃ©rivÃ©, jacobiateurCategories:17B70, 17A30

29. 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

30. CJM 2006 (vol 58 pp. 1291)

Weimar-Woods, 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, higher-order identities, and sign invariants. We characterize the equivalence classes of $G$-graded contractions by showing that our set of invariants (support, higher-order identities, and sign invariants) is complete, which yields a classification. Keywords:Lie algebras, graded contractionsCategories:17B05, 17B70

31. 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 semi-simple, reductive, and affine Kac--Moody 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

32. CJM 2006 (vol 58 pp. 3)

Ben Saïd, Salem
 The Functional Equation of Zeta Distributions Associated With Non-Euclidean Jordan Algebras This paper is devoted to the study of certain zeta distributions associated with simple non-Euclidean Jordan algebras. An explicit form of the corresponding functional equation and Bernstein-type identities is obtained. Keywords:Zeta distributions, functional equations, Bernstein polynomials, non-Euclidean Jordan algebrasCategories:11M41, 17C20, 11S90

33. 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 productCategories:17B01, 05E10, 20C30, 16W30

34. 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 semi-simple complex finite-dimensional 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 Bernstein-Gelfand-Gelfand category $\Oo$ and prove that the blocks of this parabolic category are equivalent to certain blocks of the category of Harish-Chandra 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 categoriesCategories:17B10, 22E47

35. 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 invariantsCategories:17B20, 17B45, 22E47

36. CJM 2003 (vol 55 pp. 766)

Kerler, Thomas
 Homology TQFT's and the Alexander--Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory We develop an explicit skein-theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. 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 Frohman--Nicas theory from the Hard--Lefschetz 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, Seiberg--Witten 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 Reshetikhin--Turaev 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 Reshetikhin--Turaev TQFT and such a homological TQFT. Categories:57R56, 14D20, 16W30, 17B37, 18D35, 57M27

37. CJM 2003 (vol 55 pp. 856)

Su, Yucai
 Poisson Brackets and Structure of Nongraded Hamiltonian Lie Algebras Related to Locally-Finite 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

38. CJM 2002 (vol 54 pp. 595)

Nahlus, Nazih
 Lie Algebras of Pro-Affine Algebraic Groups We extend the basic theory of Lie algebras of affine algebraic groups to the case of pro-affine algebraic groups over an algebraically closed field $K$ of characteristic 0. However, some modifications are needed in some extensions. So we introduce the pro-discrete topology on the Lie algebra $\mathcal{L}(G)$ of the pro-affine algebraic group $G$ over $K$, which is discrete in the finite-dimensional 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

39. 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

40. CJM 2001 (vol 53 pp. 195)

Mokler, Claus
 On the Steinberg Map and Steinberg Cross-Section for a Symmetrizable Indefinite Kac-Moody Group Let $G$ be a symmetrizable indefinite Kac-Moody group over $\C$. Let $\Tr_{\La_1},\dots,\Tr_{\La_{2n-l}}$ be the characters of the fundamental irreducible representations of $G$, defined as convergent series on a certain part $G^{\tralg} \subseteq G$. Following Steinberg in the classical case and Br\"uchert in the affine case, we define the Steinberg map $\chi := (\Tr_{\La_1},\dots, \Tr_{\La_{2n-l}})$ as well as the Steinberg cross section $C$, together with a natural parametrisation $\omega \colon \C^{n} \times (\C^\times)^{\,n-l} \to C$. We investigate the local behaviour of $\chi$ on $C$ near $\omega \bigl( (0,\dots,0) \times (1,\dots,1) \bigr)$, and we show that there exists a neighborhood of $(0,\dots,0) \times (1,\dots,1)$, on which $\chi \circ \omega$ is a regular analytical map, satisfying a certain functional identity. This identity has its origin in an action of the center of $G$ on~$C$. Categories:22E65, 17B65

41. CJM 2000 (vol 52 pp. 503)

Gannon, Terry
 The Level 2 and 3 Modular Invariants for the Orthogonal Algebras The 1-loop partition function' of a rational conformal field theory is a sesquilinear combination of characters, invariant under a natural action of $\SL_2(\bbZ)$, and obeying an integrality condition. Classifying these is a clearly defined mathematical problem, and at least for the affine Kac-Moody algebras tends to have interesting solutions. This paper finds for each affine algebra $B_r^{(1)}$ and $D_r^{(1)}$ all of these at level $k\le 3$. Previously, only those at level 1 were classified. An extraordinary number of exceptionals appear at level 2---the $B_r^{(1)}$, $D_r^{(1)}$ level 2 classification is easily the most anomalous one known and this uniqueness is the primary motivation for this paper. The only level 3 exceptionals occur for $B_2^{(1)} \cong C_2^{(1)}$ and $D_7^{(1)}$. The $B_{2,3}$ and $D_{7,3}$ exceptionals are cousins of the ${\cal E}_6$-exceptional and $\E_8$-exceptional, respectively, in the A-D-E classification for $A_1^{(1)}$, while the level 2 exceptionals are related to the lattice invariants of affine~$u(1)$. Keywords:Kac-Moody algebra, conformal field theory, modular invariantsCategories:17B67, 81T40

42. CJM 2000 (vol 52 pp. 141)

Li, Chi-Kwong; Tam, Tin-Yau
 Numerical Ranges Arising from Simple Lie Algebras A unified formulation is given to various generalizations of the classical numerical range including the $c$-numerical range, congruence numerical range, $q$-numerical range and von Neumann range. Attention is given to those cases having connections with classical simple real Lie algebras. Convexity and inclusion relation involving those generalized numerical ranges are investigated. The underlying geometry is emphasized. Keywords:numerical range, convexity, inclusion relationCategories:15A60, 17B20

43. CJM 1999 (vol 51 pp. 506)

Elduque, A.; Iltyakov, A. V.
 On Polynomial Invariants of Exceptional Simple Algebraic Groups We study polynomial invariants of systems of vectors with respect to exceptional simple algebraic groups in their minimal linear representations. For each type we prove that the algebra of invariants is integral over the subalgebra of trace polynomials for a suitable algebraic system (\cf\ \cite{Schw1}, \cite{Schw2}, \cite{Ilt}). Categories:15A72, 17C20

44. CJM 1999 (vol 51 pp. 658)

Shumyatsky, Pavel
 Nilpotency of Some Lie Algebras Associated with $p$-Groups Let $L=L_0+L_1$ be a $\mathbb{Z}_2$-graded Lie algebra over a commutative ring with unity in which $2$ is invertible. Suppose that $L_0$ is abelian and $L$ is generated by finitely many homogeneous elements $a_1,\dots,a_k$ such that every commutator in $a_1,\dots,a_k$ is ad-nilpotent. We prove that $L$ is nilpotent. This implies that any periodic residually finite $2'$-group $G$ admitting an involutory automorphism $\phi$ with $C_G(\phi)$ abelian is locally finite. Categories:17B70, 20F50

45. CJM 1999 (vol 51 pp. 523)

Fabbri, Marc A.; Okoh, Frank
 Representations of Virasoro-Heisenberg Algebras and Virasoro-Toroidal Algebras Virasoro-toroidal algebras, $\tilde\mathcal{T}_{[n]}$, are semi-direct products of toroidal algebras $\mathcal{T}_{[n]}$ and the Virasoro algebra. The toroidal algebras are, in turn, multi-loop versions of affine Kac-Moody algebras. Let $\Gamma$ be an extension of a simply laced lattice $\dot{Q}$ by a hyperbolic lattice of rank two. There is a Fock space $V(\Gamma)$ corresponding to $\Gamma$ with a decomposition as a complex vector space: $V(\Gamma) = \coprod_{m \in \mathbf{Z}}K(m)$. Fabbri and Moody have shown that when $m \neq 0$, $K(m)$ is an irreducible representation of $\tilde\mathcal{T}_{[2]}$. In this paper we produce a filtration of $\tilde\mathcal{T}_{[2]}$-submodules of $K(0)$. When $L$ is an arbitrary geometric lattice and $n$ is a positive integer, we construct a Virasoro-Heisenberg algebra $\tilde\mathcal{H}(L,n)$. Let $Q$ be an extension of $\dot{Q}$ by a degenerate rank one lattice. We determine the components of $V(\Gamma)$ that are irreducible $\tilde\mathcal{H}(Q,1)$-modules and we show that the reducible components have a filtration of $\tilde\mathcal{H}(Q,1)$-submodules with completely reducible quotients. Analogous results are obtained for $\tilde\mathcal{H} (\dot{Q},2)$. These results complement and extend results of Fabbri and Moody. Categories:17B65, 17B68

46. CJM 1998 (vol 50 pp. 1323)

Morales, Jorge
 L'invariant de Hasse-Witt de la forme de Killing Nous montrons que l'invariant de Hasse-Witt de la forme de Killing d'une alg{\e}bre de Lie semi-simple $L$ s'exprime {\`a} l'aide de l'invariant de Tits de la repr{\'e}sentation irr{\'e}ductible de $L$ de poids dominant $\rho=\frac{1}{2}$ (somme des racines positives), et des invariants associ{\'e}s au groupe des sym{\'e}tries du diagramme de Dynkin de $L$. Categories:11E04, 11E72, 17B10, 17B20, 11E88, 15A66

47. CJM 1998 (vol 50 pp. 929)

Broer, Abraham
 Decomposition varieties in semisimple Lie algebras The notion of decompositon class in a semisimple Lie algebra is a common generalization of nilpotent orbits and the set of regular semisimple elements. We prove that the closure of a decomposition class has many properties in common with nilpotent varieties, \eg, its normalization has rational singularities. The famous Grothendieck simultaneous resolution is related to the decomposition class of regular semisimple elements. We study the properties of the analogous commutative diagrams associated to an arbitrary decomposition class. Categories:14L30, 14M17, 15A30, 17B45

48. CJM 1998 (vol 50 pp. 972)

Brüchert, Gerd
 Trace class elements and cross-sections in Kac-Moody groups Let $G$ be an affine Kac-Moody group, $\pi_0,\dots,\pi_r,\pi_{\delta}$ its fundamental irreducible representations and $\chi_0, \dots, \chi_r, \chi_{\delta}$ their characters. We determine the set of all group elements $x$ such that all $\pi_i(x)$ act as trace class operators, \ie, such that $\chi_i(x)$ exists, then prove that the $\chi_i$ are class functions. Thus, $\chi:=(\chi_0, \dots, \chi_r, \chi_{\delta})$ factors to an adjoint quotient $\bar{\chi}$ for $G$. In a second part, following Steinberg, we define a cross-section $C$ for the potential regular classes in $G$. We prove that the restriction $\chi|_C$ behaves well algebraically. Moreover, we obtain an action of $\hbox{\Bbbvii C}^{\times}$ on $C$, which leads to a functional identity for $\chi|_C$ which shows that $\chi|_C$ is quasi-homogeneous. Categories:22E65, 17B67

49. CJM 1998 (vol 50 pp. 816)

Mazorchuk, Volodymyr
 Tableaux realization of generalized Verma modules We construct the tableaux realization of generalized Verma modules over the Lie algebra $\sl(3,{\bbd C})$. By the same procedure we construct and investigate the structure of a new family of generalized Verma modules over $\sl(n,{\bbd C})$. Categories:17B35, 17B10

50. CJM 1998 (vol 50 pp. 266)

Britten, D. J.; Lemire, F. W.
 The torsion free Pieri formula Central to the study of simple infinite dimensional $g\ell(n, \Bbb C)$-modules having finite dimensional weight spaces are the torsion free modules. All degree $1$ torsion free modules are known. Torsion free modules of arbitrary degree can be constructed by tensoring torsion free modules of degree $1$ with finite dimensional simple modules. In this paper, the central characters of such a tensor product module are shown to be given by a Pieri-like formula, complete reducibility is established when these central characters are distinct and an example is presented illustrating the existence of a nonsimple indecomposable submodule when these characters are not distinct. Category:17B10
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