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26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. CJM 1998 (vol 50 pp. 356)

Gross, Leonard
 Some norms on universal enveloping algebras The universal enveloping algebra, $U(\frak g)$, of a Lie algebra $\frak g$ supports some norms and seminorms that have arisen naturally in the context of heat kernel analysis on Lie groups. These norms and seminorms are investigated here from an algebraic viewpoint. It is shown that the norms corresponding to heat kernels on the associated Lie groups decompose as product norms under the natural isomorphism $U(\frak g_1 \oplus \frak g_2) \cong U(\frak g_1) \otimes U(\frak g_2)$. The seminorms corresponding to Green's functions are examined at a purely Lie algebra level for $\rmsl(2,\Bbb C)$. It is also shown that the algebraic dual space $U'$ is spanned by its finite rank elements if and only if $\frak g$ is nilpotent. Categories:17B35, 16S30, 22E30

34. 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

35. CJM 1998 (vol 50 pp. 225)

Benkart, Georgia
 Derivations and invariant forms of Lie algebras graded by finite root systems Lie algebras graded by finite reduced root systems have been classified up to isomorphism. In this paper we describe the derivation algebras of these Lie algebras and determine when they possess invariant bilinear forms. The results which we develop to do this are much more general and apply to Lie algebras that are completely reducible with respect to the adjoint action of a finite-dimensional subalgebra. Categories:17B20, 17B70, 17B25

36. CJM 1998 (vol 50 pp. 210)

Zhao, Kaiming
 Isomorphisms between generalized Cartan type $W$ Lie algebras in characteristic $0$ In this paper, we determine when two simple generalized Cartan type $W$ Lie algebras $W_d (A, T, \varphi)$ are isomorphic, and discuss the relationship between the Jacobian conjecture and the generalized Cartan type $W$ Lie algebras. Keywords:Simple Lie algebras, the general Lie algebra, generalized Cartan type $W$ Lie algebras, isomorphism, Jacobian conjectureCategories:17B40, 17B65, 17B56, 17B68

37. CJM 1997 (vol 49 pp. 1206)

Letzter, Gail
 Subalgebras which appear in quantum Iwasawa decompositions Let $g$ be a semisimple Lie algebra. Quantum analogs of the enveloping algebra of the fixed Lie subalgebra are introduced for involutions corresponding to the negative of a diagram automorphism. These subalgebras of the quantized enveloping algebra specialize to their classical counterparts. They are used to form an Iwasawa type decompostition and begin a study of quantum Harish-Chandra modules. Category:17B37

38. CJM 1997 (vol 49 pp. 772)

Jie, Xiao
 Finite dimensional representations of $U_t\bigl(\rmsl (2)\bigr)$ at roots of unity All finite dimensional indecomposable representations of $U_t (\rmsl (2))$ at roots of $1$ are determined. Categories:16G10, 16G70, 17B37

39. CJM 1997 (vol 49 pp. 820)

Robart, Thierry
 Sur l'intÃ©grabilitÃ© des sous-algÃ¨bres de Lie en dimension infinie Une des questions fondamentales de la th\'eorie des groupes de Lie de dimension infinie concerne l'int\'egrabilit\'e des sous-alg\ebres de Lie topologiques $\cal H$ de l'alg\ebre de Lie $\cal G$ d'un groupe de Lie $G$ de dimension infinie au sens de Milnor. Par contraste avec ce qui se passe en th\'eorie classique il peut exister des sous-alg\ebres de Lie ferm\'ees $\cal H$ de $\cal G$ non-int\'egrables en un sous-groupe de Lie. C'est le cas des alg\ebres de Lie de champs de vecteurs $C^{\infty}$ d'une vari\'et\'e compacte qui ne d\'efinissent pas un feuilletage de Stefan. Heureusement cette imperfection" de la th\'eorie n'est pas partag\'ee par tous les groupes de Lie int\'eressants. C'est ce que montre cet article en exhibant une tr\es large classe de groupes de Lie de dimension infinie exempte de cette imperfection. Cela permet de traiter compl\etement le second probl\`eme fondamental de Sophus Lie pour les groupes de jauge de la physique-math\'ematique et les groupes formels de diff\'eomorphismes lisses de $\R^n$ qui fixent l'origine. Categories:22E65, 58h05, 17B65

40. CJM 1997 (vol 49 pp. 119)

Osborn, J. Marshall
 Automorphisms of the Lie algebras $W^*$ in characteristic $0$ No abstract. Categories:17B40, 17B65, 17B66, 17B68, 17B70
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