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26. CMB 2003 (vol 46 pp. 597)

Neeb, Karl-Hermann; Penkov, Ivan
 Cartan Subalgebras of $\mathfrak{gl}_\infty$ Let $V$ be a vector space over a field $\mathbb{K}$ of characteristic zero and $V_*$ be a space of linear functionals on $V$ which separate the points of $V$. We consider $V\otimes V_*$ as a Lie algebra of finite rank operators on $V$, and set $\mathfrak{gl} (V,V_*) := V\otimes V_*$. We define a Cartan subalgebra of $\mathfrak{gl} (V,V_*)$ as the centralizer of a maximal subalgebra every element of which is semisimple, and then give the following description of all Cartan subalgebras of $\mathfrak{gl} (V,V_*)$ under the assumption that $\mathbb{K}$ is algebraically closed. A subalgebra of $\mathfrak{gl} (V,V_*)$ is a Cartan subalgebra if and only if it equals $\bigoplus_j \bigl( V_j \otimes (V_j)_* \bigr) \oplus (V^0 \otimes V_*^0)$ for some one-dimensional subspaces $V_j \subseteq V$ and $(V_j)_* \subseteq V_*$ with $(V_i)_* (V_j) = \delta_{ij} \mathbb{K}$ and such that the spaces $V_*^0 = \bigcap_j (V_j)^\bot \subseteq V_*$ and $V^0 = \bigcap_j \bigl( (V_j)_* \bigr)^\bot \subseteq V$ satisfy $V_*^0 (V^0) = \{0\}$. We then discuss explicit constructions of subspaces $V_j$ and $(V_j)_*$ as above. Our second main result claims that a Cartan subalgebra of $\mathfrak{gl} (V,V_*)$ can be described alternatively as a locally nilpotent self-normalizing subalgebra whose adjoint representation is locally finite, or as a subalgebra $\mathfrak{h}$ which coincides with the maximal locally nilpotent $\mathfrak{h}$-submodule of $\mathfrak{gl} (V,V_*)$, and such that the adjoint representation of $\mathfrak{h}$ is locally finite. Categories:17B65, 17B20

27. CMB 2003 (vol 46 pp. 529)

Billig, Yuly
 Representations of the Twisted Heisenberg--Virasoro Algebra at Level Zero We describe the structure of the irreducible highest weight modules for the twisted Heisenberg--Virasoro Lie algebra at level zero. We prove that either a Verma module is irreducible or its maximal submodule is cyclic. Categories:17B68, 17B65

28. CMB 2002 (vol 45 pp. 509)

Benkart, Georgia; Elduque, Alberto
 Lie Superalgebras Graded by the Root Systems $C(n)$, $D(m,n)$, $D(2,1;\alpha)$, $F(4)$, $G(3)$ We determine the Lie superalgebras that are graded by the root systems of the basic classical simple Lie superalgebras of type $C(n)$, $D(m,n)$, $D(2,1;\alpha)$ $(\alpha \in \mathbb{F} \setminus \{0,-1\})$, $F(4)$, and $G(3)$. Category:17A70

29. CMB 2002 (vol 45 pp. 525)

Berman, Stephen; Morita, Jun; Yoshii, Yoji
 Some Factorizations in Universal Enveloping Algebras of Three Dimensional Lie Algebras and Generalizations We introduce the notion of Lie algebras with plus-minus pairs as well as regular plus-minus pairs. These notions deal with certain factorizations in universal enveloping algebras. We show that many important Lie algebras have such pairs and we classify, and give a full treatment of, the three dimensional Lie algebras with plus-minus pairs. Categories:17B05, 17B35, 17B67, 17B70

30. CMB 2002 (vol 45 pp. 672)

Rao, S. Eswara; Batra, Punita
 A New Class of Representations of EALA Coordinated by Quantum Tori in Two Variables We study the representations of extended affine Lie algebras $s\ell_{\ell+1} (\mathbb{C}_q)$ where $q$ is $N$-th primitive root of unity ($\mathbb{C}_q$ is the quantum torus in two variables). We first prove that $\bigoplus s\ell_{\ell+1} (\mathbb{C})$ for a suitable number of copies is a quotient of $s\ell_{\ell+1} (\mathbb{C}_q)$. Thus any finite dimensional irreducible module for $\bigoplus s\ell_{\ell+1} (\mathbb{C})$ lifts to a representation of $s\ell_{\ell+1} (\mathbb{C}_q)$. Conversely, we prove that any finite dimensional irreducible module for $s\ell_{\ell+1} (\mathbb{C}_q)$ comes from above. We then construct modules for the extended affine Lie algebras $s\ell_{\ell+1} (\mathbb{C}_q) \oplus \mathbb{C} d_1 \oplus \mathbb{C} d_2$ which is integrable and has finite dimensional weight spaces. Categories:17B65, 17B66, 17B68

31. CMB 2002 (vol 45 pp. 653)

Martínez, Consuelo; Zelmanov, Efim
 Specializations of Jordan Superalgebras In this paper we study specializations and one-sided bimodules of simple Jordan superalgebras. Categories:17C70, 17C25, 17C40

32. CMB 2002 (vol 45 pp. 623)

Gao, Yun
 Fermionic and Bosonic Representations of the Extended Affine Lie Algebra $\widetilde{\mathfrak{gl}_N} (\mathbb{C}_q)$ We construct a class of fermions (or bosons) by using a Clifford (or Weyl) algebra to get two families of irreducible representations for the extended affine Lie algebra $\widetilde{\mathfrak{gl}_N (\mathbb{C}_q)}$ of level $(1,0)$ (or $(-1,0)$). Categories:17B65, 17B67

33. CMB 2002 (vol 45 pp. 606)

Gannon, Terry
 Postcards from the Edge, or Snapshots of the Theory of Generalised Moonshine We begin by reviewing Monstrous Moonshine. The impact of Moonshine on algebra has been profound, but so far it has had little to teach number theory. We introduce (using `postcards') a much larger context in which Monstrous Moonshine naturally sits. This context suggests Moonshine should indeed have consequences for number theory. We provide some humble examples of this: new generalisations of Gauss sums and quadratic reciprocity. Categories:11F22, 17B67, 81T40

34. CMB 2002 (vol 45 pp. 567)

De Sole, Alberto; Kac, Victor G.
 Subalgebras of $\gc_N$ and Jacobi Polynomials We classify the subalgebras of the general Lie conformal algebra $\gc_N$ that act irreducibly on $\mathbb{C} [\partial]^N$ and that are normalized by the sl$_2$-part of a Virasoro element. The problem turns out to be closely related to classical Jacobi polynomials $P_n^{(-\sigma,\sigma)}$, $\sigma \in \mathbb{C}$. The connection goes both ways---we use in our classification some classical properties of Jacobi polynomials, and we derive from the theory of conformal algebras some apparently new properties of Jacobi polynomials. Categories:17B65, 17B68, 17B69, 33C45

35. CMB 2001 (vol 44 pp. 27)

Goodaire, Edgar G.; Milies, César Polcino
 Normal Subloops in the Integral Loop Ring of an $\RA$ Loop We show that an $\RA$ loop has a torsion-free normal complement in the loop of normalized units of its integral loop ring. We also investigate whether an $\RA$ loop can be normal in its unit loop. Over fields, this can never happen. Categories:20N05, 17D05, 16S34, 16U60

36. CMB 2000 (vol 43 pp. 459)

Ndogmo, J. C.
 Properties of the Invariants of Solvable Lie Algebras We generalize to a field of characteristic zero certain properties of the invariant functions of the coadjoint representation of solvable Lie algebras with abelian nilradicals, previously obtained over the base field $\bbC$ of complex numbers. In particular we determine their number and the restricted type of variables on which they depend. We also determine an upper bound on the maximal number of functionally independent invariants for certain families of solvable Lie algebras with arbitrary nilradicals. Categories:17B30, 22E70

37. CMB 2000 (vol 43 pp. 79)

König, Steffen
 Cyclotomic Schur Algebras and Blocks of Cyclic Defect An explicit classification is given of blocks of cyclic defect of cyclotomic Schur algebras and of cyclotomic Hecke algebras, over discrete valuation rings. Categories:20G05, 20C20, 16G30, 17B37, 57M25

38. CMB 2000 (vol 43 pp. 3)

Adin, Ron; Blanc, David
 Resolutions of Associative and Lie Algebras Certain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees. Keywords:resolutions, homology, Lie algebras, associative algebras, non-associative algebras, Jacobi identity, leaf-labeled trees, associahedronCategories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50

39. CMB 1999 (vol 42 pp. 486)

Sawyer, P.
 Spherical Functions on $\SO_0(p,q)/\SO(p)\times \SO(q)$ An integral formula is derived for the spherical functions on the symmetric space $G/K=\break \SO_0(p,q)/\SO(p)\times \SO(q)$. This formula allows us to state some results about the analytic continuation of the spherical functions to a tubular neighbourhood of the subalgebra $\a$ of the abelian part in the decomposition $G=KAK$. The corresponding result is then obtained for the heat kernel of the symmetric space $\SO_0(p,q)/\SO (p)\times\SO (q)$ using the Plancherel formula. In the Conclusion, we discuss how this analytic continuation can be a helpful tool to study the growth of the heat kernel. Categories:33C55, 17B20, 53C35

40. CMB 1999 (vol 42 pp. 412)

Tai, Yung-Sheng
 Peirce Domains A theorem of Kor\'anyi and Wolf displays any Hermitian symmetric domain as a Siegel domain of the third kind over any of its boundary components. In this paper we give a simple proof that an analogous realization holds for any self-adjoint homogeneous cone. Category:17C27

41. CMB 1997 (vol 40 pp. 143)

Bremner, Murray
 Quantum deformations of simple Lie algebras It is shown that every simple complex Lie algebra $\fg$ admits a 1-parameter family $\fg_q$ of deformations outside the category of Lie algebras. These deformations are derived from a tensor product decomposition for $U_q(\fg)$-modules; here $U_q(\fg)$ is the quantized enveloping algebra of $\fg$. From this it follows that the multiplication on $\fg_q$ is $U_q(\fg)$-invariant. In the special case $\fg = {\ss}(2)$, the structure constants for the deformation ${\ss}(2)_q$ are obtained from the quantum Clebsch-Gordan formula applied to $V(2)_q \otimes V(2)_q$; here $V(2)_q$ is the simple 3-dimensional $U_q\bigl({\ss}(2)\bigr)$-module of highest weight $q^2$. Categories:17B37, 17A01

42. CMB 1997 (vol 40 pp. 103)

Riley, David M.; Tasić, Vladimir
 The transfer of a commutator law from a nil-ring to its adjoint group For every field $F$ of characteristic $p\geq 0$, we construct an example of a finite dimensional nilpotent $F$-algebra $R$ whose adjoint group $A(R)$ is not centre-by-metabelian, in spite of the fact that $R$ is Lie centre-by-metabelian and satisfies the identities $x^{2p}=0$ when $p>2$ and $x^8=0$ when $p=2$. The existence of such algebras answers a question raised by A.~E.~Zalesskii, and is in contrast to positive results obtained by Krasilnikov, Sharma and Srivastava for Lie metabelian rings and by Smirnov for the class Lie centre-by-metabelian nil-algebras of exponent 4 over a field of characteristic 2 of cardinality at least 4. Categories:16U60, 17B60
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