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26. CMB 2008 (vol 51 pp. 298)

Tocón, Maribel
The Kostrikin Radical and the Invariance of the Core of Reduced Extended Affine Lie Algebras
In this paper we prove that the Kostrikin radical of an extended affine Lie algebra of reduced type coincides with the center of its core, and use this characterization to get a type-free description of the core of such algebras. As a consequence we get that the core of an extended affine Lie algebra of reduced type is invariant under the automorphisms of the algebra.

Keywords:extended affine Lie algebra, Lie torus, core, Kostrikin radical
Categories:17B05, 17B65

27. CMB 2007 (vol 50 pp. 603)

Penkov, Ivan; Zuckerman, Gregg
Construction of Generalized Harish-Chandra Modules with Arbitrary Minimal $\mathfrak k$-Type
Let $\mathfrak g$ be a semisimple complex Lie algebra and $\k\subset\g$ be any algebraic subalgebra reductive in $\mathfrak g$. For any simple finite dimensional $\mathfrak k$-module $V$, we construct simple $(\mathfrak g,\mathfrak k)$-modules $M$ with finite dimensional $\mathfrak k$-isotypic components such that $V$ is a $\mathfrak k$-submodule of $M$ and the Vogan norm of any simple $\k$-submodule $V'\subset M, V'\not\simeq V$, is greater than the Vogan norm of $V$. The $(\mathfrak g,\mathfrak k)$-modules $M$ are subquotients of the fundamental series of $(\mathfrak g,\mathfrak k)$-modules.

Categories:17B10, 17B55

28. CMB 2007 (vol 50 pp. 469)

Tvalavadze, M. V.
Simple Decompositions of the Exceptional Jordan Algebra
This paper presents some results on the simple exceptional Jordan algebra over an algebraically closed field $\Phi$ of characteristic not $2$. Namely an example of simple decomposition of $H(O_3)$ into the sum of two subalgebras of the type $H(Q_3)$ is produced, and it is shown that this decomposition is the only one possible in terms of simple subalgebras.


29. CMB 2006 (vol 49 pp. 492)

Chan, Kai-Cheong; Đoković, Dragomir Ž.
Conjugacy Classes of Subalgebras of the Real Sedenions
By applying the Cayley--Dickson process to the division algebra of real octonions, one obtains a 16-dimensional real algebra known as (real) sedenions. We denote this algebra by $\bA_4$. It is a flexible quadratic algebra (with unit element 1) but not a division algebra. We classify the subalgebras of $\bA_4$ up to conjugacy (\emph{i.e.,} up to the action of the automorphism group $G$ of $\bA_4$) with one exception: we leave aside the more complicated case of classifying the quaternion subalgebras. Any nonzero subalgebra contains 1 and we show that there are no proper subalgebras of dimension 5, 7 or $>8$. The proper non-division subalgebras have dimensions 3, 6 and 8. We show that in each of these dimensions there is exactly one conjugacy class of such subalgebras. There are infinitely many conjugacy classes of subalgebras in dimensions 2 and 4, but only 4 conjugacy classes in dimension 8.

Categories:17A45, 17A36, 17A20

30. CMB 2005 (vol 48 pp. 587)

Lopes, Samuel A.
Separation of Variables for $U_{q}(\mathfrak{sl}_{n+1})^{+}$
Let $U_{q}(\SL)^{+}$ be the positive part of the quantized enveloping algebra $U_{q}(\SL)$. Using results of Alev--Dumas and Caldero related to the center of $U_{q}(\SL)^{+}$, we show that this algebra is free over its center. This is reminiscent of Kostant's separation of variables for the enveloping algebra $U(\g)$ of a complex semisimple Lie algebra $\g$, and also of an analogous result of Joseph--Letzter for the quantum algebra $\Check{U}_{q}(\g)$. Of greater importance to its representation theory is the fact that $\U{+}$ is free over a larger polynomial subalgebra $N$ in $n$ variables. Induction from $N$ to $\U{+}$ provides infinite-dimensional modules with good properties, including a grading that is inherited by submodules.

Categories:17B37, 16W35, 17B10, 16D60

31. CMB 2005 (vol 48 pp. 460)

Sommers, Eric N.
$B$-Stable Ideals in the Nilradical of a Borel Subalgebra
We count the number of strictly positive $B$-stable ideals in the nilradical of a Borel subalgebra and prove that the minimal roots of any $B$-stable ideal are conjugate by an element of the Weyl group to a subset of the simple roots. We also count the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.

Categories:20F55, 17B20, 05E99

32. CMB 2005 (vol 48 pp. 445)

Patras, Frédéric; Reutenauer, Christophe; Schocker, Manfred
On the Garsia Lie Idempotent
The orthogonal projection of the free associative algebra onto the free Lie algebra is afforded by an idempotent in the rational group algebra of the symmetric group $S_n$, in each homogenous degree $n$. We give various characterizations of this Lie idempotent and show that it is uniquely determined by a certain unit in the group algebra of $S_{n-1}$. The inverse of this unit, or, equivalently, the Gram matrix of the orthogonal projection, is described explicitly. We also show that the Garsia Lie idempotent is not constant on descent classes (in fact, not even on coplactic classes) in $S_n$.

Categories:17B01, 05A99, 16S30, 17B60

33. CMB 2005 (vol 48 pp. 394)

Đoković, D. Ž.; Szechtman, F.; Zhao, K.
Diagonal Plus Tridiagonal Representatives for Symplectic Congruence Classes of Symmetric Matrices
Let $n=2m$ be even and denote by $\Sp_n(F)$ the symplectic group of rank $m$ over an infinite field $F$ of characteristic different from $2$. We show that any $n\times n$ symmetric matrix $A$ is equivalent under symplectic congruence transformations to the direct sum of $m\times m$ matrices $B$ and $C$, with $B$ diagonal and $C$ tridiagonal. Since the $\Sp_n(F)$-module of symmetric $n\times n$ matrices over $F$ is isomorphic to the adjoint module $\sp_n(F)$, we infer that any adjoint orbit of $\Sp_n(F)$ in $\sp_n(F)$ has a representative in the sum of $3m-1$ root spaces, which we explicitly determine.

Categories:11E39, 15A63, 17B20

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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 $G(3)$.


43. 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

44. 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

45. 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

46. 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, associahedron
Categories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50

47. 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

48. 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.

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