Expand all Collapse all | Results 26 - 42 of 42 |
26. CMB 2003 (vol 46 pp. 597)
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)
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)
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)
Some Factorizations in Universal Enveloping Algebras of Three Dimensional Lie Algebras and Generalizations |
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)
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)
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)
Fermionic and Bosonic Representations of the Extended Affine Lie Algebra $\widetilde{\mathfrak{gl}_N} (\mathbb{C}_q)$ |
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)
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)
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)
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)
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)
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)
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 |
39. CMB 1999 (vol 42 pp. 486)
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)
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)
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)
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 |