Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 42 (2001)
Abstracts
A. J. Breda d'Azevedo, G. A. Jones: Platonic Hypermaps, 42 (2001) 001--037
- We classify the regular hypermaps (orientable or non-orientable)
whose full automorphism group is isomorphic to the symmetry group
of a Platonic solid. There are 185 of them, of which 93 are maps.
We also classify the regular hypermaps with automorphism group
A5: there are 19 of these, all non-orientable, and 9 of
them are maps. These hypermaps are constructed as combinatorial and
topological objects, many of them arising as coverings of Platonic
solids and Kepler-Poinsot polyhedra (viewed as hypermaps), or their
associates. We conclude by showing that any rotary Platonic hypermap
is regular, so there are no chiral Platonic hypermaps.
- We classify the rotary hypermaps (sometimes called regular
hypermaps) on an orientable surface of genus 2. There are 43 of
them, of which 10 are maps (classified by Threlfall), 20 more can
be obtained from the 10 maps by applying Machi's operations, and
the remaining 13 may be obtained from the maps by using Walsh's
bijection between maps and hypermaps. As a corollary, we deduce
that there are no non-orientable reflexible hypermaps of
characteristic -1.
- We construct the q-analogue of a certain class of group
cohomology and introduce the action of Hecke operators on such
cohomology. We also show that such an action determines a
representation of a Hecke ring in each of the associated group
cohomology spaces.
- We give a classification of the surfaces that can be generated
by a moving conic in more than one way. It turns out that these
surfaces belong to classes which have been thoroughly studied in
other contexts (ruled surfaces, Veronese surfaces, del Pezzo
surfaces).
- For each v, k and m such that v >= k >= 2m+2 >= 8, we
construct a periodically-cyclic Gale 2m-polytope with v
vertices and the period k. For such a polytope, there is a
complete description of each of its facets based upon a labelling
(total ordering) of the vertices so that every subset of k
successive vertices generates a cyclic 2m-polytope.
- The multiplicity of an m-primary ideal I of a
Cohen-Macaulay local ring (A, m) of dimension d can be
written as e(I) = l(I/I2) - (d-1) l(A/I)
+ K - 1 for some
integer K >= 1. In the case K=1 or 2, the Hilbert function of I
and the depth of the associated graded ring of A with respect to I are very well understood.
We are dealing with
the case K=3 and we determine the possible Hilbert functions of
stretched ideals whose Cohen-Macaulay type is not too big. Our main
result extends to a considerable extent a deep result of J. Sally
who proved that the associated graded ring of a Gorenstein local
ring with embedding dimension equal to e(m) + d - 3, is
Cohen-Macaulay.
- If C is a compact subset of Rd and H is a halfspace bounded
by a hyperplane p, the set T(C) obtained by shaking
C on p is defined as the set contained in H, such
that for every line l orthogonal to p, the intersection
of T(C) and l is a segment of the same length as the intersection of C and l, and
one of its endpoints is on p. It is shown that there
exist d + 1 hyperplanes such that every compact set can be reduced to a simplex,
via repeated shaking processes on these hyperplanes.
- We study some properties of submanifolds in Euclidean space
concerning their generic contacts with straight lines and hyperplanes and
expose some duality relation associated to these contacts.
- We classify n-dimensional multi-helicoidal submanifolds of nonzero
constant sectional curvature and cohomogeneity one in the Euclidean space
R2n-1, that is, n-dimensional submanifolds of nonzero constant
sectional curvature in R2n-1 that are invariant under the action
of an (n-1)-parameter subgroup of isometries of R2n-1 with no
pure translations. This is accomplished by first giving a complete
description of all these subgroups and then deriving a multidimensional
version of a lemma due to Bour. We also prove that such submanifolds are
precisely the ones that correspond to solutions of the generalized sine-Gordon
and elliptic sinh-Gordon equations that are invariant by an (n-1)-dimensional
subgroup of translations of the symmetry group of these equations.
- We introduce a series of invariants on Kähler manifolds and
prove a series of general inequalities involving these invariants
for Kähler submanifolds in complex space forms. We also
determine Kähler submanifolds in complex space forms which
satisfy the equality cases of these inequalities.
-
- In a previous paper ["On a strong property of the weak subalgebra lattice", Alg.
Univ. 40 (1998) 477--495] we showed that for each locally finite unary (total) algebra
of finite type, its weak subalgebra lattice uniquely determines its (strong) subalgebra
lattice. Now we generalize this fact to algebras having also finitely many binary
operations (for example, groupoids, semigroups, semilattices). More precisely, we
generalize some ideas from the above-mentioned paper to prove: Let A be a locally
finite (total) algebra with m unary operations k1A, ... ,
kmA and n binary operations f1A, ... ,
fnA and let A satisfy the following formula: for any x, y and
1 <= i <= n, x not eqal to y, implies that fi(x, y) is not
equal to x and that fi(x, y) is not equal to y. Then for every
partial algebra B with m unary and n binary operations,
if the weak subalgebra lattices of A and B are
isomorphic, then their (strong) subalgebra lattices are also
isomorphic and moreover, B is total and locally finite and
satisfies the same formula.
- Let C = ( C1, C2, ... , Cn) be a finite
sequence of unit cubes in the d-dimensional space. The sequence C is called
a facet-to-facet snake if the intersection D(i, i+1) of Ci and Ci+1
is a common facet of Ci and Ci+1, 1 <= i <= n-1, and
dim D(i, j) <= max{-1, d + i - j}, 1 <= i < j <= n.
A facet-to-facet snake of unit cubes is called maximal if it is
not a proper subset of another facet-to-facet snake of unit cubes. We prove that the minimum number of
d-dimensional
unit cubes which can form a maximal facet-to-facet snake is 8d - 1 for all d
>= 3.
- All rings are assumed to be commutative with identity.
A generalized GCD ring (G-GCD ring) is a ring (zero-divisors
admitted) in which the intersection of every two finitely
generated (f.g.) faithful multiplication ideals is a f.g. faithful
multiplication ideal. Various properties of G-GCD rings are
considered. We generalize some of Jäger's and Lüneburg's
results to f.g. faithful multiplication ideals.
- The geometrization of 3-manifolds plays an important role in
various topological investigations and in the geometry as well. Thurston
classified the eight simply connected 3-dimensional maximal homogeneous
Riemannian geometries [see: "Three-dimensional manifolds, Kleinian groups and
hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982) 357--381; "Three-dimensional
geometry and topology, Vol.1, ed. by S. Levy, Princeton University Press 1997,
(Ch.3.8,4.7)]. One of these is S2´R,
i.e. the direct product of the spherical plane S2 and the real
line R. Our purpose is the classification of the space groups of
S2´R, i.e. discrete transformation
groups which act on S2´R with a lattice
on R (see Section 3), analogously to that of the classical Euclidean geometry
E3.
- We continue to investigate the algebro-geometric structure of Drinfeld-Anderson
motives introduced in previous papers of the author ["Arithmetique des corps globaux
de fonctions et geometrie des schemas modulaires de Drinfeld", Ph. D. Thesis, Joseph
Fourier University, Grenoble (France), January 1997, 87 p.; and "Drinfeld-Anderson
motives and multicomponent KP hierarchy, in: Recent progress in algebra (Taejon/Seoul
1997), 213--227, Contemp. Math. 224, Amer. Math. Soc., Providence, RI, 1999]. In the
first part we construct shtukas related to Drinfeld-Anderson motives. The main result
of the second part is a uniformization Theorem.
- Among the different notions of area in a Minkowski space, those
due to Busemann and to Holmes and Thompson, respectively, have
found particular attention. In recent papers it was shown that
the Holmes-Thompson area is integral-geometric, in the sense
that certain integral-geometric formulas of Crofton-type, well
known for the area in Euclidean space, can be carried over to
Minkowski spaces and the Holmes-Thompson area. In the present
paper, the Busemann area is investigated from this point of view.
- We explore the properties of being hereditary and
being strong among the radicals of associative rings, and prove
certain results such as a relationship between Brown-McCoy and
Behrens radicals.
- We present some geometric properties of isoptics
of pairs of nested closed strictly convex curves. The theory of
isoptics provides a simple geometric method to prove some
generalizations of well-known integral formulas of Crofton-type.
- We study the problem concerning the angle between
two subspaces of arbitrary dimensions in Euclidean space En.
It is proven that the angle between two subspaces is equal to the
angle between their orthogonal subspaces. Using the eigenvalues
and eigenvectors of corresponding matrix representations, there
are introduced principal values and principal subspaces. Their
geometrical interpretation is also given together with the
canonical representation of the two subspaces. The canonical
matrix for the two subspaces is introduced and its properties of
duality are obtained. Here obtained results expand the classic
results of P. R. Halmos ["Finite Dimensional Vector Spaces", 2nd ed.
Van Nostrand Reinhold, New York 1958] and S. Kurepa ["Finite Dimensional
Vector Spaces and Applications", Sveucilisna Naklada Liber, Zagreb 1979].
- We consider embedding classes of hexagonal unknots with edges of fixed length.
J. Cantarella and H. Johnston [Nontrivial embeddings of polygonal intervals and
unknots in 3-space, Journal of Knot Theory and its Ramifications 7 (1998) 1027--1039]
recently showed that there exist "stuck" hexagonal unknots which cannot be reconfigured
to convex hexagons for suitable choices of edge lengths. Here we uncover a new
class of stuck unknotted hexagons, thereby proving that there exist at least five
classes of nontrivial embeddings of the unknot. Furthermore, this new class is
stuck in a stronger way than the class described in the paper mentioned above.
- Given a simple polygon in the plane, a deflation is defined as the inverse of a
flip in the Erdös-Nagy sense. B. Wegner [Partial inflation of closed polygons in the
plane, Beiträge Algebra Geom. 34 (1993) 77--85] conjectured that every
simple polygon admits only a finite number of deflations. We describe a counter-example
to this conjecture by exhibiting a family of polygons on which deflations go on forever.
- Let b(K) denote the minimal number of smaller homothetical copies of a
convex body K contained in Rn, n > 1, covering K. For the class
B of belt bodies, which is dense in the set of all convex bodies (in the
Hausdorff metric), 3 ´ 2n-2 is known to
be an upper bound on b(K) if K is different from a parallelotope. We will
show that (except for all parallelotopes and two particular cases, each
satisfying b(K) = 3 ´ 2n-2 ) within B
this bound can be improved to 5 ´ 2n-3.
-
-
- In Section 1 we introduce Frobenius coordinates in the general setting that
includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories
of Frobenius algebras and augmented Frobenius algebras with some new material
in Section 3. In Section 4 we study the Frobenius structure of an FH-algebra H
[P] and extend two recent theorems in [E]. We obtain two Radford formulas for
the antipode in H and generalize in Section 7 the results on its order in [F].
We study the Frobenius structure on an FH-subalgebra pair in Sections 5 and 6.
In Section 8 we show that the quantum double of H is symmetric and unimodular.
[P] B. Pareigis: On the cohomology of modules over Hopf algebras, J. Algebra 22 (1972) 161-182.
[E] P. Etingof, S. Gelaki: On finite-dimensional semisimple and cosemisimple Hopf
algebras in positive characteristic, Internat. Math. Res. Notices 16 (1998) 851--864.
[F] D. Fischman, S. Montgomery, H.-J. Schneider: Frobenius extensions of subalgebras
of Hopf algebras, Trans. Amer. Math. Soc. 349 (1997) 4857--4895.
- We give a complete description of s-rings. Indeed,
we define and study a more general class of rings with involution that we call
s-semisimple rings. In particular, we prove that for
the left artinian rings with involution, this new definition coincides with the
classical definition of semisimple rings.
- Let k be a commutative ring, H a finitely generated projective Hopf algebra
over k and R a k-algebra which is a left H-module algebra. Assume that for every
H-invariant left ideal I of R and every x + I in (R/I)H there exists
s in RH, such that s-x belongs to I. The main
result of the paper is that R is left FBN if and only if R is left Noetherian
and RH is left FBN. This result generalizes [D, Theorem 8] and [G,
Theorem 2.3].
[D] S. Dascalescu, A. V. Kelarev, B. Torrecillas: FBN Hopf module algebras, Comm.
in Algebra 25 (1997) 3521--3529.
[G] J. J. Garcia, A. Del Rio: Actions of groups on fully bounded Noetherian rings,
Comm. in Algebra 22 (1994) 1495--1505.
- We prove, using the minimalist axiom system for convex geometry proposed by
W. A. Coppel [Foundations of convex geometry, Cambridge Univ. Press (1998)], that,
given n red and n blue points, such that no three are collinear, one can pair each
of the red points with a blue point such that the n segments which have these
paired points as endpoints are disjoint.
- We give an idea of constructing irreducible unitary representations of Lie groups
by using Fedosov deformation quantization in the concrete case of the group Aff(R) of
affine transformations of the real line. By an exact computation of the star-product
and the operator lZ, we show that the resulting representations exhausted
all the irreducible representations of this groups.
- We construct star-products on the co-adjoint orbit of the Lie group Aff(C) of
affine transformations of the complex line and apply them to obtain the irreducible
unitary representations of this group. These results show the effectiveness of the
Fedosov quantization even for groups which are neither nilpotent nor exponential.
Together with the result for the group Aff(R) proved in another paper of the authors
[Quantum half-plane via deformation quantization, Math. QA/9905002, 2 May 1999.], we
thus have a description of quantum MD co-adjoint orbits.
- Let A be an arrangement of n open halfspaces in Rr-1. J. Linhart [The
upper bound conjecture for arrangements of halfspaces, Beiträge Algebra Geom. 35 (1994)
29--35] proved that for r < 6, the numbers of vertices of A contained in at most k
halfspaces are bounded from above by the corresponding numbers of C(n, r), where
C(n, r) is an arrangement realizing the alternating oriented matroid of rank r on n
elements. In the present paper Linhart's result is generalized to faces of dimension
s-1 for 1 ≤ s ≤ 4.
- Given a point M in Euclidean 3-space we show that there exists a polygon without
self-intersection and not containing M such that viewed from M each vertex of the
polygon is hidden behind an edge of the polygon. As an application, we construct a
toric 4-variety which has peculiar compactification properties.
- We study the ring of global sections G (U, O) of an open
subset U = D(I) contained in Spec A, where A is a two-dimensional noetherian ring. The
main concern is to give a geometric criterion when these rings are finitely generated,
in order to correct an invalid statement of P. Schenzel [Flatness and ideal-transforms
of finite type, in: Commutative Algebra (W. Bruns and A. Simis, eds.), 1988].
- Let H be a closed, noncompact subgroup of a simple Lie group G, such that G/H
admits an invariant Lorentz metric. We show that if G = SO(2, n), with n ≥ 3,
then the identity component Ho of H is conjugate to SO(1, n)o.
Also, if G = SO(1, n), with n ≥ 3, then Ho is conjugate to
SO(1, n-1)o.
-
- A entirely new and independent enumeration of the crystallographic spacegroups
is given, based on obtaining the groups as fibrations over the plane crystallographic
groups, when this is possible. For the 35 "irreducible" groups for which it is not, an
independent method is used that has the advantage of elucidating their subgroup
relationships. Each space group is given a short "fibrifold name" which, much like the
orbifold names for two-dimensional groups, while being only specified up to isotopy,
contains enough information to allow the construction of the group from the name.
- A pseudo-line of a real plane curve C is a global real branch of C(R) that is not
homologically trivial in P2(R). A geometrically integral real plane curve C
of degree d has at most d-2 pseudo-lines, provided that C is not a real projective line.
Let C be a real plane curve of degree d having exactly d-2 pseudo-lines. Suppose that
the genus of the normalization of C is equal to d-2. We show that each pseudo-line of C
contains exactly 3 inflection points. This generalizes the fact that a nonsingular real
cubic has exactly 3 real inflection points.
- We introduce for convex domains of constant width a measure of asymmetry and show
that the most asymmetric domains are Reuleaux triangles.
- Chen initiated the study of the tensor product immersion of two immersions
of a given Riemannian manifold [Ch]. Inspired by Chen's definition, Decruyenaere,
Dillen, Verstraelen and Vrancken [D] studied the tensor product of two immersions
of, in general, different manifolds; under certain conditions, this realizes an
immersion of the product manifold. In [M] tensor product surfaces of Euclidean
plane curves were investigated.
We deal here with tensor product surfaces of a Euclidean space curve and a Euclidean
plane curve. We classify the minimal, totally real and slant such surfaces, respectively.
[Ch] B. Y. Chen: Differential geometry of semiring of immersions, I: General theory,
Bull. Inst. Math. Acad. Sinica 21 (1993) 1--34.
[D] F. Decruyenaere, F. Dillen, L. Verstraelen, L. Vrancken: The semiring of immersions
of manifolds, Beiträge Algebra Geom. 34 (1993) 209--215.
[M] I. Mihai, R. Rosca, L. Verstraelen, L. Vrancken: Tensor product surfaces of Euclidean
planar curves, Rend. Sem. Mat. Messina 3 (1994/1995) 173--184.
- In a recent paper [R] a family F = {ck | k real} of conics was
presented. The conics ck are obtained by offsetting from a given
conic c0 with proportional distance functions k d(t).
We investigate certain properties of F and give the correct version of a result
claimed in [R]: The distance function is unique (up to a constant factor) only
if c0 is not a parabola.
Furthermore we deal with the surfaces that are obtained by giving each conic
ck from F the z-coordinate l k with a
fixed real l. We find special metric properties of
these surfaces and show that they already appeared in other context.
[R] F. Granero Rodriguez, F. Jimenez Hernandez, J. J. Doria Iriarte: Constructing
a family of conics by curvature-depending offsetting from a given conic, Comput.
Aided Geom. Design 16 (1999) 793--815.
- The following conjecture is proved: If G is a subgroup of the permutation
group Sn and M is a 2-dimensional real manifold, then Mn/G
is a manifold if and only if G = Sm1 ´
Sm2 ´ ... ´
Smr where Sm1, ..., Smr
are permutation groups of partition of {1, 2, ... ,n} into r subsets with cardinalities
m1, ..., mr, and Mn is the topological product of n
copies of M.
- All rings are commutative with identity and all modules are unitary. In this
note we give some properties of a finite collection of submodules such that the sum
of any two distinct members is multiplication, generalizing those which characterize
arithmetical rings. Using these properties we are able to give a concise proof of
Patrick Smith's theorem stating conditions ensuring that the sum and intersection of
a finite collection of multiplication submodules is a multiplication module. We give
necessary and sufficient conditions for the intersection of a collection (not
necessarily finite) of multiplication modules to be a multiplication module,
generalizing Smith's result. We also give sufficient conditions on the sum and
intersection of a collection (not necessarily finite) for them to be multiplication.
We apply D. D. Anderson's new characterization of multiplication modules to investigate
the residual of multiplication modules.
- We prove several improvements and analogs of Leon Bankoff's theorem on
asymmetric propellers from directly similar triangles.
- Geodesic (totally geodesic, D-geodesic, D'-geodesic and mixed geodesic)
CR-lightlike submanifolds in indefinite Kaehler manifold are investigated.
Some necessary and sufficient conditions on totally geodesic, D-geodesic,
D'-geodesic and mixed geodesic CR-lightlike submanifolds are obtained. We
find geometric properties of CR-lightlike submanifolds of an indefinite
Kaehler manifold.
- Some curves with low geodesic curvature on a connected, simply connected
riemannian manifold with pinched negative curvature happen to be weakly convex.
It results that on a surface with 1/4-pinched negative curvature, the bisectors
and Dirichlet domains of discrete groups of isometries with a finite number of
faces are weakly convex.
- Codes with minimum distance at least d and covering radius at most
d-1 are considered. The minimal cardinality of
such codes is investigated. Herewith, their connection to covering problems
is applied and a new construction theorem is given. Additionally, a new
lower bound for the covering problem is proved. A necessary condition on
an existence problem is presented by using a multiple covering of the
farthest-off points.
Editorial Remark: This article replaces the version published by the same
author in Beiträge zur Algebra und Geometrie 41, No. 2, 469-478 (2000). Due
to an error in the files transmission the publication of that version was not
based on the final TEX-file for the article. Hence some improvements suggested
by the referee were missing.