GatevaIvanova, T.
Binomial skew polynomial rings, ArtinSchelter regularity,
and binomial solutions of the YangBaxter equation
(pp. 431470)
A B S T R A C T S
FIBRÉE VECTOLIELS SEMISTABLES SUR UNE COURBE DE GENDRE DEUX ET
ASSOCIATION DES POINTS DANS L'ESPACE PROJECTIF
Cristian Anghel
Cristian.Anghel@imar.ro
2000 Mathematics Subject Classification:
14D20, 14J60.
Key words:
courbe projective, fibres semistables,
association des points.
The aim of this paper is to prove that a certain involution on
the moduli space of stable bundles on a curve of genus two, can be viewed
as a geometric operation on the corresponding set of points in projective
space.
Z_{2}GRADED POLYNOMIAL IDENTITIES FOR
SUPERALGEBRAS OF BLOCKTRIANGULAR MATRICES
Onofrio M. Di Vincenzo
divincenzo@dm.uniba.it
2000 Mathematics Subject Classification:
Primary 16R50, Secondary 16W55.
Key words:
Graded cocharacter sequence,
triangular matrices, superalgebra, polynomial identity.
We present some results about the Z_{2}graded polynomial identities of
blocktriangular matrix superalgebras
In particular, we describe
conditions for the
T_{2}ideal of a such superalgebra to be factorable as the product
T_{2}(A)T_{2}(B). Moreover, we give formulas for computing the
sequence of the graded cocharacters of
R in some interesting case.
AUTOMORPHISMS OF THE PLANAR TREE POWER SERIES ALGEBRA
AND THE NONASSOCIATIVE LOGARITHM
L. Gerritzen
lothar.gerritzen@ruhrunibochum.de
2000 Mathematics Subject Classification:
17A50, 05C05.
Key words:
planar rooted tree, planar formal power
series, nonassociative exponential series, substitution
endomorphisms and
automorphisms for planar power series, Hopf algebra.
In this note we present the formula for
the coefficients of the substitution
series f(g(x)) of planar tree power series
g(x) into f(x). The coefficient c_{T}(f(g(x)))
relative to a finite planar reduced rooted tree T is the sum

å
 c_{S}(f) c_{(TIn(S))} (g) 

which is extended over all open subtrees S of T. In this
expression c_{S}(f) is the coefficient of f with respect to S and
TIn(S) is the complement of the inner vertices of S in T while
c_{(TIn(S))}(g) is the product of the coefficients of g relative to
the connected components of the forest TIn(S).
Also we give the formula for the compositional inverse of a planar tree
power series g in terms of coefficients of g. It is shown that the
generic nonassociative exponential series EXP has an inverse LOG.
Results about coefficients of LOG are presented.
The group G = Aut(K {{x}}_{¥}) of automorphisms of the
algebra K{{x}}_{¥} of planar tree power series
in a single variable x over a field K, see [6], is
an affine group. The finite planar reduced rooted trees T are
indexing a set of variables y_{T} of K[G] and the composition
in G induces a coproduct
for which
D(y_{T}) = 
å
 y_{S} Äy_{(TIn(S))} 

where the summation is as in the formula above.
It is an interesting question how
(K[G], D) which is indeed a Hopf algebra is related
to the Hopf algebra of Kreimer, see [12], in which the
coproduct on a variable y_{T} is defined by constructing
admissible cuts on the set of edges of the planar rooted
tree T while the above D(y_{T}) can be described
by admissible cuts on the set of vertices of T.
ON THE FACTORIZATION OF THE POINCARÉ POLYNOMIAL: A SURVEY
Ersan Akyildiz
ersan@metu.edu.tr
2000 Mathematics Subject Classification:
13P05, 14M15, 14M17, 14L30.
Key words:
Factorization, Poincaré polynomial, Algebraic homogeneous spaces.
Factorization is an important and very difficult problem in mathematics.
Finding prime factors of a given positive integer n, or finding the
roots of the polynomials in the complex plane
are some of the important problems not only in algorithmic
mathematics but also in cryptography. For a given smooth mdimensional
real manifold X, one
has the associated Poincaré polynomial
P (X,t) = å_{i = 0}^{m} b_{i} (X) t^{i} of X, where
b_{i} (X) = dim_{R} H^{i} (X ; R)
is the ith Betti number of X. It is clear that the factorization
of P (X,t) as series over the complex numbers C will carry lots
of information about the topological and geometric invariants of X. This
is possibly why a factorization of even such
a special polynomial P (X,t) is expected to be hard. However we can still
search for algorithms to write
P (X,t) as a product of some nontrivial power series. One notes that the
factorizations
P (P^{n}, t^{1/2}) = 
n å
i = 0

t^{i} = 
1t^{n+1}
1t

, 

P (G L_{n} /B, t^{1/2}) = 
n Õ
i = 1


1t^{i}
1t



are examples of such kind. Here P^{n} is the ndimensional
complex projective space and GL_{n} / B is the complex full flag
manifold associated to the upper triangular matrices B in the
invertible complex matrices GL_{n}. The aim of this survey article is to
give first a direct selfcontained elementary algebraic treatment of the
problem and then provide examples of nonsingular complex projective
varieties X so that the Calgebra H^{*} (X; C)
fits into this treatment. This will allow us to factorize P (X,t) as
above for such a variety X. These varieties X will include all the
homogeneous spaces G/P, their smooth Schubert subvarieties
and more. It is also interesting to note that in this approach, one
can read off smoothness of a Schubert variety from the factorization
of its Poincaré polynomial, which
is discussed in Section 2 and 3.
DICKSON POLYNOMIALS THAT ARE PERMUTATIONS
Mihai Cipu
mihai.cipu@imar.ro
2000 Mathematics Subject Classification:
11T06, 13P10.
Key words:
Dickson polynomial, Gröbner basis, permutation
polynomial.
A theorem of S.D. Cohen gives a characterization for Dickson polynomials
of the second kind that permutes the elements of a finite field of
cardinality the square of the characteristic. Here, a different proof is
presented for this result.
EQUIMULTIPLE LOCUS
OF EMBEDDED ALGEBROID SURFACES
AND BLOWINGUP IN CHARACTERISTIC ZERO
R. PiedraSánchez
piedra@algebra.us.es
J. M. Tornero
tornero@algebra.us.es
2000 Mathematics Subject Classification:
14B05, 32S25.
Key words:
Resolution of surface singularities, blowingup,
equimultiple locust.
The smooth equimultiple locus of embedded algebroid
surfaces appears naturally in many resolution processes, both
classical and modern. In this paper we explore how it changes by
blowingup.
COMPLEX HYPERBOLIC SURFACES OF ABELIAN TYPE
R.P. Holzapfel
holzapfl@mathematik.huberlin.de
2000 Mathematics Subject Classification:
11G15, 11G18, 14H52, 14J25, 32L07.
Key words:
algebraic curve, elliptic curve, algebraic surface, Shimura
variety, arithmetic group, Picard modular group, Gauß numbers,
congruence numbers, KählerEinstein metric, negative constant
curvature, unit ball.
We call a complex (quasiprojective) surface of hyperbolic type,
iff  after removing finitely many points and/or curves  the universal
cover
is the complex twodimensional unit ball. We characterize abelian surfaces
which have a birational transform of hyperbolic type by the existence of a
reduced
divisor with only elliptic curve components and maximal singularity
rate (equal to 4).
We discover a Picard modular surface of Gauß numbers of bielliptic type
connected with the rational cuboid problem. This paper is also necessary to
understand
new constructions of Picard modular forms of 3divisible weights by
special abelian theta functions.
INVOLUTION MATRIX ALGEBRAS  IDENTITIES AND GROWTH
Tsetska Grigorova Rashkova
tcetcka@ami.ru.acad.bg
2000 Mathematics Subject Classification:
16R50, 16R10.
Key words:
involution, polynomial identities, symmetric and
skewsymmetric variables, Bergman type polynomials, characters, Hilbert series,
codimensions, growth
The paper is a survey on involutions
(antiautomorphisms of order two) of different
kinds. Starting with the first systematic
investigations on involutions of central
simple algebras due to Albert the author
emphasizes on their basic properties, the
conditions on their existence and their
correspondence with structural characteristics
of the algebras.
Focusing on matrix algebras a complete
description of involutions of the first kind
on M_{n}(F) is given. The full correspondence
between an involution of any kind for
an arbitrary central simple algebra A
over a field F of characteristic 0 and an
involution on M_{n}(A) specially defined is studied.
The research mainly in the last 40 years concerning
the basic properties of involutions applied to
identities for matrix algebras is reviewed starting with the
works of Amitsur, Rowen and including the newest results on
the topic. The cocharactes, codimensions and growth of algebras
with involutions are considered as well.
A SMOOTH FOURDIMENSIONAL GHILBERT SCHEME
Magda Sebestean
sebes@math.jussieu.fr
2000 Mathematics Subject Classification:
14C05, 14L30, 14E15, 14J35.
Key words:
quotient singularities, crepant resolutions, toric varieties,
$G$Hilbert scheme, $G$graph.
When the cyclic group G of order 15 acts with some specific weights on
affine fourdimensional space, the GHilbert scheme is a crepant
resolution of the quotient A^{4}/G. We give an explicit description
of this resolution using Ggraphs.
COHOMOLOGY OF THE GHILBERT SCHEME FOR
^{1}/_{r}(1,1,r1)
Oskar Kedzierski
oskar@impan.gov.pl
2000 Mathematics Subject Classification:
Primary 14E15; Secondary 14C05,14L30.
Key words:
McKay correspondence;
resolutions of terminal quotient
singularities;$G$Hilbert scheme.
In this note we attempt to generalize a few statements drawn from the 3dimensional
McKay correspondence to the case of a cyclic group not in SL(3,C). We
construct
a smooth,
discrepant resolution of the cyclic, terminal
quotient singularity of type ^{1}/_{r} (1,1,r1), which turns out to
be isomorphic to Nakamura's GHilbert
scheme. Moreover we explicitly describe tautological
bundles and use them to construct a dual basis to the integral cohomology on
the resolution.
MINIMAL CODEWORDS IN LINEAR CODES
Yuri Borissov
youri@moi.math.bas.bg
Nickolai Manev
nlmanev@moi.math.bas.bg
2000 Mathematics Subject Classification:
94B05, 94B15.
Key words:
minimal codewords; cyclic codes; binary ReedMuller code.
Cyclic binary codes C of block length n = 2^{m}1 and
generator polynomial g(x) = m_{1}(x)m_{2s+1}(x), (s,m) = 1,
are considered. The cardinalities of the sets of minimal
codewords of weights 10 and 11 in codes C and of
weight 12 in their extended codes ^(C) are determined.
The weight distributions of minimal codewords in the binary
ReedMuller codes RM(3,6) and RM(3,7) are determined.
The applied method enables codes with
larger parameters to be attacked.
KNESER AND HEREDITARILY KNESER SUBGROUPS
OF A PROFINITE GROUP
Serban A. Basarab
Serban.Basarab@imar.ro
2000 Mathematics Subject Classification:
20E18, 12G05, 12F10, 12F99.
Key words:
Profinite group,
Cogalois group of a field extension, Cogalois theory,
continuous 1cocycle, Kneser group of cocycles,
Cogalois group of cocycles, radical subgroup,
hereditarily radical subgroup, Kneser subgroup,
almost Kneser subgroup, hereditarily Kneser subgroup,
spectral space, coherent map.
Given a profinite group G
acting continuously on a
discrete quasicyclic group A, certain classes of closed
subgroups of G
(radical, hereditarily radical, Kneser,
almost Kneser, and hereditarily Kneser) having natural field
theoretic interpretations are defined and investigated.
One proves that the hereditarily Kneser subgroups of
G form a closed subspace of the irreducible
spectral space of all closed subgroups of G, and
a hereditarily Kneser criterion for hereditarily
radical subgroups is provided.
LINEARLY NORMAL CURVES IN P^{n}
Ovidiu Pasarescu
ovidiu.pasarescu@imar.ro
2000 Mathematics Subject Classification:
14H45, 14H50, 14J26.
Key words:
linearly normal curves, rational surfaces.
We construct linearly normal curves covering a big range from
P^{n}, n ³ 6 (Theorems 1.7, 1.9). The problem of existence
of such algebraic curves in P^{3} has been solved in [4],
and extended to P^{4} and P^{5} in [10]. In both these
papers is used the idea appearing in [4] and consisting in
adding hyperplane sections to the curves constructed in [6]
(for P^{3}) and [15, 11] (for P^{4} and P^{5}) on some
special surfaces. In the present paper we apply the same idea to
the curves lying on some rational surfaces from P^{n}, constructed
in [12, 3, 2] (see [13, 14] also).
HENSELIAN DISCRETE VALUED FIELDS ADMITTING
ONEDIMENSIONAL LOCAL CLASS FIELD THEORY
I. D. Chipchakov
chipchak@math.bas.bg
2000 Mathematics Subject Classification:
11S31 12E15 12F10 12J20.
Key words:
field admitting (onedimensional) local class field
theory, strictly primarily quasilocal field, Henselian valued field, Brauer
group, character group, norm group, Galois extension, regular group formation.
This paper gives a characterization of Henselian discrete valued
fields whose finite abelian extensions are uniquely determined by their norm
groups and related essentially in the same way as in the classical local class
field theory. It determines the structure of the Brauer groups and character
groups of Henselian discrete valued strictly
primary quasilocal (or PQL) fields, and thereby,
describes the forms of the local reciprocity law for such fields. It shows
that, in contrast to the special cases of local fields or strictly
PQLfields algebraic over a given global field, the norm groups of
finite separable extensions of the considered fields are not necessarily equal
to norm groups of finite Galois extensions with Galois groups of easily
accessible structure.
INVARIANTS OF UNIPOTENT TRANSFORMATIONS
ACTING ON NOETHERIAN RELATIVELY FREE ALGEBRAS
Vesselin Drensky
drensky@math.bas.bg
2000 Mathematics Subject Classification:
16R10, 16R30.
Key words:
Noncommutative invariant theory;
unipotent transformations;
relatively free algebras.
The classical theorem of Weitzenböck states that the algebra
of invariants K[X]^{g}
of a single unipotent transformation g Î GL_{m}(K)
acting on the polynomial algebra K[X] = K[x_{1},¼,x_{m}]
over a field K of characteristic 0
is finitely generated. This algebra coincides with the algebra of constants
K[X]^{d} of a linear locally nilpotent derivation d of K[X].
Recently the author and C. K. Gupta have started the study
of the algebra of invariants F_{m}(V)^{g}
where F_{m}(V) is the relatively free algebra of rank m in a variety
V of associative algebras. They have shown that
F_{m}(V)^{g} is not finitely generated if V contains
the algebra UT_{2}(K) of 2×2 upper triangular matrices
(and g ¹ 1). The main result of the present paper is that
the algebra F_{m}(V)^{g} is finitely generated if
and only if the variety V does not contain the algebra UT_{2}(K).
As a byproduct of the proof we have established also the finite generation
of the algebra of invariants T_{nm}^{g} where T_{nm} is the mixed trace algebra
generated by m generic n×n matrices and the traces of their products.
REMARKS ON THE NAGATA CONJECTURE
Beata StrycharzSzemberg
bm0061@uniessen.de
Tomasz Szemberg
mat905@uniessen.de
2000 Mathematics Subject Classification:
14C20, 14E25, 14J26.
Key words:
Nagata Conjecture,
linear series, Seshadri constants,
HarbourneHirschowitz Conjecture, big divisors.
The famous Nagata Conjecture predicts the lowest degree of a plane curve
passing
with prescribed multiplicities through given points in general position. We
explain how
this conjecture extends naturally via multiple point Seshadri constants to
ample line
bundles on arbitrary surfaces. We show that if there exist curves of
unpredictable low
degree, then they must have equal multiplicities in all but possibly one of
the given points.
We use this restriction in order to obtain lower bounds on multiple point
Seshadri
constants on a surface. We discuss also briefly a seemingly new point of
view on the
Nagata Conjecture via the bigness of the involved linear series.
BINOMIAL SKEW POLYNOMIAL RINGS, ARTINSCHELTER REGULARITY,
AND BINOMIAL SOLUTIONS OF THE YANGBAXTER EQUATION
Tatiana GatevaIvanova
tatianagateva@yahoo.com,
tatyana@aubg.bg,
tatiana@math.bas.bg
2000 Mathematics Subject Classification:
Primary 81R50, 16W50, 16S36, 16S37.
Key words:
YangBaxter equation,
Quadratic algebras, ArtinSchelter regular rings, Quantum groups.
Let k be a field and X be a set of n elements. We introduce
and study a class of quadratic kalgebras
called quantum binomial algebras. Our main result shows
that such an
algebra A defines a solution of the classical YangBaxter
equation (YBE), if and only if its Koszul dual A^{!} is
Frobenius of dimension n, with a regular socle and for
each x,y Î X an equality of the type xyy = azzt, where
a Î k \{0}, and z,t Î
X is satisfied in
A. We prove the equivalence of the notions a binomial skew
polynomial ring and a binomial solution of YBE. This
implies that the YangBaxter algebra of such a solution is of
PoincaréBirkhoffWitt type, and possesses a number of other
nice properties such as being Koszul, Noetherian, and an
ArtinSchelter regular domain.
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