2,095 research outputs found
Finitely generated algebras with involution and their identities
Associative algebras with involution over a field of zero characteristic are
considered. It is proved that in this case for any finitely generated
associative algebra with involution there exists a finite dimensional algebra
with involution which satisfies exactly the same identities with involution
Finite basis problem for identities with involution
We consider associative algebras with involution over a field of
characteristic zero. We proved that any algebra with involution satisfies the
same identities with involution as the Grassmann envelope of some finite
dimensional -graded algebra with graded involution. As a consequence we
obtain the positive solution of the Specht problem for identities with
involution: any associative algebra with involution over a field of
characteristic zero has a finite basis of identities with involution. These
results are analogs of theorems of A.R.Kemer for ordinary identities
Shimura curve computations via K3 surfaces of Neron-Severi rank at least 19
It is known that K3 surfaces S whose Picard number rho (= rank of the
Neron-Severi group of S) is at least 19 are parametrized by modular curves X,
and these modular curves X include various Shimura modular curves associated
with congruence subgroups of quaternion algebras over Q. In a family of such K3
surfaces, a surface has rho=20 if and only if it corresponds to a CM point on
X. We use this to compute equations for Shimura curves, natural maps between
them, and CM coordinates well beyond what could be done by working with the
curves directly as we did in ``Shimura Curve Computations'' (1998) =
Comment: 16 pages (1 figure drawn with the LaTeX picture environment); To
appear in the proceedings of ANTS-VIII, Banff, May 200
Identities of finitely generated graded algebras with involution
We consider associative algebras with involution graded by a finite abelian
group G over a field of characteristic zero. Suppose that the involution is
compatible with the grading. We represent conditions permitting
PI-representability of such algebras. Particularly, it is proved that a
finitely generated (Z/qZ)-graded associative PI-algebra with involution
satisfies exactly the same graded identities with involution as some finite
dimensional (Z/qZ)-graded algebra with involution for any prime q or q = 4.
This is an analogue of the theorem of A.Kemer for ordinary identities, and an
extension of the result of the author for identities with involution. The
similar results were proved also recentely for graded identities
Families of explicitly isogenous Jacobians of variable-separated curves
We construct six infinite series of families of pairs of curves (X,Y) of
arbitrarily high genus, defined over number fields, together with an explicit
isogeny from the Jacobian of X to the Jacobian of Y splitting multiplication by
2, 3, or 4. For each family, we compute the isomorphism type of the isogeny
kernel and the dimension of the image of the family in the appropriate moduli
space. The families are derived from Cassou--Nogu\`es and Couveignes' explicit
classification of pairs (f,g) of polynomials such that f(x_1) - g(x_2) is
reducible
Toric anti-self-dual Einstein metrics via complex geometry
Using the twistor correspondence, we give a classification of toric
anti-self-dual Einstein metrics: each such metric is essentially determined by
an odd holomorphic function. This explains how the Einstein metrics fit into
the classification of general toric anti-self-dual metrics given in an earlier
paper (math.DG/0602423). The results complement the work of Calderbank-Pedersen
(math.DG/0105263), who describe where the Einstein metrics appear amongst the
Joyce spaces, leading to a different classification. Taking the twistor
transform of our result gives a new proof of their theorem.Comment: v2. Published version. Additional references. 14 page
Crossings, Motzkin paths and Moments
Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain
-analogues of Laguerre and Charlier polynomials. The moments of these
orthogonal polynomials have combinatorial models in terms of crossings in
permutations and set partitions. The aim of this article is to prove simple
formulas for the moments of the -Laguerre and the -Charlier polynomials,
in the style of the Touchard-Riordan formula (which gives the moments of some
-Hermite polynomials, and also the distribution of crossings in matchings).
Our method mainly consists in the enumeration of weighted Motzkin paths, which
are naturally associated with the moments. Some steps are bijective, in
particular we describe a decomposition of paths which generalises a previous
construction of Penaud for the case of the Touchard-Riordan formula. There are
also some non-bijective steps using basic hypergeometric series, and continued
fractions or, alternatively, functional equations.Comment: 21 page
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