171 research outputs found
Asymptotic Behaviour of Colength of Varieties of Lie Algebras
This project was partially supported by RFBR, grants 99-01-00233, 98-01-01020 and
00-15-96128.We study the asymptotic behaviour of numerical characteristics
of polynomial identities of Lie algebras over a field of characteristic 0. In
particular we investigate the colength for the cocharacters of polynilpotent
varieties of Lie algebras. We prove that there exist polynilpotent Lie varieties
with exponential and overexponential colength growth. We give the exact
asymptotics for the colength of a product of two nilpotent varieties
The Variety of Leibniz Algebras Defined by the Identity x(y(zt)) ≡ 0
2000 Mathematics Subject Classification: Primary: 17A32; Secondary: 16R10, 16P99,
17B01, 17B30, 20C30Let F be a field of characteristic zero. In this paper we study
the variety of Leibniz algebras 3N determined by the identity x(y(zt)) ≡ 0.
The algebras of this variety are left nilpotent of class not more than 3. We
give a complete description of the vector space of multilinear identities in
the language of representation theory of the symmetric group Sn
and Young
diagrams. We also show that the variety
3N is generated by an abelian
extension of the Heisenberg Lie algebra. It has turned out that
3N has many
properties which are similar to the properties of the variety of the abelian-by-nilpotent of class 2 Lie algebras. It has overexponential growth of the
codimension sequence and subexponential growth of the colength sequence.This project was partially supported by RFBR, grants 01-01-00728, 02-01-00219 and
UR 04.01.036
Hilbert schemes of 8 points
The Hilbert scheme H^d_n of n points in A^d contains an irreducible component
R^d_n which generically represents n distinct points in A^d. We show that when
n is at most 8, the Hilbert scheme H^d_n is reducible if and only if n = 8 and
d >= 4. In the simplest case of reducibility, the component R^4_8 \subset H^4_8
is defined by a single explicit equation which serves as a criterion for
deciding whether a given ideal is a limit of distinct points.
To understand the components of the Hilbert scheme, we study the closed
subschemes of H_n^d which parametrize those ideals which are homogeneous and
have a fixed Hilbert function. These subschemes are a special case of
multigraded Hilbert schemes, and we describe their components when the colength
is at most 8. In particular, we show that the scheme corresponding to the
Hilbert function (1,3,2,1) is the minimal reducible example.Comment: 28 pages; Rewrote introduction and reorganized parts of the paper,
some minor errors have been fixe
Some Numerical Invariants of Multilinear Identities
2010 Mathematics Subject Classification: Primary 16R10, 16A30, 16A50, 17B01, 17C05, 17D05, 16P90, 17A, 17D.We consider non-necessarily associative algebras over a field of characteristic zero and their polynomial identities. Here we describe most of the results obtained in recent years on two numerical sequences that can be attached to the multilinear identities satisfied by an algebra: the sequence of codimensions and the sequence of colengths
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