201 research outputs found
The partially alternating ternary sum in an associative dialgebra
The alternating ternary sum in an associative algebra, , gives rise to the partially alternating ternary sum in an
associative dialgebra with products and by making the
argument the center of each term: . We use computer algebra to determine the polynomial identities in
degree satisfied by this new trilinear operation. In degrees 3 and 5 we
obtain and ; these identities define a new variety of partially alternating ternary
algebras. We show that there is a 49-dimensional space of multilinear
identities in degree 7, and we find equivalent nonlinear identities. We use the
representation theory of the symmetric group to show that there are no new
identities in degree 9.Comment: 14 page
ifCNV: A novel isolation-forest-based package to detect copy-number variations from various targeted NGS datasets
Copy-number variations (CNVs) are an essential component of genetic variation distributed across large parts of the human genome. CNV detection from next-generation sequencing data and artificial intelligence algorithms have progressed in recent years. However, only a few tools have taken advantage of machine-learning algorithms for CNV detection, and none propose using artificial intelligence to automatically detect probable CNV-positive samples. The most developed approach is to use a reference or normal dataset to compare with the samples of interest, and it is well known that selecting appropriate normal samples represents a challenging task that dramatically influences the precision of results in all CNV-detecting tools. With careful consideration of these issues, we propose here ifCNV, a new software based on isolation forests that creates its own reference, available in R and python with customizable parameters. ifCNV combines artificial intelligence using two isolation forests and a comprehensive scoring method to faithfully detect CNVs among various samples. It was validated using targeted next-generation sequencing (NGS) datasets from diverse origins (capture and amplicon, germline and somatic), and it exhibits high sensitivity, specificity, and accuracy. ifCNV is a publicly available open-source software (https://github.com/SimCab-CHU/ifCNV) that allows the detection of CNVs in many clinical situations
Hom-Lie color algebra structures
This paper introduces the notion of Hom-Lie color algebra, which is a natural
general- ization of Hom-Lie (super)algebras. Hom-Lie color algebras include
also as special cases Lie (super) algebras and Lie color algebras. We study the
homomorphism relation of Hom-Lie color algebras, and construct new algebras of
such kind by a \sigma-twist. Hom-Lie color admissible algebras are also defined
and investigated. They are finally classified via G-Hom-associative color
algebras, where G is a subgroup of the symmetric group S_3.Comment: 16 page
Invariants of solvable rigid Lie algebras up to dimension 8
The invariants of all complex solvable rigid Lie algebras up to dimension
eight are computed. Moreover we show, for rank one solvable algebras, some
criteria to deduce to non-existence of non-trivial invariants or the existence
of fundamental sets of invariants formed by rational functions of the Casimir
invariants of the associated nilradical.Comment: 16 pages, 7 table
On the structure of maximal solvable extensions and of Levi extensions of nilpotent algebras
We establish an improved upper estimate on dimension of any solvable algebra
s with its nilradical isomorphic to a given nilpotent Lie algebra n. Next we
consider Levi decomposable algebras with a given nilradical n and investigate
restrictions on possible Levi factors originating from the structure of
characteristic ideals of n. We present a new perspective on Turkowski's
classification of Levi decomposable algebras up to dimension 9.Comment: 21 pages; major revision - one section added, another erased;
author's version of the published pape
Quasi-classical Lie algebras and their contractions
After classifying indecomposable quasi-classical Lie algebras in low
dimension, and showing the existence of non-reductive stable quasi-classical
Lie algebras, we focus on the problem of obtaining sufficient conditions for a
quasi-classical Lie algebras to be the contraction of another quasi-classical
algebra. It is illustrated how this allows to recover the Yang-Mills equations
of a contraction by a limiting process, and how the contractions of an algebra
may generate a parameterized families of Lagrangians for pairwise
non-isomorphic Lie algebras.Comment: 17 pages, 2 Table
All solvable extensions of a class of nilpotent Lie algebras of dimension n and degree of nilpotency n-1
We construct all solvable Lie algebras with a specific n-dimensional
nilradical n_(n,2) (of degree of nilpotency (n-1) and with an (n-2)-dimensional
maximal Abelian ideal). We find that for given n such a solvable algebra is
unique up to isomorphisms. Using the method of moving frames we construct a
basis for the Casimir invariants of the nilradical n_(n,2). We also construct a
basis for the generalized Casimir invariants of its solvable extension s_(n+1)
consisting entirely of rational functions of the chosen invariants of the
nilradical.Comment: 19 pages; added references, changes mainly in introduction and
conclusions, typos corrected; submitted to J. Phys. A, version to be
publishe
Contractions of Low-Dimensional Lie Algebras
Theoretical background of continuous contractions of finite-dimensional Lie
algebras is rigorously formulated and developed. In particular, known necessary
criteria of contractions are collected and new criteria are proposed. A number
of requisite invariant and semi-invariant quantities are calculated for wide
classes of Lie algebras including all low-dimensional Lie algebras.
An algorithm that allows one to handle one-parametric contractions is
presented and applied to low-dimensional Lie algebras. As a result, all
one-parametric continuous contractions for the both complex and real Lie
algebras of dimensions not greater than four are constructed with intensive
usage of necessary criteria of contractions and with studying correspondence
between real and complex cases.
Levels and co-levels of low-dimensional Lie algebras are discussed in detail.
Properties of multi-parametric and repeated contractions are also investigated.Comment: 47 pages, 4 figures, revised versio
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