Pasha Zusmanovich

Many sequence variants affecting diversity of adult human height

Adult human height is one of the classical complex human traits. We searched for sequence variants that affect height by scanning the genomes of 25,174 Icelanders, 2,876 Dutch, 1,770 European Americans and 1,148 African Americans. We then combined these results with previously published results from the Diabetes Genetics Initiative on 3,024 Scandinavians and tested a selected subset of SNPs in 5,517 Danes. We identified 27 regions of the genome with one or more sequence variants showing significant association with height. The estimated effects per allele of these variants ranged between 0.3 and 0.6 cm and, taken together, they explain around 3.7% of the population variation in height. The genes neighboring the identified loci cluster in biological processes related to skeletal development and mitosis. Association to three previously reported loci are replicated in our analyses, and the strongest association was with SNPs in the ZBTB38 gene.

Detection of sharing by descent, long-range phasing and haplotype imputation

Uncertainty about the phase of strings of SNPs creates complications in genetic analysis, although methods have been developed for phasing population-based samples. However, these methods can only phase a small number of SNPs effectively and become unreliable when applied to SNPs spanning many linkage disequilibrium (LD) blocks. Here we show how to phase more than 1,000 SNPs simultaneously for a large fraction of the 35,528 Icelanders genotyped by Illumina chips. Moreover, haplotypes that are identical by descent (IBD) between close and distant relatives, for example, those separated by ten meioses or more, can often be reliably detected. This method is particularly powerful in studies of the inheritance of recurrent mutations and fine-scale recombinations in large sample sets. A further extension of the method allows us to impute long haplotypes for individuals who are not genotyped.

The Alternative Operad Is Not Koszul

In the online compendium [Loday 07], it is asked whether the alternative operad is Koszul. The purpose of this note is to demonstrate that the answer to this question is negative. In doing so, we have been aided by the programs Albert and PARI/GP.

Low-dimensional cohomology of current lie algebras and analogs of the Riemann tensor for loop manifolds

Abstract We obtain formulas for the first and second cohomology groups of a general current Lie algebra with coefficients in the “current” module, and apply them to compute structure functions for manifolds of loops with values in compact Hermitian symmetric spaces.

Central extensions of current algebras

The second cohomology group of Lie algebras of kind L⊗U with trivial coefficients is investigated, where L admits a decomposition with onedimensional root spaces and U is an arbitrary associative commutative algebra with unit. This paper gives a unification of some recent results of C. Kassel and A. Haddi and provides a determination of central extensions of certain modular semisimple Lie algebras

INVARIANTS OF LIE ALGEBRAS EXTENDED OVER COMMUTATIVE ALGEBRAS WITHOUT UNIT

We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms, and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac–Moody algebras.

A Compendium of Lie Structures on Tensor Products

It is demonstrated how a simple linear-algebraic technique used earlier to compute the low-degree cohomology of current Lie algebras, can be utilized to compute other kinds of structures on such Lie algebras, and discuss further generalizations, applications, and related questions. While doing so, seemingly diverse topics are touched upon such as associative algebras of infinite representation type, Hom-Lie structures, Poisson brackets of hydrodynamic type, Novikov algebras, simple Lie algebras in small characteristics, and Koszul dual operads.

Commutative 2-cocycles on Lie algebras☆

Abstract On Lie algebras, we study commutative 2-cocycles, i.e., symmetric bilinear forms satisfying the usual cocycle equation. We note their relationship with antiderivations and compute them for some classes of Lie algebras, including finite-dimensional semisimple, current and Kac–Moody algebras.

Deformations of W1(n)⊗A and modular semisimple Lie algebras with a solvable maximal subalgebra

Abstract In one of his last papers, Boris Weisfeiler proved that if modular semisimple Lie algebra possesses a solvable maximal subalgebra which defines in it a long filtration, then associated graded algebra is isomorphic to one constructed from the Zassenhaus algebra tensored with the divided powers algebra. We completely determine such class of algebras, calculating in process low-dimensional cohomology groups of Zassenhaus algebra tensored with any associative commutative algebra.

A Lie algebra that can be written as a sum of two nilpotent subalgebras is solvable

This is an old paper put here for archeological purposes. It is proved that a finite-dimensional Lie algebra over a field of characteristic p>5, that can be written as a vector space (not necessarily direct) sum of two nilpotent subalgebras, is solvable. The same result (but covering also the cases of low characteristics) was established independently by V. Panyukov (Russ. Math. Surv. 45 (1990), N4, 181-182), and the homological methods utilized in the proof were developed later in arXiv:math/0204004. Many inaccuracies in the English translation are corrected, otherwise the text is identical to the published version.