On near and the nearest correlation matrix
We present an elementary heuristic reasoning based on Arnold's theory of versal deformations in support of a straightforward algorithm for finding a correlation matrix near a given symmetric one.
stat.AP↗
arXiv subjects
Pasha Zusmanovich
Publications and source records attributed to Pasha Zusmanovich.
Lie algebras with given properties of subalgebras and elements
Results about the following classes of finite-dimensional Lie algebras over a field of characteristic zero are presented: anisotropic (i.e., Lie algebras for which each adjoint operator is semisimple), regular (i.e., Lie algebras in which each nonzero element is regular in the sense of Bourbaki), minimal nonabelian (i.e., nonabelian Lie algebras all whose proper subalgebras are abelian), and algebras of depth 2 (i.e., Lie algebras all whose proper subalgebras are abelian or minimal nonabelian).
math.RA↗
On Shirshov bases of graded algebras
We prove that if the neutral component in a finitely-generated associative algebra graded by a finite group has a Shirshov base, then so does the whole algebra.
math.RA↗
$ω$-Lie algebras
We study a certain generalization of Lie algebras where the Jacobian of three elements does not vanish but is equal to an expression depending on a skew-symmetric bilinear form.
math.RA↗
The alternative operad is not Koszul
Using computer calculations, we prove the statement in the title.
math.RA↗
How Euler would compute the Euler-Poincaré characteristic of a Lie superalgebra
The Euler-Poincaré characteristic of a finite-dimensional Lie algebra vanishes. If we want to extend this result to Lie superalgebras, we should deal with infinite sums. We observe that a suitable method of summation, which goes back to Euler, allows to do that, to a certain degree. The mathematics behind it is simple, we just glue the pieces of elementary homological algebra, first-year calculus and pedestrian combinatorics together, and present them in a (hopefully) coherent manner.
math.KT↗
The second homology group of current Lie algebras
This is an old paper put here for archeological purposes. We derive a general formula expressing the second homology of a Lie algebra of the form L\otimes A with coefficients in the trivial module through homology of $L$, cyclic homology of $A$, and other invariants of $L$ and $A$. This is achieved by using the Hopf formula expressing the second homology of a Lie algebra in terms of its presentation. We also derive a similar formula for the associated Lie algebra of the tensor product of two associative algebras.
math.KT↗
A converse to the Whitehead Theorem
We show that finite-dimensional Lie algebras over a field of characteristic zero such that their high-degree cohomology in any finite-dimensional non-trivial irreducible module vanishes, are, essentially, direct sums of semisimple and nilpotent algebras.
math.RA↗