Related papers: A note on a theorem of Bourbaki
In view of a previous work, we explicitly give the Poincare polinomials of 19 Hyperbolic Lie algebras of rank 3. It is seen that every one of these polinomials is expressed as the ratio of Poincare polinomial of $B_3$ Lie algebra and a…
Poincare polinomials of hyperbolic Lie algebras, which are given by $HA_2$ and $HA_3$ in the Kac's notation, are calculated explicitly. The results show that there is a significant form for hyperbolic Poincare polinomials. Their explicit…
We have general frameworks to obtain Poincare polynomials for Finite and also Affine types of Kac-Moody Lie algebras. Very little is known however beyond Affine ones, though we have a constructive theorem which can be applied both for…
The notion of the characteristic Lie algebra of the discrete hyperbolic type equation is introduced. An effective algorithm to compute the algebra for the equation given is suggested. Examples and further applications are discussed.
We single out a large class of semisimple singularities with the property that all roots of the Poincar\'e polynomial of the Lie algebra of derivations of the corresponding suitably (not necessarily quasihomogeneously) graded moduli algebra…
In this note, we study the Hilbert-Poincar\'e polynomials for the PBW-graded of simple modules for a simple complex Lie algebra. The computation of their degree can be reduced to modules of fundamental highest weight. We provide these…
A class of representations of a Lie superalgebra (over a commutative superring) in its symmetric algebra is studied. As an application we get a direct and natural proof of a strong form of the Poincare'-Birkhoff-Witt theorem, extending this…
We deal with the algebraicity of a Puiseux series in terms of the properties of its coefficients. We show that the algebraicity of a Puiseux series for given bounded degree is determined by a finite number of explicit polynomial formulae.…
We prove a formula relating the fermionic forms and the Poincare polynomials of quiver varieties associated to a finite quiver. Applied to quivers of type ADE, our result implies a version of the fermionic conjecture of Lusztig.
The real type of a finite family of univariate polynomials characterizes the combined sign behavior of the polynomials over the real line. We derive an explicit formula for the number of real types subject to given degree bounds. For the…
We introduce and study the Bourbaki degree as a numerical invariant for \(2 \times 4\) matrices $\Theta$ of homogeneous polynomials over a polynomial ring \(R = k[x_1, \dots, x_n]\). This invariant, defined via a Bourbaki sequence for the…
Hyperquot schemes are generalizations of Grothendieck's Quot scheme to partial flags. Using a Bialynicki-Birula decomposition, we obtain combinatorial data for the Betti numbers, and collect this information into the form of rational…
A well-known and old result of Hazewinkel and Koszul states that the cohomology of a finite-dimensional Lie algebra is isomorphic, up to a suitable shift, to its twisted homology, a Lie-theoretical version of Poincare duality. This paper…
We give criteria for finite dimensionality or infinite dimensionality of the polynomial centralizer of the Lie algebra of a linear Lie group, in terms of invariants and relative invariants of the group. In the finite dimensional scenario…
In this paper, we study Whittaker modules for a Lie algebras of Block type. We define Whittaker modules and under some conditions, obtain a one to one correspondence between the set of isomorphic classes of Whittaker modules over this…
In the first, mostly expository, part of this paper, a graded Lie algebra is associated to every group G given with an N-series of subgroups. The asymptotics of the Poincare series of this algebra give estimates on the growth of the group…
We prove a version of the Poincar\'e-Birkhoff-Witt Theorem for profinite pronilpotent Lie algebras in which their symmetric and universal enveloping algebras are replaced with appropriate formal analogues and discuss some immediate…
We deal with the algebraicity of an iterated Puiseux series in several variables in terms of the properties of its coefficients. Our aim is to generalize to several variables the results from [HM15]. We show that the algebraicity of such a…
In these lectures we study some possible higher order (of degree greater than two) extensions of the Poincar\'e algebra. We first give some general properties of Lie superalgebras with some emphasis on the supersymmetric extension of the…
In this paper, we study first the relationship between Pommaret bases and Hilbert series. Given a finite Pommaret basis, we derive new explicit formulas for the Hilbert series and for the degree of the ideal generated by it which exhibit…