اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدCohomology of Oriented Tree Diagram Lie Algebras
279
0
0.0
(
0
)
تأليف
Li Luo
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are natural analogues of the maximal nilpotent Lie subalgebras of finite-dimensional simple Lie algebras. In this paper, we use Hodge Laplacian to study the cohomology of these Lie algebras. The total rank conjecture and $b_2$-conjecture for the algebras are proved. Moreover, we find the generating functions of the Betti numbers by means of Young tableaux for the Lie algebras associated with certain tree diagrams of single branch point. By these functions and Euler-Poincare principle, we obtain analogues of the denominator identity for finite-dimensional simple Lie algebras. The result is a natural generalization of the Botts classical result in the case of special linear Lie algebras.
قيم البحث
اقرأ أيضاً
We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with the structure of the anticyclic PreLie operad.
A Lie algebra $L$ over a field $mathbb{F}$ is said to be zero product determined (zpd) if every bilinear map $f:Ltimes Lto mathbb{F}$ with the property that $f(x,y)=0$ whenever $x$ and $y$ commute is a coboundary. The main goal of the paper is to det
ermine whether or not some important Lie algebras are zpd. We show that the Galilei Lie algebra $mathfrak{sl}_2ltimes V$, where $V$ is a simple $mathfrak{sl}_2$-module, is zpd if and only if $dim V =2$ or $dim V$ is odd. The class of zpd Lie algebras also includes the quantum torus Lie algebras $mathcal{L}_q$ and $mathcal{L}^+_q$, the untwisted affine Lie algebras, the Heisenberg Lie algebras, and all Lie algebras of dimension at most $3$, while the class of non-zpd Lie algebras includes the ($4$-dimensional) aging Lie algebra $mathfrak {age}(1)$ and all Lie algebras of dimension more than $3$ in which only linearly dependent elements commute. We also give some evidence of the usefulness of the concept of a zpd Lie algebra by using it in the study of commutativity preserving linear maps.
Quantum Drinfeld Hecke algebras are generalizations of Drinfeld Hecke algebras in which polynomial rings are replaced by quantum polynomial rings. We identify these algebras as deformations of skew group algebras, giving an explicit connection to Hoc
hschild cohomology. We compute the relevant part of Hochschild cohomology for actions of many reflection groups and we exploit computations from our paper with Shroff for diagonal actions. By combining our work with recent results of Levandovskyy and Shepler, we produce examples of quantum Drinfeld Hecke algebras. These algebras generalize the braided Cherednik algebras of Bazlov and Berenstein.
Quantum symmetric algebras (or noncommutative polynomial rings) arise in many places in mathematics. In this article we find the multiplicative structure of their Hochschild cohomology when the coefficients are in an arbitrary bimodule algebra. When
this bimodule algebra is a finite group extension (under a diagonal action) of a quantum symmetric algebra, we give explicitly the graded vector space structure. This yields a complete description of the Hochschild cohomology ring of the corresponding skew group algebra.
In this paper, first we show that under the assumption of the center of h being zero, diagonal non-abelian extensions of a regular Hom-Lie algebra g by a regular Hom-Lie algebra h are in one-to-one correspondence with Hom-Lie algebra morphisms from g
to Out(h). Then for a general Hom-Lie algebra morphism from g to Out(h), we construct a cohomology class as the obstruction of existence of a non-abelian extension that induce the given Hom-Lie algebra morphism.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
