اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدStructures of Nichols (braided) Lie algebras of diagonal type
100
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Let $V$ be a braided vector space of diagonal type. Let $mathfrak B(V)$, $mathfrak L^-(V)$ and $mathfrak L(V)$ be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over $V$, respectively. We show that a monomial belongs to $mathfrak L(V)$ if and only if that this monomial is connected. We obtain the basis for $mathfrak L(V)$ of arithmetic root systems and the dimension for $mathfrak L(V)$ of finite Cartan type. We give the sufficient and necessary conditions for $mathfrak B(V) = Foplus mathfrak L^-(V)$ and $mathfrak L^-(V)= mathfrak L(V)$. We obtain an explicit basis of $mathfrak L^ - (V)$ over quantum linear space $V$ with $dim V=2$.
قيم البحث
اقرأ أيضاً
It is shown that if $mathfrak B(V) $ is connected Nichols algebra of diagonal type with $dim V>1$, then $dim (mathfrak L^-(V)) = infty$ $($resp. $ dim (mathfrak L(V)) = infty $$)$ $($ resp. $ dim (mathfrak B(V)) = infty $$)$ if and only if $Delta(mat
hfrak B(V)) $ is an arithmetic root system and the quantum numbers (i.e. the fixed parameters) of generalized Dynkin diagrams of $V$ are of finite order. Sufficient and necessary conditions for $m$-fold adjoint action in $mathfrak B(V)$ equal to zero, viz. $overline{l}_{x_{i}}^{m}[x_{j}]^ -=0$ for $x_i,~x_jin mathfrak B(V)$, are given.
Assume that $V$ is a braided vector space with diagonal type. It is shown that a monomial belongs to Nichols braided Lie algebra $mathfrak L(V)$ if and only if this monomial is connected. A basis of Nichols braided Lie algebra and dimension of Nichol
s braided Lie algebra of finite Cartan type are obtained.
We prove {rm (i)} Nichols algebra $mathfrak B(V)$ of vector space $V$ is finite-dimensional if and only if Nichols braided Lie algebra $mathfrak L(V)$ is finite-dimensional; {rm (ii)} If the rank of connected $V$ is $2$ and $mathfrak B(V)$ is an arit
hmetic root system, then $mathfrak B(V) = F oplus mathfrak L(V);$ and {rm (iii)} if $Delta (mathfrak B(V))$ is an arithmetic root system and there does not exist any $m$-infinity element with $p_{uu} ot= 1$ for any $u in D(V)$, then $dim (mathfrak B(V) ) = infty$ if and only if there exists $V$, which is twisting equivalent to $V$, such that $ dim (mathfrak L^ - (V)) = infty.$ Furthermore we give an estimation of dimensions of Nichols Lie algebras and two examples of Lie algebras which do not have maximal solvable ideals.
We establish the relationship among Nichols algebras, Nichols braided Lie algebras and Nichols Lie algebras. We prove two results: (i) Nichols algebra $mathfrak B(V)$ is finite-dimensional if and only if Nichols braided Lie algebra $mathfrak L(V)$ is
finite-dimensional if there does not exist any $m$-infinity element in $mathfrak B(V)$; (ii) Nichols Lie algebra $mathfrak L^-(V)$ is infinite dimensional if $ D^-$ is infinite. We give the sufficient conditions for Nichols braided Lie algebra $mathfrak L(V)$ to be a homomorphic image of a braided Lie algebra generated by $V$ with defining relations.
In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by Lusztig led to the first general construction of PBW bases of these algebras. Also, they have important applications to the representation theory of thes
e and related algebras. In the present paper the Drinfeld double for a class of graded Hopf algebras is investigated. Various quantum algebras, including multiparameter quantizations of semisimple Lie algebras and of Lie superalgebras, are covered by the given definition. For these Drinfeld doubles Lusztig maps are defined. It is shown that these maps induce isomorphisms between doubles of Nichols algebras of diagonal type. Further, the obtained isomorphisms satisfy Coxeter type relations in a generalized sense. As an application, the Lusztig isomorphisms are used to give a characterization of Nichols algebras of diagonal type with finite arithmetic root system. Key words: Hopf algebra, quantum group, Weyl groupoid
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
