اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSpecial classes of homomorphisms between generalized Verma modules for ${mathcal U}_q(su(n,n))$
48
0
0.0
(
0
)
تأليف
Hans Plesner Jakobsen
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We study homomorphisms between quantized generalized Verma modules $M(V_{Lambda})stackrel{phi_{Lambda,Lambda_1}}{rightarrow}M(V_{Lambda_1})$ for ${mathcal U}_q(su(n,n))$. There is a natural notion of degree for such maps, and if the map is of degree $k$, we write $phi^k_{Lambda,Lambda_1}$. We examine when one can have a series of such homomorphisms $phi^1_{Lambda_{n-1},Lambda_{n}} circ phi^1_{Lambda_{n-2}, Lambda_{n-1}} circcdotscirc phi^1_{Lambda,Lambda_1} = textrm{Det}_q$, where $textrm{Det}_q$ denotes the map $M(V_{Lambda}) i prightarrow textrm{Det}_qcdot pin M(V_{Lambda_n})$. If, classically, $su(n,n)^{mathbb C}={mathfrak p}^-oplus(su(n)oplus su(n)oplus {mathbb C})oplus {mathfrak p}^+$, then $Lambda = (Lambda_L,Lambda_R,lambda)$ and $Lambda_n =(Lambda_L,Lambda_R,lambda+2)$. The answer is then that $Lambda$ must be one-sided in the sense that either $Lambda_L=0$ or $Lambda_R=0$ (non-exclusively). There are further demands on $lambda$ if we insist on ${mathcal U}_q({mathfrak g}^{mathbb C})$ homomorphisms. However, it is also interesting to loosen this to considering only ${mathcal U}^-_q({mathfrak g}^{mathbb C})$ homomorphisms, in which case the conditions on $lambda$ disappear. By duality, there result have implications on covariant quantized differential operators. We finish by giving an explicit, though sketched, determination of the full set of ${mathcal U}_q({mathfrak g}^{mathbb C})$ homomorphisms $phi^1_{Lambda,Lambda_1}$.
قيم البحث
اقرأ أيضاً
We determine the center of a localization of ${mathcal U}_q({mathfrak n}_omega)subseteq {mathcal U}^+_q({mathfrak g})$ by the covariant elements (non-mutable elements) by means of constructions and results from quantum cluster algebras. In our set-up
, ${mathfrak g}$ is any finite-dimensional complex Lie algebra and $omega$ is any element in the Weyl group $W$. The non-zero complex parameter $q$ is mostly assumed not to be a root of unity, but our method also gives many details in case $q$ is a primitive root of unity. We point to a new and very useful direction of approach to a general set of problems which we exemplify here by obtaining the result that the center is determined by the null space of $1+omega$. Further, we use this to give a generalization to double Schubert Cell algebras where the center is proved to be given by $omega^{mathfrak a}+omega^{mathfrak c}$. Another family of quadratic algebras is also considered and the centers determined.
Let $mathbb{F}$ be a field of characteristic 0, $G$ an additive subgroup of $mathbb{F}$, $alphain mathbb{F}$ satisfying $alpha otin G, 2alphain G$. We define a class of infinite-dimensional Lie algebras which are called generalized Schr{o}dinger-Vira
soro algebras and use $mathfrak{gsv}[G,alpha]$ to denote the one corresponding to $G$ and $alpha$. In this paper the automorphism group and irreducibility of Verma modules for $mathfrak{gsv}[G,alpha]$ are completely determined.
We find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra $hat{mathfrak{g}}$, from $mathfrak{g}$-module homomorphisms. When $mathfrak{g}=mathfrak{sl}_2$, t
hese results extend previous joint work with J. Yang, but the method used here is different. Here, we construct intertwining operators by solving Knizhnik-Zamolodchikov equations for three-point correlation functions associated to $hat{mathfrak{g}}$, and we identify obstructions to the construction arising from the possible non-existence of series solutions having a prescribed form.
The exact expressions for integrated maximal $U(1)_Y$ violating (MUV) $n$-point correlators in $SU(N)$ ${mathcal N}=4$ supersymmetric Yang--Mills theory are determined. The analysis generalises previous results on the integrated correlator of four su
perconformal primaries and is based on supersymmetric localisation. The integrated correlators are functions of $N$ and $tau=theta/(2pi)+4pi i/g_{_{YM}}^2$, and are expressed as two-dimensional lattice sums that are modular forms with holomorphic and anti-holomorphic weights $(w,-w)$ where $w=n-4$. The correlators satisfy Laplace-difference equations that relate the $SU(N+1)$, $SU(N)$ and $SU(N-1)$ expressions and generalise the equations previously found in the $w=0$ case. The correlators can be expressed as infinite sums of Eisenstein modular forms of weight $(w,-w)$. For any fixed value of $N$ the perturbation expansion of this correlator is found to start at order $( g_{_{YM}}^2 N)^w$. The contributions of Yang--Mills instantons of charge $k>0$ are of the form $q^k, f(g_{_{YM}})$, where $q=e^{2pi i tau}$ and $f(g_{_{YM}})= O(g_{_{YM}}^{-2w})$ when $g_{_{YM}}^2 ll 1$ anti-instanton contributions have charge $k<0$ and are of the form $bar q^{|k|} , hat f(g_{_{YM}})$, where $hat f(g_{_{YM}}) = O(g_{_{YM}}^{2w})$ when $g_{_{YM}}^2 ll 1$. Properties of the large-$N$ expansion are in agreement with expectations based on the low energy expansion of flat-space type IIB superstring amplitudes. We also comment on the relation of $n$-point MUV correlators to $(n-4)$-loop contributions to the four-point correlator. In particular, we argue that it is important to ensure the $SL(2, mathbb{Z})$-covariance even in the construction of perturbative loop integrands.
A recent result of N. Abe implies that the Gabber-Joseph conjecture is true for the first-degree extensions between Verma modules with regular integral highest weights.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
