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Register a new userThe center of ${mathcal U}_q({mathfrak n}_omega)$
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Authors
Hans Plesner Jakobsen
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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.
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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 $phi^1_{Lambda,Lambda_1}$.
We introduce and define the quantum affine $(m|n)$-superspace (or say quantum Manin superspace) $A_q^{m|n}$ and its dual object, the quantum Grassmann superalgebra $Omega_q(m|n)$. Correspondingly, a quantum Weyl algebra $mathcal W_q(2(m|n))$ of $(m|n)$-type is introduced as the quantum differential operators (QDO for short) algebra $textrm{Diff}_q(Omega_q)$ defined over $Omega_q(m|n)$, which is a smash product of the quantum differential Hopf algebra $mathfrak D_q(m|n)$ (isomorphic to the bosonization of the quantum Manin superspace) and the quantum Grassmann superalgebra $Omega_q(m|n)$. An interested point of this approach here is that even though $mathcal W_q(2(m|n))$ itself is in general no longer a Hopf algebra, so are some interesting sub-quotients existed inside. This point of view gives us one of main expected results, that is, the quantum (restricted) Grassmann superalgebra $Omega_q$ is made into the $mathcal U_q(mathfrak g)$-module (super)algebra structure,$Omega_q=Omega_q(m|n)$ for $q$ generic, or $Omega_q(m|n, bold 1)$ for $q$ root of unity, and $mathfrak g=mathfrak{gl}(m|n)$ or $mathfrak {sl}(m|n)$, the general or special linear Lie superalgebra. This QDO approach provides us with explicit realization models for some simple $mathcal U_q(mathfrak g)$-modules, together with the concrete information on their dimensions. Similar results hold for the quantum dual Grassmann superalgebra $Omega_q^!$ as $mathcal U_q(mathfrak g)$-module algebra.In the paper some examples of pointed Hopf algebras can arise from the QDOs, whose idea is an expansion of the spirit noted by Manin in cite{Ma}, & cite{Ma1}.
Let $V$ be the two-dimensional simple module and $M$ be a projective Verma module for the quantum group of $mathfrak{sl}_2$ at generic $q$. We show that for any $rge 1$, the endomorphism algebra of $Motimes V^{otimes r}$ is isomorphic to the type $B$ Temperley-Lieb algebra $rm{TLB}_r(q, Q)$ for an appropriate parameter $Q$ depending on $M$. The parameter $Q$ is determined explicitly. We also use the cellular structure to determine precisely for which values of $r$ the endomorphism algebra is semisimple. A key element of our method is to identify the algebras $rm{TLB}_r(q,Q)$ as the endomorphism algebras of the objects in a quotient category of the category of coloured ribbon graphs of Freyd-Yetter or the tangle diagrams of Turaev and Reshitikhin.
In this paper, we show that every singular fiber of the Gelfand--Cetlin system on coadjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a $2$-stage quotient of a compact Lie group by free actions of two other compact Lie groups. In many cases, these singular fibers can be shown to be homogeneous spaces or even diffeomorphic to compact Lie groups. We also give a combinatorial formula for computing the dimensions of all singular fibers, and give a detailed description of these singular fibers in many cases, including the so-called (multi-)diamond singularities. These (multi-)diamond singular fibers are degenerate for the Gelfand--Cetlin system, but they are Lagrangian submanifolds diffeomorphic to direct products of special unitary groups and tori. Our methods of study are based on different ideas involving complex ellipsoids, Lie groupoids, and also general ideas coming from the theory of singularities of integrable Hamiltonian systems.
We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian $mathrm Y(mathfrak{gl}_{m|n})$. To a solution we associate a rational difference operator $mathcal D$ and a superspace of rational functions $W$. We show that the set of complete factorizations of $mathcal D$ is in canonical bijection with the variety of superflags in $W$ and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains.
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