Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDouble Quantum Schubert Cells and Quantum Mutations
69
0
0.0
(
0
)
Authors
Hans P. Jakobsen
Ask ChatGPT about the research
No Arabic abstract
Let ${mathfrak p}subset {mathfrak g}$ be a parabolic subalgebra of s simple finite dimensional Lie algebra over ${mathbb C}$. To each pair $w^{mathfrak a}leq w^{mathfrak c}$ of minimal left coset representatives in the quotient space $W_pbackslash W$ we construct explicitly a quantum seed ${mathcal Q}_q({mathfrak a},{mathfrak c})$. We define Schubert creation and annihilation mutations and show that our seeds are related by such mutations. We also introduce more elaborate seeds to accommodate our mutations. The quantized Schubert Cell decomposition of the quantized generalized flag manifold can be viewed as the result of such mutations having their origins in the pair $({mathfrak a},{mathfrak c})= ({mathfrak e},{mathfrak p})$, where the empty string ${mathfrak e}$ corresponds to the neutral element. This makes it possible to give simple proofs by induction. We exemplify this in three directions: Prime ideals, upper cluster algebras, and the diagonal of a quantized minor.
rate research
Read More
The goal of this work is to provide an elementary construction of the canonical basis $mathbf B(w)$ in each quantum Schubert cell~$U_q(w)$ and to establish its invariance under modified Lusztigs symmetries. To that effect, we obtain a direct characterization of the upper global basis $mathbf B^{up}$ in terms of a suitable bilinear form and show that $mathbf B(w)$ is contained in $mathbf B^{up}$ and its large part is preserved by modified Lusztigs symmetries.
The aim of this paper is to study the representation theory of quantum Schubert cells. Let $g$ be a simple complex Lie algebra. To each element $w$ of the Weyl group $W$ of $g$, De Concini, Kac and Procesi have attached a subalgebra $U_q[w]$ of the quantised enveloping algebra $U_q(g)$. Recently, Yakimov showed that these algebras can be interpreted as the quantum Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of $U_q[w]$. More precisely, it follows from the Stratification Theorem of Goodearl and Letzter that the primitive spectrum of $U_q[w]$ admits a stratification indexed by those primes that are invariant under a natural torus action. Moreover each stratum is homeomorphic to the spectrum of maximal ideals of a torus. The main result of this paper gives an explicit formula for the dimension of the stratum associated to a given torus-invariant prime.
A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but subclasses have been studied previously by other authors. The algebras are indexed by double partitions or double flag varieties. Equivalently, they are indexed by broken lines $L$. By grouping together neighboring mutations into quantum line mutations we can mutate from the cluster algebra of one broken line to another. Compatible pairs can be written down. The algebras are equal to their upper cluster algebras. The variables of the quantum seeds are given by elements of the dual canonical basis. This is the final version, where some arguments have been expanded and/or improved and several typos corrected. Full bibliographic details: Journal of Algebra (2012), pp. 172-203 DOI information: 10.1016/j.jalgebra.2012.09.015
We define even dimensional quantum spheres Sigma_q^2n that generalize to higher dimension the standard quantum two-sphere of Podles and the four-sphere Sigma_q^4 obtained in the quantization of the Hopf bundle. The construction relies on an iterated Poisson double suspension of the standard Podles two-sphere. The Poisson spheres that we get have the same symplectic foliation consisting of a degenerate point and a symplectic plane and, after quantization, have the same C^*-algebraic completion. We investigate their K-homology and K-theory by introducing Fredholm modules and projectors.
We construct a canonical isomorphism between the Bethe algebra acting on a multiplicity space of a tensor product of evaluation gl_N[t]-modules and the scheme-theoretic intersection of suitable Schubert varieties. Moreover, we prove that the multiplicity space as a module over the Bethe algebra is isomorphic to the coregular representation of the scheme-theoretic intersection. In particular, this result implies the simplicity of the spectrum of the Bethe algebra for real values of evaluation parameters and the transversality of the intersection of the corresponding Schubert varieties.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
