Masatoshi Noumi, Yasuhiko Yamada
We propose a general method to realize an arbitrary Weyl group of Kac-Moody type as a group of birational canonical transformations, by means of a nilpotent Poisson algebra. We also give a Lie theoretic interpretation of this realization in terms of Kac-Moody Lie algebras and Kac-Moody groups.
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We canonically quantize the tau-functions for the birational Weyl group action arising from a nilpotent Poisson algebra proposed by Noumi and Yamada. We also construct the q-difference deformation of the canonical quantization of the tau-functions. Using the translation functors for the symmetrizable Kac-Moody algebras, we prove the regularity of the quantum tau-functions, namely, we show that the quantum tau-functions are polynomials in dependent variables.
Birational Equivalences and Generalized Weyl Algebras
本研究将量子群与Podleś球面实现为广义Weyl代数（GWA），利用GWA与1-环面smash积的双有理等价性计算其局部化Hochschild同调，并解决了GWA的双有理等价与光滑性问题。
We shall construct the quantized q-analogues of the birational Weyl group actions arising from nilpotent Poisson algebras, which are conceptual generalizations, proposed by Noumi and Yamada, of the Bäcklund transformations for Painlevé equations. Consider a quotient Ore domain of the lower nilpotent part of a quantized universal enveloping algebra of arbitrary symmetrizable Kac-Moody type. Then non-integral powers of the image of the Chevalley generators generate the quantized q-analogue of the birational Weyl group action. Using the same method, we shall reconstruct the quantized Bäcklund tra…
It is known that the set of irreducible components of nilpotent varieties provides a geometric realization of the crystal basis for quantum groups. For each reduced expression of a Weyl group element, Geiß, Leclerc and Schröer has recently given a parametrization of irreducible components of nilpotent varieties in studying cluster algebras. In this paper we show that their parametrization coincides with Lusztig's parametrization of the canonical basis.
Birational Weyl Group Action on the Symplectic Groupoid and Cluster Algebras
本文研究了辛群oid上的理查德·韦伊群作用，通过簇变换关联于quiver中的某些环，证明了韦伊群不变子代数是矩阵条目生成代数的有限中心扩张，并展示了该作用在经典情况下的嵌入图像与韦伊群不变子代数的商代数的泊松同构性。
We show that endomorphisms of Weyl algebras over fields of characteristic zero are flat and that birational endomorphisms are automorphisms by reducing to positive characteristic. We also give examples showing that endomorphisms of Weyl algebras are not in general flat over fields of positive characteristic.
This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue of the Kostant partition function, and propose the corresponding two-parameter analogue of the weight multiplicity formula. In a different direction, each principal nilpotent pair gives rise to a harmon…
We attach a Dixmier algebra B to the closure of any nilpotent orbit of G where G is GL(n,C), O(n,C) or Sp(2n,C). This algebra B is a noncommutative analog of the coordinate ring R of the orbit closure, in the sense that B has a G-invariant algebra filtration and gr B=R. We obtain B by making a noncommutative analog of the Kraft-Procesi construction which modeled the orbit closure as the algebraic symplectic reduction of a finite-dimensional symplectic vector space L. Indeed B is a subquotient of the Weyl algebra for L.
We suggest new realizations of quantum groups corresponding to complex simple Lie algebras, and of affine quantum groups. These new realizations are labeled by Coxeter elements of the corresponding Weyl group and have the following key feature: The natural counterparts of the subalgebras U(n), where n is a maximal nilpotent subalgebra in g, possess non--singular characters.
We define algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g. The new slices are transversal to the conjugacy classes in an algebraic group G with Lie algebra g. These slices are associated to (the conjugacy classes of) elements s of the Weyl group W of g. For such slices we prove an analogue of the Kostant cross-section theorem for the action of a unipotent group.
We study the maximal abelian ad-nilpotent (mad) subalgebras of the domains D Morita equivalent to the first Weyl algebra. We give a complete description both of the individual mad subalgebras and of the space of all such. A surprising consequence is that this last space is independent of D. Our results generalize some classic theorems of Dixmier about the Weyl algebra.
We study a quantum (non-commutative) representation of the affine Weyl group mainly of type $E_8^{(1)}$, where the representation is given by birational actions on two variables $x$, $y$ with $q$-commutation relations. Using the tau variables, we also construct quantum "fundamental" polynomials $F(x,y)$ which completely control the Weyl group actions. The geometric properties of the polynomials $F(x,y)$ for the commutative case is lifted distinctively in the quantum case to certain singularity structures as the $q$-difference operators. This property is further utilized as the characterization…
Is $A_1$ of type $B_2$
本文研究 $U_q^+(B_2)$ 的 Gelfand-Kirillov 维数为 2 的单商的导子，证明特定族的所有导子均为内导子，其余为量子广义 Weyl 代数且一阶 Hochschild 上同调维数为 1。
Consider a complex simple Lie algebra g of rank n. Denote by Πa system of simple roots, by W the corresponding Weyl group, consider a reduced expression w = s_{α_{1}} ... s_{α_{t}} (each α_{i} in Π) of some w \in W and call diagram any subset of {1, ..., t}. We denote by U_{q}^{w}(g) the "quantum nilpotent" algebra defined by J. C. Jantzen. We prove (theorem 5.3. 1) that the positive diagrams naturally associated with the positive subexpressions (of the reduced expression of w) in the sense of R. Marsh and K. Rietsch, coincide with the admissible diagrams constructed by G. Cauchon which descri…
A generalized cluster structure on $GL_n$ via birational Poisson maps
本文通过理性泊松映射在GL_n上构造广义聚类结构，探讨了当非周期性条件不满足时的构造方法，结合了Cremmer-Gervais解与Weyl群最长元素的共轭关系。
For any module $M$ over small quantum group one defines the support variety using construction from the theory of restricted Lie algebras. It is a closed conical subset of nilpotent cone of the corresponding Lie algebra. If module $M$ is a module over the quantum group $U_ξ$ with divided powers then its support variety is invariant under the action of the corresponding algebraic group. In this case we relate codimension $2a$ of the support variety of $M$ in nilpotent cone and dimension of $M$. Namely, we prove that $\dim M$ is `almost' divisible by $l^a$. Further, we give an a priori estimate …
All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always equal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quantized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have a broad range of applications to thes…
According to Kazhdan-Lusztig and Ginzburg, the Hecke algebra of an affine Weyl group is identified with the equivariant $K$-group of Steinberg's triple variety. The $K$-group is equipped with a filtration indexed by closed $G$-stable subvarieties of the nilpotent variety, where $G$ is the corresponding reductive algebraic group over $\mathbb{C}$. In this paper we will show in the case of type $A$ that the filtration is compatible with the Kazhdan-Lusztig basis of the Hecke algebra.
In one of our recent papers, the associative and the Lie algebras of Weyl type $A[D]=A\otimes F[D]$ were defined and studied, where $A$ is a commutative associative algebra with an identity element over a field $F$ of any characteristic, and $F[D]$ is the polynomial algebra of a commutative derivation subalgebra $D$ of $A$. In the present paper, a class of the above associative and Lie algebras $A[D]$ with $F$ being a field of characteristic 0, $D$ consisting of locally finite but not locally nilpotent derivations of $A$, are studied. The isomorphism classes and automorphism groups of these as…
De Concini, Kac and Procesi defined a family of subalgebras U^w_+ of a quantized universal enveloping algebra U_q(g), associated to the elements of the corresponding Weyl group W. They are deformations of the universal enveloping algebras U(n_+ \cap Ad_w(n_-)) where n_\pm are the nilradicals of a pair of dual Borel subalgebras. Based on results of Gorelik and Joseph and an interpretation of U^w_+ as quantized algebras of functions on Schubert cells, we construct explicitly the H invariant prime ideals of each U^w_+ and show that the corresponding poset is isomorphic to W^{\leq w}, where H is t…
On the category $\mathcal{O}$ for generalized Weyl algebras
本文研究广义Weyl代数的范畴$\mathcal{O}$，在特定幂零性和理想生成条件下，证明了原代数与其第$l$次扭转代数上的范畴$\mathcal{O}$等价，推广了$\mathbb{Z}/p^n\mathbb{Z}$上Weyl代数的相应结果。
For a finite-dimensional simple Lie algebra $\mathfrak{g}$, let $U^+_q(\mathfrak{g})$ be the positive part of the quantized universal enveloping algebra, and $A_q(\mathfrak{g})$ be the quantized algebra of functions. We show that the transition matrix of the PBW bases of $U^+_q(\mathfrak{g})$ coincides with the intertwiner between the irreducible $A_q(\mathfrak{g})$-modules labeled by two different reduced expressions of the longest element of the Weyl group of $\mathfrak{g}$. This generalizes the earlier result by Sergeev on $A_2$ related to the tetrahedron equation and endows a new represent…
In this paper, we construct the Lusztig symmetries for quantum Borcherds-Bozec algebra $U_q(\mathscr g)$ and its weight module $M\in \mathcal O$, on which the generators with real indices of $U_q(\mathscr g)$ act nilpotently. We show that these symmetries satisfy the defining relations of the braid group, associated to the Weyl group $W$ of $U_q(\mathscr g)$, which gives a braid group action.
It is known that the semi-infinite cohomology spaces of the infinitely twisted nilpotent subalgebra in an affine Lie algebra $g$ with coefficients in an integrable simple module over the affine Lie algebra have a base enumerated by elements of the corresponding affine Weyl group graded by the semiinfinite length function. Let $U$ be the affine quantum group corresponding to $g$. It is possible to define a subalgebra in $U$ being the quantum analogue of the universal enveloping algebra of the infinitely twisted nilpotent subalgebra in $g$. In this paper we prove that for general values of the p…
For symplectic Lie algebras $\mathfrak{sp}(2n,\mathbb{C})$, denote by $\mathfrak{b}$ and $\mathfrak{n}$ its Borel subalgebra and maximal nilpotent subalgebra, respectively. We construct a relationship between the abelian ideals of $\mathfrak{b}$ and the cohomology of $\mathfrak{n}$ with trivial coefficients. By this relationship, we can enumerate the number of abelian ideals of $\mathfrak{b}$ with certain dimension via the Poincare polynomials of Weyl groups of type $A_{n-1}$ and $C_n$.
We study a class of representations of the Lie algebra of Laurent polynomials with values in the nilpotent subalgebra of sl(3). We derive Weyl-type (bosonic) character formulas for these representations. We establish a connection between the bosonic formulas and the Whittaker vector in the Verma module for the quantum group $U_v sl(3)$. We also obtain a fermionic formula for an eigenfunction of the sl(3) quantum Toda Hamiltonian.
In this article we formulate a group of birational transformations which is isomorphic to an extended affine Weyl group of type $(A_{2n+1}+A_1+A_1)^{(1)}$ with the aid of mutations and permutations of vertices to a mutation-periodic quiver on a torus. This group provides a class of higher order generalizations of Jimbo-Sakai's $q$-Painlevé VI equation as translations on a root lattice. Then the known three systems are obtained again; the $q$-Garnier system, a similarity reduction of the lattice $q$-UC hierarchy and a similarity reduction of the $q$-Drinfeld-Sokolov hierarchy.
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