arXiv subjects
Publications and source records attributed to Vyacheslav Futorny.
We construct realizations of quantum generalized Verma modules for U_q(sl_n(C)) by quan- tum differential operators. Taking the classical limit q ! 1 provides a realization of classical generalized Verma modules for sl_n(C) by differential operators.
In the present paper we study Gelfand-Tsetlin modules defined in terms of BGG differential operators. The structure of these modules is described with the aid of the Postnikov-Stanley polynomials introduced in [PS09]. These polynomials are used to identify the action of the Gelfand-Tsetlin subalgebra on the BGG operators. We also provide explicit bases of the corresponding Gelfand-Tsetlin modules and prove a simplicity criterion for these modules. The results hold for modules defined over standard Galois orders of type $A$ - a large class of rings that include the universal enveloping algebra of $\mathfrak{gl} (n)$ and the finite $W$-algebras of type $A$.
We introduce the notion of virtual endomorphisms of Lie algebras and use it as an approach for constructing self-similarity of Lie algebras. This is done in particular for a class of metabelian Lie algebras having homological type F Pn, which are variants of lamp-lighter groups. We establish several criteria when the existence of virtual endomorphism implies a self-similar Lie structure. Furthermore, we prove that the classical Lie algebra sln(k), where char(k) does not divide n affords non-trivial faithful self-similarity.
We reprove the results of Jordan [18] and Siebert [31] and show that the Lie algebra of polynomial vector fields on an irreducible affine variety X is simple if and only if X is a smooth variety. Given proof is self-contained and does not depend on papers mentioned above. Besides, the structure of the module of polynomial functions on an irreducible smooth affine variety over the Lie algebra of vector fields is studied. Examples of Lie algebras of polynomial vector fields on an N-dimensional sphere, non-singular hyperelliptic curves and linear algebraic groups are considered.
In this note we compute the leading term with respect to the De Concini-Kac filtration of $U_q(\mathfrak{gl}_n)$ of a generating set for the quantum Gelfand-Tsetlin subalgebra.
For each two-dimensional vector space $V$ of commuting $n\times n$ matrices over a field $\mathbb F$ with at least 3 elements, we denote by $\widetilde V$ the vector space of all $(n+1)\times(n+1)$ matrices of the form $\left[\begin{smallmatrix}A&*\\0&0\end{smallmatrix}\right]$ with $A\in V$. We prove the wildness of the problem of classifying Lie algebras $\widetilde V$ with the bracket operation $[u,v]:=uv-vu$. We also prove the wildness of the problem of classifying two-dimensional vector spaces consisting of commuting linear operators on a vector space over a field.
For an irreducible affine variety $X$ over an algebraically closed field of characteristic zero we define two new classes of modules over the Lie algebra of vector fields on $X$ - gauge modules and Rudakov modules, which admit a compatible action of the algebra of functions. Gauge modules are generalizations of modules of tensor densities whose construction was inspired by non-abelian gauge theory. Rudakov modules are generalizations of a family of induced modules over the Lie algebra of derivations of a polynomial ring studied by Rudakov. We prove general simplicity theorems for these two types of modules and establish a pairing between them.
The purpose of this paper is to construct new families of irreducible Gelfand-Tsetlin modules for U_q(gl_n). These modules have arbitrary singularity and Gelfand-Tsetlin multiplicities bounded by 2. Most previously known irreducible modules had all Gelfand-Tsetlin multiplicities bounded by 1 \cite{FRZ1}, \cite{FRZ2}. In particular, our method works for q=1 providing new families of irreducible Gelfand-Tsetlin modules for gl_n. This generalizes the results of \cite{FGR3} and \cite{FRZ}.
We prove a uniqueness theorem for irreducible non-critical Gelfand-Tsetlin modules. The uniqueness result leads to a complete classification of the irreducible Gelfand-Tsetlin modules with 1-singularity. An explicit construction of such modules was given in \cite{FGR2}. In particular, we show that the modules constructed in \cite{FGR2} exhaust all irreducible Gelfand-Tsetlin modules with 1-singularity. To prove the result we introduce a new category of modules (called Drinfeld category) related to the Drinfeld generators of the Yangian Y(gl_n) and define a functor from the category of non-critical Gelfand-Tsetlin modules to the Drinfeld category.
Singular Gelfand-Tsetlin modules of index 2 are modules whose tableaux bases may have singular pairs but no singular triples of entries on each row. In this paper we construct singular Gelfand-Tsetlin modules for arbitrary singular character of index 2. Explicit bases of derivative tableaux and the action of the generators of $\mathfrak{gl}(n)$ are given for these modules. Our construction leads to new families of irreducible Gelfand-Tsetlin modules and also provides tableaux bases for some simple Verma modules.
Let R be a ring (not necessarily with 1) and G be a finite group of automorphisms of R. The set B(R, G) of primes p such that p | |G| and R is not p-torsion free, is called the set of bad primes. When the ring is |G|-torsion free, i.e., B(R, G) is empty set, the properties of the rings R and R^G are closely connected. The aim of the paper is to show that this is also true when B(R, G) is not empty set under natural conditions on bad primes. In particular, it is shown that the Jacobson radical (resp., the prime radical) of the ring R^G is equal to the intersection of the Jacobson radical (resp., the prime radical) of R with R^G; if the ring R is semiprime then so is R^G; if the trace of the ring R is nilpotent then the ring itself is nilpotent; if R is a semiprime ring then R is left Goldie iff the ring R^G is so, and in this case, the ring of G-invariants of the left quotient ring of R is isomorphic to the left quotient ring of R^G and \udim (R^G)\leq \udim (R)\leq |G| \udim (R^G).
W.E. Roth (1952) proved that the matrix equation $AX-XB=C$ has a solution if and only if the matrices $\left[\begin{matrix}A&C\\0&B\end{matrix}\right]$ and $\left[\begin{matrix}A&0\\0&B\end{matrix}\right]$ are similar. A. Dmytryshyn and B. Kågström (2015) extended Roth's criterion to systems of matrix equations $A_iX_{i'}M_i-N_iX_{i''}^{σ_i} B_i=C_i$ $(i=1,\dots,s)$ with unknown matrices $X_1,\dots,X_t$, in which every $X^σ$ is $X$, $X^T$, or $X^*$. We extend their criterion to systems of complex matrix equations that include the complex conjugation of unknown matrices. We also prove an analogous criterion for systems of quaternion matrix equations.
We construct new families of U_q(gl_n)-modules by continuation from finite dimensional representations. Each such module is associated with a combinatorial object - admissible set of relations defined in \cite{FRZ}. More precisely, we prove that any admissible set of relations leads to a family of irreducible U_q(gl_n)-modules. Finite dimensional and generic modules are particular cases of this construction.
For an admissible affine vertex algebra $V_k(\mathfrak{g})$ of type $A$, we describe a new family of relaxed highest weight representations of $V_k(\mathfrak{g})$. They are simple quotients of representations of the affine Kac-Moody algebra $\widehat{\mathfrak{g}}$ induced from the following $\mathfrak{g}$-modules: 1) generic Gelfand-Tsetlin modules in the principal nilpotent orbit, in particular all such modules induced from $\mathfrak{sl}_2$; 2) all Gelfand-Tsetlin modules in the principal nilpotent orbit which are induced from $\mathfrak{sl}_3$; 3) all simple Gelfand-Tsetlin modules over $\mathfrak{sl}_3$. This in particular gives the classification of all simple positive energy weight representations of $V_k(\mathfrak{g})$ with finite dimensional weight spaces for $\mathfrak{g}=\mathfrak{sl}_3$.
W.Specht (1940) proved that two $n\times n$ complex matrices $A$ and $B$ are unitarily similar if and only if $\operatorname{trace} w(A,A^{\ast}) = \operatorname{trace} w(B,B^{\ast})$ for every word $w(x,y)$ in two noncommuting variables. We extend his criterion and its generalizations by N.A.Wiegmann (1961) and N.Jing (2015) to an arbitrary system $\mathcal A$ consisting of complex or real inner product spaces and linear mappings among them. We represent such a system by the directed graph $Q(\mathcal A)$, whose vertices are inner product spaces and arrows are linear mappings. Denote by $\widetilde Q(\mathcal A)$ the directed graph obtained by enlarging to $Q(\mathcal A)$ the adjoint linear mappings. We prove that a system $\mathcal A$ is transformed by isometries of its spaces to a system $\mathcal B$ if and only if the traces of all closed directed walks in $\widetilde Q(\mathcal A)$ and $\widetilde Q(\mathcal B)$ coincide.
For the algebra L= K of polynomial integro-differential operators over a field K of characteristic zero, a classification of indecomposable, generalized weight L-modules of finite length is given. Each such module is an infinite dimensional uniserial module. Ext-groups are found between indecomposable generalized weight modules, it is proven that they are finite dimensional vector spaces.
Let $n>1$ be an integer, $α\in{\mathbb C}^n$, $b\in{\mathbb C}$, and $V$ a $\mathfrak{gl}_n$-module. We define a class of weight modules $F^α_{b}(V)$ over $\sl_{n+1}$ using the restriction of modules of tensor fields over the Lie algebra of vector fields on $n$-dimensional torus. In this paper we consider the case $n=2$ and prove the irreducibility of such 5-parameter $\mathfrak{sl}_{3}$-modules $F^α_{b}(V)$ generically. All such modules have infinite dimensional weight spaces and lie outside of the category of Gelfand-Tsetlin modules. Hence, this construction yields new families of irreducible $\mathfrak{sl}_{3}$-modules.
We solve the noncommutative Noether's problem for the reflection groups by showing that the skew field of the invariants of the Weyl algebra under the action of any reection group is a Weyl field, that is isomorphic to a skew field of some Weyl algebra over a transcendental extension of the ground field. We also extend this result to the invariants of the ring of differential operators on any dimensional torus.The results are applied to obtain analogs of the Gelfand-Kirillov Conjecture for Cherednik algebras and Galois algebras.