arXiv subjects
Publications and source records attributed to Vyacheslav Futorny.
We study how small perturbations of a 2-by-2 complex matrix can change its canonical form for *congruence. We construct the Hasse diagram for the closure ordering on the set of *congruence classes of 2-by-2 matrices.
We classify all simple $W_n$-modules with finite-dimensional weight spaces. Every such module is either of a highest weight type or is a quotient of a module of tensor fields on a torus, which was conjectured by Eswara Rao. This generalizes the classical result of Mathieu on simple weight modules for the Virasoro algebra. In our proof of the classification we construct a functor from the category of cuspidal $W_n$-modules to the category of $W_n$-modules with a compatible action of the algebra of functions on a torus. We also present a new identity for certain quadratic elements in the universal enveloping algebra of $W_1$, which provides important information about cuspidal $W_1$-modules.
We classify simple weight modules over infinite dimensional Weyl algebras and realize them using the action on certain localizations of the polynomial ring. We describe indecomposable projective and injective weight modules and deduce from this a description of blocks of the category of weight modules by quivers and relations. As a corollary we establish Koszulity for all blocks.
For a given poset, we consider its representations by systems of subspaces of a unitary space ordered by inclusion. We classify such systems for all posets for which an explicit classification is possible.
We give a canonical form of matrices of a cycle of linear or semilinear mapping V_1 --- V_2 --- ... --- V_t --- V_1 in which all V_i are complex vector spaces, each line is an arrow ---> or <---, and each arrow denotes a linear or semilinear mapping.
We study $\mathbb Z$-graded modules of nonzero level with arbitrary weight multiplicities over Heisenberg Lie algebras and the associated generalized loop modules over affine Kac-Moody Lie algebras. We construct new families of such irreducible modules over Heisenberg Lie algebras. Our main result establishes the irreducibility of the corresponding generalized loop modules providing an explicit construction of many new examples of irreducible modules for affine Lie algebras. In particular, to any function $ϕ:\mathbb N\rightarrow \{\pm\}$ we associate a $ϕ$-highest weight module over the Heisenberg Lie algebra and a $ϕ$-imaginary Verma module over the affine Lie algebra. We show that any $ϕ$-imaginary Verma module of nonzero level is irreducible.
We study representations of the Loop Kac-Moody Lie algebra g \otimes A, where g is any Kac-Moody algebra and A is a ring of Laurent polynomials in n commuting variables. In particular, we study representations with finite dimensional weight spaces and their graded versions. When we specialize g to be a finite dimensional or affine Lie algebra we obtain modules for toroidal Lie algebras.
A number of recent papers treated the representation theory of partially ordered sets in unitary spaces with the so called orthoscalar relation. Such theory generalizes the classical theory which studies the representations of partially ordered sets in linear spaces. It happens that the results in the unitary case are well-correlated with those in the linear case. The purpose of this article is to shed light on this phenomena.
We define the Virasoro algebra action on imaginary Verma modules for affine sl(2) and construct the analogs of Knizhnik-Zamolodchikov equation in the operator form. Both these results are based on a free field realization of imaginary Verma modules.
A square matrix is nonderogatory if its Jordan blocks have distinct eigenvalues. We give canonical forms (i) for nonderogatory complex matrices up to unitary similarity and (ii) for pairs of complex matrices up to similarity, in which one matrix has distinct eigenvalues. The types of these canonical forms are given by undirected and, respectively, directed graphs with no undirected cycles.
V.I. Arnold [Russian Math. Surveys 26 (2) (1971) 29-43] constructed a miniversal deformation of matrices under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We construct a miniversal deformation of matrices under congruence.
It is shown that the q-difference Noether problem for all classical Weyl groups has a positive solution, simultaneously generalizing well known results on multisymmetric functions of Mattuck and Miyata in the case q=1, and q-deforming the noncommutative Noether problem for the symmetric group. It is also shown that the quantum Gelfand-Kirillov conjecture for gl_N (for a generic q) follows from the positive solution of the q-difference Noether problem for the Weyl group of type D_n. The proof is based on the theory of Galois rings developed by the first author and Ovsienko. From here we obtain a new proof of the quantum Gelfand-Kirillov conjecture for sl_N, thus recovering the result of Fauquant-Millet. Moreover, we provide an explicit description of skew fields of fractions for quantized gl_N and sl_N generalizing Alev and Dumas.
The goal of this paper is to study the representation theory of a classical infinite-dimensional Lie algebra - the Lie algebra of vector fields on an N-dimensional torus for N > 1. The case N=1 gives a famous Virasoro algebra (or its centerless version - the Witt algebra). The algebra of vector fields has an important class of tensor modules parametrized by finite-dimensional modules of gl(N). Tensor modules can be used in turn to construct bounded irreducible modules for the vector fields on N+1-dimensional torus, which are the central objects of our study. We solve two problems regarding these bounded modules: we construct their free field realizations and determine their characters. To solve these problems we analyze the structure of the irreducible modules for the semidirect product of vector fields with the quotient of 1-forms by the differentials of functions. These modules remain irreducible when restricted to the subalgebra of vector fields, unless they belongs to the chiral de Rham complex, introduced by Malikov-Schechtman-Vaintrob.
A twisted generalized Weyl algebra A of degree n depends on a base algebra R, n commuting automorphisms s_i of R, n central elements t_i of R and on some additional scalar parameters. In a paper by V.Mazorchuk and L.Turowska (1999) it is claimed that certain consistency conditions for s_i and t_i are sufficient for the algebra to be nontrivial. However, in this paper we give an example which shows that this is false. We also correct the statement by finding a new set of consistency conditions and prove that the old and new conditions together are necessary and sufficient for the base algebra R to map injectively into A. In particular they are sufficient for the algebra A to be nontrivial. We speculate that these consistency relations may play a role in other areas of mathematics, analogous to the role played by the Yang-Baxter equation in the theory of integrable systems.
We introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordan's simple localization of the multiparameter quantized Weyl algebra and Hayashi's q-analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus, we also describe all Whittaker pairs up to isomorphism over a class of twisted generalized Weyl algebras which includes the multiparameter twisted Weyl algebras.
We prove that any twisted generalized Weyl algebra satisfying certain consistency conditions can be embedded into a crossed product. We also introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordan's simple localization of the multiparameter quantized Weyl algebra and Hayashi's $q$-analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus, we also describe all Whittaker pairs up to isomorphism over a class of twisted generalized Weyl algebras which includes the multiparameter twisted Weyl algebras.
Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A and B be upper triangular n-by-n matrices that (i) are not similar to direct sums of matrices of smaller sizes, or (ii) are in general position and have the same main diagonal. We prove that A and B are unitarily similar if and only if ||h(A_k)||=||h(B_k)|| for all complex polynomials h(x) and k=1, 2, . . , n, where A_k and B_k are the principal k-by-k submatrices of A and B, and ||M|| is the Frobenius norm of M.
We describe (braided-)commutative algebras with non-degenerate multiplicative form in certain braided monoidal categories, corresponding to abelian metric Lie algebras (so-called Drinfeld categories). We also describe local modules over these algebras and classify commutative algebras with finite number of simple local modules.