Alexandre Kirillov

Alexandre Kirillov
University of Pennsylvania | UP · Department of Mathematics

Doctor of sciences

About

121
Publications
8,327
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3,389
Citations
Citations since 2016
1 Research Item
689 Citations
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2016201720182019202020212022020406080100
2016201720182019202020212022020406080100
2016201720182019202020212022020406080100

Publications

Publications (121)
Article
We consider tangent cones of Schubert varieties in the complete flag variety, and investigate the problem when the tangent cones of two different Schubert varieties coincide. We give a sufficient condition for such coincidence, and formulate a conjecture that provides a necessary condition. In particular, we show that all Schubert varieties corresp...
Article
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We consider the Lie algebra \(\mathfrak{g} = \mathfrak{p}_n \) of (n + 1) × (n + 1) matrices with zeros in the last row. This algebra has received the name of mirabolic; it has many remarkable properties and plays an important role in representation theory. In this paper we study open coadjoint orbits for the corresponding Lie group P n .
Book
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Since Benoit Mandelbrot's pioneering work in the late 1970s, scores of research articles and books have been published on the topic of fractals. Despite the volume of literature in the field, the general level of theoretical understanding has remained low; most work is aimed either at too mainstream an audience to achieve any depth or at too specia...
Chapter
The Laplace operator on the Sierpiński gasket. The Laplace operator on Euclidean space and its analogue on graphs. Maximum principle for harmonic functions. Eigenfunctions of the Laplace operator on the Sierpiński gasket. Comparing the spectra of the Laplace operator on different approximations.
Article
We give an explicit formula for the exterior powers ∧ k π 1 of the defining representation π 1 of the simple Lie algebra ςο(2n + 1, ℂ). We use the technique of family algebras. All representations in question are children of the spinor representation σ of g2ο(2n + 1, ℂ). We also give a survey of main results on family algebras.
Article
I tell the story of may acquaintance with George Mackey as mathematician and as a man. Next, I compare the “golden years of representation theory” in Russian with those of the West. Finally, I discuss Mackey’s Imprimitivity theorem and igve a natural proof of it in the case of lie groups.
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In this article we investigate the structure of local Lie algebras with a one-dimensional fibre. We show that all such Lie algebras are essentially exhausted by the classical examples of the Hamiltonian and contact Poisson bracket algebras. We give some examples, unsolved problems, and applications of Lie superalgebras.
Chapter
A new class of associative algebras related to simple complex Lie algebras (or root systems) was introduced and studied in tikya[K1] and tikya[K2]. They were named classical and quantum family algebras. The aim of this paper is to introduce the odd analogue of these algebras and expose some results about their structure. In particular, we describe...
Article
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This is an overview article on compact Lie groups and their representations, written for the Encyclopedia of Mathematical Physics to be published by Elsevier.
Article
In this paper, we study $G$-equivariant tensor categories for a finite group $G$. These categories were introduced by Turaev under the name of $G$-crossed categories; the motivating example of such a category is the category of twisted modules over a vertex operator algebra $V$ with a finite group of automorphisms $G$. We discuss the notion of "orb...
Article
In this note, we give a description of the modular functor associated to the Chern-Simons theory with a finite group from the complex-analytic point of view, i.e. as a vector bundle with a flat connection on the moduli space of punctured curves. We show that it can be obtained from the trivial local system on the moduli space of "admissible G-cover...
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The orbit method was created 40 years ago (see [K1]) in the attempt to describe the unitary dual N for the group N of upper triangular matrices with units on the main diagonal.
Article
This is a continuation of the paper "Modular tensor categories and orbifold theories", arXiv:math.QA/0104242. It discusses orbifold models of conformal filed theory, or, in mathematical language, question of constructing the category of representations of the fixed point algebra $V^G$ for a given vertex operator algebra $V$ with an action of a fini...
Article
The goal of this paper is to classify ``finite subgroups in U_q sl(2)'' where $q=e^{\pi\i/l}$ is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of U_q sl(2); we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related...
Article
Classical and quantum family algebras, previously introduced by the author and playing an important role in the theory of semi-smiple Lie algebras and their representations are studied. Basic properties, structure theorems and explicit fomulas are obtained for both types of family algebras in many significant cases. Exact formulas (based on experim...
Article
Full-text available
A new class of associative algebras is introduced and studied. These algebras are related to simple complex Lie algebras (or root systems). Roughly speaking, they are finite dimensional approximations to the envelop-ing algebra U (g) viewed as a module over its center. It seems that several important questions on semisimple algebras and their repre...
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We present a solution to the problem of finding the index of parabolic subalgebras of GL(n) and expose their relationship to interesting combinatorial objects, namely, Meanders.
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This survey is the expanded version of my talk at the AMS meet- ing in April 1997. I explain to non-experts how to use the orbit method, discuss its strong and weak points and advertise some open problems.
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We want to realize the discrete series of unirreps for the Virasoro-Bott group Vir (= the central extension of Diff+(S1)) in the space of holomorphic functions on the infinite dimensional Kähler manifold M = Diff+(S1)/S1. The explicit formulae are given for the action of Vir in the space of polynomial functions in the natural complex coordinates on...
Article
Full-text available
A remarkable sequence of polynomials is considered. These polynomials in q describe in particular the number of solution to the equation X 2 = 0 in triangular n × n matrices over a field q with q elements. They have at least three other important interpretations and a conjectural explicit expression in terms of the entries of the Catalan triangle....
Book
This EMS volume contains two contributions: the first one, "Harmonic Analysis on Homogeneous Spaces", is written by V.F.Molchanov, the second one, "Representations of Lie Groups and Special Functions", by N.Ya.Vilenkin and A.U.Klimyk. Molchanov focuses on harmonic analysis on semi-simple spaces, whereas Vilenkin and Klimyk treat group theoretical m...
Chapter
Noncommutative harmonic analysis and its basic tool — the theory of group representations — has existed as an independent domain of mathematics for about 100 years.
Article
Not Available Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints
Chapter
Below we consider coordinate systems on a torus T2 that associate with each point P a pair of real numbers (x, y) defined modulo 2π.
Article
The word “quantization” is used both in physical und in mathematical works in many different senses. In recent times this has come to be reflected explicitly in the terminology: the terms “asymptotic”, “deformational”, “geometric” quantization, etc., have emerged.
Article
A geometric background is given for representation with a highest weight of the Virasoro algebra. The representation space consists of holomorphic sections of an analytic line bundle over the manifold M = Diff+S1/Rot S1 or over its factor manifold M1 = Diff+S1/PSL (2, {A figure is presented}). A class of polynomial sections of such a bundle is intr...
Article
In this paper, we study an approach by Gelfand-Tsetlin to the representation of symplectic groups.
Article
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Article
The operations on geometric objects presently known which commute with changes of variables are described, and their properties are discussed.
Article
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Article
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This survey is timed to coincide with the sixtieth birthday of I. M. Gel'fand. The authors have confined themselves to those branches of mathematics in which he has been engaged during the last decade, and in the various branches the chronology of the articles covered by the survey is different. Gel'fand's research in the theory of group representa...
Article
. We want to realize the discrete series of unirreps for the Virasoro-Bott group V ir (= the central extension of Diff+ (S 1 ) ) in the space of holomorphic functions on the infinite dimensional Kahler manifold M = Diff+ (S 1 )=S 1 . The explicit formulae are given for the action of V ir in the space of polynomial functions in the natural complex c...
Article
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. We continue the study of family algebras introduced by the author. In this paper we describe completely the structure of quantum family algebras for two cases of representations with a simple spectrum. 1. Generalities about family algebras 1.1. Basic denitions. A new class of associative algebras related to simple complex Lie algebras (or root sy...
Article
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In a role of numbers can occur not only elements of a field or a skew-field. In this part I talk about other mathematical objects which are used as numbers. It is interesting that all of them had appeared first in "pure" mathematics and later were used in mathematical physics and some other applications. It support the thesis that there is only one...
Article
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The aim of this course is to show, what meaning has the notion of number in modern mathematics; tell about the problems arising in connection with dierent understanding of numbers and how these problems are being solved. Of course, I can explain only first steps of corresponding theories. For those who want to know more, I indicate the appropriate...

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