scholarly journals Generalised holonomies and K(E9)

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Axel Kleinschmidt ◽  
Hermann Nicolai
Keyword(s):  
Infinite Sequence ◽  
Moody Algebra ◽  
Consistency Conditions ◽  
Direct Sum ◽  
Holonomy Groups ◽  
Levi Subalgebra

Abstract The involutory subalgebra K($$ \mathfrak{e} $$ e 9) of the affine Kac-Moody algebra $$ \mathfrak{e} $$ e 9 was recently shown to admit an infinite sequence of unfaithful representations of ever increasing dimensions [1]. We revisit these representations and describe their associated ideals in more detail, with particular emphasis on two chiral versions that can be constructed for each such representation. For every such unfaithful representation we show that the action of K($$ \mathfrak{e} $$ e 9) decomposes into a direct sum of two mutually commuting (‘chiral’ and ‘anti-chiral’) parabolic algebras with Levi subalgebra $$ \mathfrak{so} $$ so (16)+ ⊕ $$ \mathfrak{so} $$ so (16)−. We also spell out the consistency conditions for uplifting such representations to unfaithful representations of K($$ \mathfrak{e} $$ e 10). From these results it is evident that the holonomy groups so far discussed in the literature are mere shadows (in a Platonic sense) of a much larger structure.

scholarly journals On indecomposable decompositions of CS-modules

1996 ◽  
Vol 61 (1) ◽  
pp. 30-41 ◽  
Author(s):  
Nguyen V. Dung
Keyword(s):  
Direct Summand ◽  
Infinite Sequence ◽  
Exchange Property ◽  
Endomorphism Rings ◽  
Direct Sum

AbstractIt is shown that, over any ring R, the direct sum M = ⊕i∈IMi of uniform right R-modules Mi with local endomorphism rings is a CS-module if and only if every uniform submodule of M is essential in a direct summand of M and there does not exist an infinite sequence of non-isomorphic monomorphisms , with distinct in ∈ I. As a consequence, any CS-module which is a direct sum of submodules with local endomorphism rings has the exchange property.

scholarly journals Poisson-Lie T-duality of WZW model via current algebra deformation

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Francesco Bascone ◽  
Franco Pezzella ◽  
Patrizia Vitale
Keyword(s):  
Phase Space ◽  
Configuration Space ◽  
Current Algebra ◽  
Degrees Of Freedom ◽  
Target Space ◽  
Moody Algebra ◽  
Hamiltonian Description ◽  
Direct Sum ◽  
Group Manifold ◽  
Two Parameter

Abstract Poisson-Lie T-duality of the Wess-Zumino-Witten (WZW) model having the group manifold of SU(2) as target space is investigated. The whole construction relies on the deformation of the affine current algebra of the model, the semi-direct sum $$ \mathfrak{su}(2)\left(\mathrm{\mathbb{R}}\right)\overset{\cdot }{\oplus}\mathfrak{a} $$ su 2 ℝ ⊕ ⋅ a , to the fully semisimple Kac-Moody algebra $$ \mathfrak{sl}\left(2,\mathrm{\mathbb{C}}\right)\left(\mathrm{\mathbb{R}}\right) $$ sl 2 ℂ ℝ . A two-parameter family of models with SL(2, ℂ) as target phase space is obtained so that Poisson-Lie T-duality is realised as an O(3, 3) rotation in the phase space. The dual family shares the same phase space but its configuration space is SB(2, ℂ), the Poisson-Lie dual of the group SU(2). A parent action with doubled degrees of freedom on SL(2, ℂ) is defined, together with its Hamiltonian description.

POISSON-NIJENHUIS STRUCTURES AND SATO HIERARCHY

1991 ◽  
Vol 03 (04) ◽  
pp. 403-466 ◽  
Author(s):  
G. MAGNANO ◽  
F. MAGRI
Keyword(s):  
Finite Number ◽  
Lie Algebra ◽  
Vector Fields ◽  
Hamiltonian Structure ◽  
Differential Operators ◽  
Hamiltonian Vector ◽  
Poisson Brackets ◽  
Moody Algebra ◽  
Hamiltonian Vector Fields ◽  
Direct Sum

We show that the direct sum of n copies of a Lie algebra is endowed with a sequence of affine Lie-Poisson brackets, which are pairwise compatible and define a multi-Hamiltonian structure; to this structure one can associate a recursion operator and a Kac-Moody algebra of Hamiltonian vector fields. If the initial Lie algebra is taken to be an associative algebra of differential operators, a suitable family of Hamiltonian vector fields reproduce either the n-th Gel'fand-Dikii hierarchy (for n finite) or Sato's hierarchy (for n = ∞). Within the same framework, it is also possible to recover a class of integro-differential hierarchies involving a finite number of fields, which generalize the Gel'fand-Dikii equations and are equivalent to Sato's hierarchy.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

scholarly journals Nonsingular bilinear forms on direct sums of ideals

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Beata Rothkegel
Keyword(s):  
Bilinear Form ◽  
Dedekind Domain ◽  
Bilinear Forms ◽  
Direct Sums ◽  
Direct Sum

AbstractIn the paper we formulate a criterion for the nonsingularity of a bilinear form on a direct sum of finitely many invertible ideals of a domain. We classify these forms up to isometry and, in the case of a Dedekind domain, up to similarity.

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

scholarly journals Expressiveness and Nash Equilibrium in Iterated Boolean Games

2021 ◽  
Vol 22 (2) ◽  
pp. 1-38
Author(s):  
Julian Gutierrez ◽  
Paul Harrenstein ◽  
Giuseppe Perelli ◽  
Michael Wooldridge
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Infinite Sequence ◽  
Multi Agent Systems ◽  
Temporal Logics ◽  
Agent Systems ◽  
Temporal Properties ◽  
Multi Agent ◽  
Game Theoretic ◽  
Boolean Games

We define and investigate a novel notion of expressiveness for temporal logics that is based on game theoretic equilibria of multi-agent systems. We use iterated Boolean games as our abstract model of multi-agent systems [Gutierrez et al. 2013, 2015a]. In such a game, each agent  has a goal  , represented using (a fragment of) Linear Temporal Logic ( ) . The goal  captures agent  ’s preferences, in the sense that the models of  represent system behaviours that would satisfy  . Each player controls a subset of Boolean variables , and at each round in the game, player is at liberty to choose values for variables in any way that she sees fit. Play continues for an infinite sequence of rounds, and so as players act they collectively trace out a model for , which for every player will either satisfy or fail to satisfy their goal. Players are assumed to act strategically, taking into account the goals of other players, in an attempt to bring about computations satisfying their goal. In this setting, we apply the standard game-theoretic concept of (pure) Nash equilibria. The (possibly empty) set of Nash equilibria of an iterated Boolean game can be understood as inducing a set of computations, each computation representing one way the system could evolve if players chose strategies that together constitute a Nash equilibrium. Such a set of equilibrium computations expresses a temporal property—which may or may not be expressible within a particular fragment. The new notion of expressiveness that we formally define and investigate is then as follows: What temporal properties are characterised by the Nash equilibria of games in which agent goals are expressed in specific fragments of  ? We formally define and investigate this notion of expressiveness for a range of fragments. For example, a very natural question is the following: Suppose we have an iterated Boolean game in which every goal is represented using a particular fragment of : is it then always the case that the equilibria of the game can be characterised within ? We show that this is not true in general.

scholarly journals Tropical ideals do not realise all Bergman fans

2021 ◽  
Vol 8 (3) ◽  
Author(s):  
Jan Draisma ◽  
Felipe Rincón
Keyword(s):  
Tropical Geometry ◽  
Tensor Products ◽  
Tropical Variety ◽  
Polyhedral Complex ◽  
Direct Sum ◽  
Weight One ◽  
Polyhedral Complexes

AbstractEvery tropical ideal in the sense of Maclagan–Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements and in which all maximal cones have weight one.

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