scholarly journals COSET REALIZATION OF UNIFYING ${\mathcal W}$ ALGEBRAS

1995 ◽  
Vol 10 (16) ◽  
pp. 2367-2430 ◽  
Author(s):  
R. BLUMENHAGEN ◽  
W. EHOLZER ◽  
A. HONECKER ◽  
R. HÜBEL ◽  
K. HORNFECK
Keyword(s):  
Lie Algebras ◽  
Classical Limit ◽  
Classical Case ◽  
Quantum Level ◽  
Minimal Models ◽  
Classical Formula

We construct several quantum coset [Formula: see text] algebras, e.g. [Formula: see text] and [Formula: see text] and argue that they are finitely nonfreely generated. Furthermore, we discuss in detail their role as unifying [Formula: see text] algebras of Casimir [Formula: see text] algebras. We show that it is possible to give coset realizations of various types of unifying [Formula: see text] algebras; for example, the diagonal cosets based on the symplectic Lie algebras sp (2n) realize the unifying [Formula: see text] algebras which have previously been introduced as [Formula: see text]. In addition, minimal models of [Formula: see text] are studied. The coset realizations provide a generalization of level-rank duality of dual coset pairs. As further examples of finitely nonfreely generated quantum [Formula: see text] algebras, we discuss orbifolding of [Formula: see text] algebras which on the quantum level has different properties than in the classical case. We demonstrate through some examples that the classical limit — according to Bowcock and Watts — of these finitely nonfreely generated quantum [Formula: see text] algebras probably yields infinitely nonfreely generated classical [Formula: see text] algebras.

CLASSICAL NILPOTENT ORBITS AS HYPERKÄHLER QUOTIENTS

1996 ◽  
Vol 07 (02) ◽  
pp. 193-210 ◽  
Author(s):  
PIOTR Z. KOBAK ◽  
ANDREW SWANN
Keyword(s):  
Lie Algebras ◽  
Direct Proof ◽  
Classical Case ◽  
Nilpotent Orbits ◽  
Kahler Geometry ◽  
Complex Semisimple ◽  
Low Dimensional ◽  
Hyperkahler Quotients ◽  
Kahler Quotients ◽  
Quaternionic Kähler

We show that on an arbitrary nilpotent orbit [Formula: see text] in [Formula: see text] where [Formula: see text] is a direct sum of classical simple Lie algebras, there is a G-invariant hyperKähler structure obtainable as a hyperKäher quotient of the flat hyperKähler manifold ℝ4N≅ℍN. Coïncidences between various low-dimensional simple Lie groups lead to some nilpotent orbits being described as hyperKähler quotients (in some cases in fact finite quotients) of other nilpotent orbits. For example, from the construction we are able to read off pairs of orbits [Formula: see text] in different classical Lie algebras [Formula: see text] such that there is a finite [Formula: see text]-equivariant surjection [Formula: see text] between the orbit closures. We include a table listing examples of hyperKähler quotients between small nilpotent orbits. The above-mentioned results have consequences in quaternionic Kähler geometry: it is known that nilpotent orbits in complex semisimple Lie algebras give rise to quaternionic Kähler manifolds. Our approach gives a more direct proof of this in the classical case as these manifolds turn out to be quaternionic Kähler quotients of quaternionic projective spaces. We find that many of these manifolds can also be constructed as quaternionic Kähler quotients of complex Grassmannians [Formula: see text].

scholarly journals Formality conjecture for minimal surfaces of Kodaira dimension 0

2021 ◽  
Vol 157 (2) ◽  
pp. 215-235
Author(s):  
Ruggero Bandiera ◽  
Marco Manetti ◽  
Francesco Meazzini
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Minimal Surfaces ◽  
K3 Surfaces ◽  
Kodaira Dimension ◽  
Projective Surface ◽  
Cyclic Structure ◽  
Minimal Models

Let $\mathcal {F}$ be a polystable sheaf on a smooth minimal projective surface of Kodaira dimension 0. Then the differential graded (DG) Lie algebra $R\operatorname {Hom}(\mathcal {F},\mathcal {F})$ of derived endomorphisms of $\mathcal {F}$ is formal. The proof is based on the study of equivariant $L_{\infty }$ minimal models of DG Lie algebras equipped with a cyclic structure of degree 2 which is non-degenerate in cohomology, and does not rely (even for K3 surfaces) on previous results on the same subject.

scholarly journals DUALISM BETWEEN PHYSICAL FRAMES AND TIME IN QUANTUM GRAVITY

2004 ◽  
Vol 19 (20) ◽  
pp. 1519-1527 ◽  
Author(s):  
SIMONE MERCURI ◽  
GIOVANNI MONTANI
Keyword(s):  
Quantum Gravity ◽  
Reference Frame ◽  
Path Integral ◽  
Classical Limit ◽  
Quantum Dynamics ◽  
Einstein Equations ◽  
Quantum Level ◽  
Dust Fluid ◽  
Lapse Function ◽  
Canonical Quantum

In this work we present a discussion of the existing links between the procedures of endowing the quantum gravity with a real time and of including in the theory a physical reference frame. More precisely, as a first step, we develop the canonical quantum dynamics, starting from the Einstein equations in presence of a dust fluid and arrive at a Schrödinger evolution. Then, by fixing the lapse function in the path-integral of gravity, we get a Schrödinger quantum dynamics, of which eigenvalues problem provides the appearance of a dust fluid in the classical limit. The main issue of our analysis is to claim that a theory, in which the time displacement invariance, on a quantum level, is broken, is indistinguishable from a theory for which this symmetry holds, but a real reference fluid is included.

scholarly journals QUANTUM RIEMANN SURFACES, 2-D GRAVITY AND THE GEOMETRICAL ORIGIN OF MINIMAL MODELS

1994 ◽  
Vol 09 (31) ◽  
pp. 2871-2878 ◽  
Author(s):  
MARCO MATONE
Keyword(s):  
Quantum Gravity ◽  
Classical Limit ◽  
Riemann Surfaces ◽  
Ward Identity ◽  
Riemann Sphere ◽  
Projective Connection ◽  
Minimal Models ◽  
Standard Representation ◽  
Basic Object ◽  
Liouville Action

Based on a recent paper by Takhtajan, we propose a formulation of 2-D quantum gravity whose basic object is the Liouville action on the Riemann sphere Σ0, m+n with both parabolic and elliptic points. The identification of the classical limit of the conformal Ward identity with the Fuchsian projective connection on Σ0, m+n implies a relation between conformal weights and ramification indices. This formulation works for arbitrary d and admits a standard representation only for d ≤ 1. Furthermore, it turns out that the integerness of the ramification number constrains d = 1 − 24/(n2 − 1) that for n = 2m + 1 coincides with the unitary minimal series of CFT.

NEW SOLUTIONS OF THE YANG-BAXTER EQUATION AND THEIR YANG-BAXTERIZATION

1991 ◽  
Vol 06 (04) ◽  
pp. 559-576 ◽  
Author(s):  
M. COUTURE ◽  
Y. CHENG ◽  
M.L. GE ◽  
K. XUE
Keyword(s):  
Lie Algebras ◽  
Braid Group ◽  
Classical Limit ◽  
Usual Sense ◽  
Baxter Equation ◽  
Group Representations ◽  
Simple Lie Algebras

Through the examples associated with simple Lie algebras B2, C2 and D2, an approach of constructing new solutions of the spectral-independent Yang-Baxter equation (i.e. the braid group representations) was shown. These new solutions possess some particular features, such as that they do not have the classical limit in the usual sense of Refs. 2, 5, etc., and give rise to the new solutions of the x-dependent Yang-Baxter equation by using the Yang-Baxterization prescription.

SCHEMES OF TORI AND THE STRUCTURE OF TAME RESTRICTED LIE ALGEBRAS

2001 ◽  
Vol 63 (3) ◽  
pp. 553-570 ◽  
Author(s):  
ROLF FARNSTEINER ◽  
DETLEF VOIGT
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Classical Case ◽  
Forthcoming Paper ◽  
Structure Theory ◽  
Group Schemes ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras ◽  
Geometric Techniques ◽  
Support Varieties

Much of the recent progress in the representation theory of infinitesimal group schemes rests on the application of algebro-geometric techniques related to the notion of cohomological support varieties (cf. [6, 8–10]). The noncohomological characterization of these varieties via the so-called rank varieties (see [21, 22]) involves schemes of additive subgroups that are the infinitesimal counterparts of the elementary abelian groups. In this note we introduce another geometric tool by considering schemes of tori of restricted Lie algebras. Our interest in these derives from the study of infinitesimal groups of tame representation type, whose determination [12] necessitates the results to be presented in §4 and §5 as well as techniques from abstract representation theory.In contrast to the classical case of complex Lie algebras, the information on the structure of a restricted Lie algebra that can be extracted from its root systems is highly sensitive to the choice of the underlying maximal torus. Schemes of tori obviate this defect by allowing us to study algebraic families of root spaces. Accordingly, these schemes may also shed new light on various aspects of the structure theory of restricted Lie algebras. We intend to pursue these questions in a forthcoming paper [13], and focus here on first applications within representation theory.

scholarly journals Derivation double Lie algebras

2016 ◽  
Vol 15 (06) ◽  
pp. 1650114
Author(s):  
Dietrich Burde
Keyword(s):  
Recent Work ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Classical Formula ◽  
Simple Derivation ◽  
Affine Actions

We study classical [Formula: see text]-matrices [Formula: see text] for Lie algebras [Formula: see text] such that [Formula: see text] is also a derivation of [Formula: see text]. This yields derivation double Lie algebras [Formula: see text]. The motivation comes from recent work on post-Lie algebra structures on pairs of Lie algebras arising in the study of nil-affine actions of Lie groups. We prove that there are no nontrivial simple derivation double Lie algebras, and study related Lie algebra identities for arbitrary Lie algebras.

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