scholarly journals Log smooth deformation theory via Gerstenhaber algebras

2020 ◽  
Author(s):  
Simon Felten
Keyword(s):  
Lie Algebra ◽  
Mirror Symmetry ◽  
Deformation Theory ◽  
Geometric Interpretation ◽  
Algebraic Version ◽  
Gerstenhaber Algebra ◽  
Recent Developments ◽  
Graded Lie Algebra ◽  
Gerstenhaber Algebras ◽  
Smooth Deformation

AbstractWe construct a $$k\left[ \!\left[ Q\right] \!\right] $$ k Q -linear predifferential graded Lie algebra $$L^{\bullet }_{X_0/S_0}$$ L X 0 / S 0 ∙ associated to a log smooth and saturated morphism $$f_0: X_0 \rightarrow S_0$$ f 0 : X 0 → S 0 and prove that it controls the log smooth deformation functor. This provides a geometric interpretation of a construction in Chan et al. (Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties, 2019. arXiv:1902.11174) whereof $$L^{\bullet }_{X_0/S_0}$$ L X 0 / S 0 ∙ is a purely algebraic version. Our proof crucially relies on studying deformations of the Gerstenhaber algebra of polyvector fields; this method is closely related to recent developments in mirror symmetry.

scholarly journals Nijenhuis operators on pre-Lie algebras

2019 ◽  
Vol 21 (07) ◽  
pp. 1850050 ◽  
Author(s):  
Qi Wang ◽  
Yunhe Sheng ◽  
Chengming Bai ◽  
Jiefeng Liu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Regular Representation ◽  
Complex Structure ◽  
Geometric Interpretation ◽  
Dual Representation ◽  
New Approach ◽  
Lie Bracket ◽  
Graded Lie Algebra ◽  
Dendriform Algebras

First we use a new approach to define a graded Lie algebra whose Maurer–Cartan elements characterize pre-Lie algebra structures. Then using this graded Lie bracket, we define the notion of a Nijenhuis operator on a pre-Lie algebra which generates a trivial deformation of this pre-Lie algebra. There are close relationships between [Formula: see text]-operators, Rota–Baxter operators and Nijenhuis operators on a pre-Lie algebra. In particular, a Nijenhuis operator “connects” two [Formula: see text]-operators on a pre-Lie algebra whose any linear combination is still an [Formula: see text]-operator in certain sense and hence compatible [Formula: see text]-dendriform algebras appear naturally as the induced algebraic structures. For the case of the dual representation of the regular representation of a pre-Lie algebra, there is a geometric interpretation by introducing the notion of a pseudo-Hessian–Nijenhuis structure which gives rise to a sequence of pseudo-Hessian and pseudo-Hessian–Nijenhuis structures. Another application of Nijenhuis operators on pre-Lie algebras in geometry is illustrated by introducing the notion of a para-complex structure on a pre-Lie algebra and then studying para-complex quadratic pre-Lie algebras and para-complex pseudo-Hessian pre-Lie algebras in detail. Finally, we give some examples of Nijenhuis operators on pre-Lie algebras.

scholarly journals Refined Scattering Diagrams and Theta Functions From Asymptotic Analysis of Maurer–Cartan Equations

2019 ◽  
Author(s):  
Naichung Conan Leung ◽  
Ziming Nikolas Ma ◽  
Matthew B Young
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Lie Algebra ◽  
Theta Functions ◽  
Geometric Interpretation ◽  
Analytic Approach ◽  
Hall Algebra ◽  
Wall Crossing ◽  
Graded Lie Algebra ◽  
Do So

Abstract We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer–Cartan elements of a (dg) Lie algebra constructed from a (not necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differential geometric proofs of the consistent completion of scattering diagrams, originally proved by Kontsevich–Soibelman, Gross–Siebert, and Bridgeland. We also give a geometric interpretation of theta functions and their wall-crossing. In the tropical setting, we interpret Maurer–Cartan elements, and therefore consistent scattering diagrams, in terms of the refined counting of tropical disks. We also describe theta functions, in both their tropical and Hall algebraic settings, in terms of distinguished flat sections of the Maurer–Cartan-deformed differential. In particular, this allows us to give a combinatorial description of Hall algebra theta functions for acyclic quivers with nondegenerate skew-symmetrized Euler forms.

SPECTRAL FLOW AND FREE FIELD REALIZATIONS OF BOSONIC STRINGS ON NAPPI–WITTEN GROUP MANIFOLD

2011 ◽  
Vol 26 (01) ◽  
pp. 149-160
Author(s):  
GANG CHEN
Keyword(s):  
Lie Algebra ◽  
Bosonic Strings ◽  
Free Field ◽  
Geometric Interpretation ◽  
Closed String ◽  
Space Time ◽  
Spectral Flow ◽  
Affine Lie Algebra ◽  
Group Manifold ◽  
Free Field Realization

In this paper we study some aspects of closed string theories in the Nappi–Witten space–time. The effects of spectral flow on the geodesics are studied in terms of an explicit parametrization of the group manifold. The worldsheets of the closed strings under the spectral flow of the geodesics can be classified into four classes, each with a geometric interpretation. We also obtain a free field realization of the Nappi–Witten affine Lie algebra in the most general conditions using a different but equivalent parametrization of the group manifold.

Symbols in Berezin quantization for representation operators

2021 ◽  
pp. 296-304
Author(s):  
Vladimir F. Molchanov ◽  
Svetlana V. Tsykina
Keyword(s):  
Homogeneous Space ◽  
Lie Algebra ◽  
Grassmann Manifold ◽  
Homogeneous Spaces ◽  
Finite Multiplicity ◽  
Basic Notion ◽  
Berezin Quantization ◽  
Algebraic Version ◽  
Enveloping Algebra ◽  
Algebra Of Operators

The basic notion of the Berezin quantization on a manifold M is a correspondence which to an operator A from a class assigns the pair of functions F and F^♮ defined on M. These functions are called covariant and contravariant symbols of A. We are interested in homogeneous space M=G/H and classes of operators related to the representation theory. The most algebraic version of quantization — we call it the polynomial quantization — is obtained when operators belong to the algebra of operators corresponding in a representation T of G to elements X of the universal enveloping algebra Env g of the Lie algebra g of G. In this case symbols turn out to be polynomials on the Lie algebra g. In this paper we offer a new theme in the Berezin quantization on G/H: as an initial class of operators we take operators corresponding to elements of the group G itself in a representation T of this group. In the paper we consider two examples, here homogeneous spaces are para-Hermitian spaces of rank 1 and 2: a) G=SL(2;R), H — the subgroup of diagonal matrices, G/H — a hyperboloid of one sheet in R^3; b) G — the pseudoorthogonal group SO_0 (p; q), the subgroup H covers with finite multiplicity the group SO_0 (p-1,q -1)×SO_0 (1;1); the space G/H (a pseudo-Grassmann manifold) is an orbit in the Lie algebra g of the group G.

The Kinematics of Cyclotron Emission for Quantized Electrons

1982 ◽  
Vol 4 (4) ◽  
pp. 359-362 ◽  
Author(s):  
D.B. Melrose ◽  
R.G. Hewitt ◽  
A.J. Parle
Keyword(s):  
Magnetic Field ◽  
Classical Theory ◽  
Major Axis ◽  
Lorentz Factor ◽  
Geometric Interpretation ◽  
Semi Major Axis ◽  
Electron Cyclotron Masers ◽  
Recent Developments ◽  
Cyclotron Emission ◽  
The Magnetic Field

Since the work of Wu and Lee (1979) there has been renewed interest in the classical theory of electron cyclotron masers (Lee and Wu 1980, Lee et al. 1980, Wu et al. 1981, 1982, Hewitt et al. 1981, 1982, Melrose et al. 1982, Omidi and Gurnett 1982, Melrose and Dulk 1982). A useful idea in these recent developments of the classical theory concerns a geometric interpretation of the classical gyroresonance conditionwhere Ωe, is the nonrelativistic gyrofrequency, s = 0, ± 1, ± 2,… is the harmonic number, is the Lorentz factor and ║ and ┴ denote components parallel and perpendicular to the magnetic field. In v┴ − v║ space (1) represents an ellipse with centre v║ = vc, v┴ = 0, eccentricity e0 and semi-major axis V parallel to the v┴ axis, with

scholarly journals Parallel submanifolds of complex space forms II

1983 ◽  
Vol 91 ◽  
pp. 119-149 ◽  
Author(s):  
Hiroo Naitoh
Keyword(s):  
Lie Algebra ◽  
Jordan Algebra ◽  
Triple System ◽  
Complex Space ◽  
Space Forms ◽  
Jordan Triple System ◽  
Graded Lie Algebra ◽  
Complex Space Forms

This is a continuation of Part I, which appeared in this journal.In the previous paper I we have defined the following notions: orthogonal Jordan triple system (OJTS), orthogonal symmetric graded Lie algebra (OSGLA), orthogonal Jordan algebra (OJA), Hermitian symmetric graded Lie algebra (HSGLA). And we have shown that equivalent classes of OJTS naturally correspond to equivalent classes of OSGLA and through this correspondence we have naturally constructed HSGLA’s from the OJTS’s associated with OJA’s with unity.

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