scholarly journals Center of Schrödinger algebra and annihilators of Verma modules for Schrödinger algebra

2012 ◽  
Vol 437 (1) ◽  
pp. 184-188 ◽  
Author(s):  
Yuezhu Wu ◽  
Linsheng Zhu
Keyword(s):  
Verma Modules ◽  
Schrödinger Algebra

DIFFERENCE ANALOGUES OF THE FREE SCHRÖDINGER EQUATION

1999 ◽  
Vol 14 (17) ◽  
pp. 1113-1122 ◽  
Author(s):  
V. K. DOBREV ◽  
H.-D. DOEBNER ◽  
C. MRUGALLA
Keyword(s):  
Schrödinger Equation ◽  
Difference Equations ◽  
Schrodinger Equation ◽  
Vector Fields ◽  
Differential Operators ◽  
Infinite Family ◽  
Difference Operators ◽  
Verma Modules ◽  
Algebraic Construction ◽  
Schrödinger Algebra

We propose an infinite family of difference equations, which are derived from the first principle that they are invariant with respect to the Schrödinger algebra. The first member of this family is a difference analogue of the free Schrödinger equation. These equations are obtained via a purely algebraic construction from a corresponding family of singular vectors in Verma modules over the Schrödinger algebra. The crucial moment in the construction is the realization of the Schrödinger algebra through additive difference vector fields, i.e. vector fields with difference operators instead of differential operators. Our method produces also differential-difference equations in which only space- or time-differentiation is replaced with the corresponding difference operators.

scholarly journals Classification of Degenerate Verma Modules for E(5, 10)

2021 ◽  
Author(s):  
Nicoletta Cantarini ◽  
Fabrizio Caselli ◽  
Victor Kac
Keyword(s):  
Infinite Number ◽  
Verma Module ◽  
Lie Superalgebra ◽  
Verma Modules ◽  
Singular Vectors ◽  
Finite Dimensional ◽  
De Rham ◽  
Via Classification

AbstractGiven a Lie superalgebra $${\mathfrak {g}}$$ g with a subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 , and a finite-dimensional irreducible $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 -module F, the induced $${\mathfrak {g}}$$ g -module $$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$ M ( F ) = U ( g ) ⊗ U ( g ≥ 0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${\mathfrak {g}}=E(5,10)$$ g = E ( 5 , 10 ) with the subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

scholarly journals NONRELATIVISTIC HOLOGRAPHY — A GROUP-THEORETICAL PERSPECTIVE

2014 ◽  
Vol 29 (03n04) ◽  
pp. 1430001 ◽  
Author(s):  
V. K. DOBREV
Keyword(s):  
Representation Theory ◽  
Theoretical Perspective ◽  
Explicit Construction ◽  
Intertwining Operators ◽  
Difference Operators ◽  
Semisimple Lie Groups ◽  
Boundary Operators ◽  
Boundary Fields ◽  
Theoretical Results ◽  
Schrödinger Algebra

We give a review of some group-theoretical results related to nonrelativistic holography. Our main playgrounds are the Schrödinger equation and the Schrödinger algebra. We first recall the interpretation of nonrelativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. One important result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory, and that these operators and the bulk-to-boundary operators are intertwining operators. Further, we recall the fact that there is a hierarchy of equations on the boundary, invariant with respect to Schrödinger algebra. We also review the explicit construction of an analogous hierarchy of invariant equations in the bulk, and that the two hierarchies are equivalent via the bulk-to-boundary intertwining operators. The derivation of these hierarchies uses a mechanism introduced first for semisimple Lie groups and adapted to the nonsemisimple Schrödinger algebra. These require development of the representation theory of the Schrödinger algebra which is reviewed in some detail. We also recall the q-deformation of the Schrödinger algebra. Finally, the realization of the Schrödinger algebra via difference operators is reviewed.

Automorphisms and Verma Modules for Generalized 2-dim Affine-Virasoro Algebra

2017 ◽  
Vol 24 (02) ◽  
pp. 285-296 ◽  
Author(s):  
Wenlan Ruan ◽  
Honglian Zhang ◽  
Jiancai Sun
Keyword(s):  
Automorphism Group ◽  
Verma Module ◽  
Virasoro Algebra ◽  
Verma Modules

We study the structure of the generalized 2-dim affine-Virasoro algebra, and describe its automorphism group. Furthermore, we also determine the irreducibility of a Verma module over the generalized 2-dim affine-Virasoro algebra.

scholarly journals RINGS OF DEFINABLE SCALARS OF VERMA MODULES

2007 ◽  
Vol 06 (05) ◽  
pp. 779-787 ◽  
Author(s):  
SONIA L'INNOCENTE ◽  
MIKE PREST
Keyword(s):  
Lie Algebra ◽  
Model Theory ◽  
Regular Ring ◽  
Verma Module ◽  
Closed Field ◽  
Verma Modules ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Canonical Map ◽  
Von Neumann Regular

Let M be a Verma module over the Lie algebra, sl 2(k), of trace zero 2 × 2 matrices over the algebraically closed field k. We show that the ring, RM, of definable scalars of M is a von Neumann regular ring and that the canonical map from U( sl 2(k)) to RM is an epimorphism of rings. We also describe the Ziegler closure of M. The proofs make use of ideas from the model theory of modules.

scholarly journals Verma modules over Virasoro-like algebras

2006 ◽  
Vol 80 (2) ◽  
pp. 179-191 ◽  
Author(s):  
Xian-Dong Wang ◽  
Kaiming Zhao
Keyword(s):  
Additive Group ◽  
Total Order ◽  
Verma Modules ◽  
Direct Sum ◽  
Highest Weight ◽  
Highest Weight Modules ◽  
Weight Modules

AbstractLet K be a field of characteristic 0, G the direct sum of two copies of the additive group of integers. For a total order ≺ on G, which is compatible with the addition, and for any ċ1, ċ2 ∈ K, we define G-graded highest weight modules M(ċ1, ċ2, ≺) over the Virasoro-like algebra , indexed by G. It is natural to call them Verma modules. In the present paper, the irreducibility of M (ċ1, ċ2, ≺) is completely determined and the structure of reducible module M (ċ1, ċ2, ≺)is also described.

scholarly journals Quasi-Whittaker modules for the Schrödinger algebra

2014 ◽  
Vol 463 ◽  
pp. 16-32 ◽  
Author(s):  
Yan-an Cai ◽  
Yongsheng Cheng ◽  
Ran Shen
Keyword(s):  
Whittaker Modules ◽  
Schrödinger Algebra
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