JORDAN SYMMETRIC SPACES

2009 ◽  
Vol 02 (03) ◽  
pp. 407-415
Author(s):  
Cho-Ho Chu
Keyword(s):  
Symmetric Spaces ◽  
Hermitian Symmetric Spaces ◽  
Triple Systems ◽  
Infinite Dimensional ◽  
Jordan Triple Systems ◽  
Riemannian Symmetric Spaces

We introduce a class of Riemannian symmetric spaces, called Jordan symmetric spaces, which correspond to real Jordan triple systems and may be infinite dimensional. This class includes the symmetric R-spaces as well as the Hermitian symmetric spaces.

TERNARY ALGEBRAIC CONSTRUCTION OF EXTENDED SUPERCONFORMAL ALGEBRAS

1991 ◽  
Vol 06 (19) ◽  
pp. 1733-1743 ◽  
Author(s):  
MURAT GÜNAYDIN ◽  
SEUNGJOON HYUN
Keyword(s):  
Lie Groups ◽  
Symmetric Spaces ◽  
Coset Space ◽  
Algebraic Construction ◽  
Hermitian Symmetric Spaces ◽  
Magic Square ◽  
Triple Systems ◽  
Coset Spaces ◽  
Jordan Triple Systems ◽  
Ternary Algebras

We give a construction of extended (N = 2 and N = 4) superconformal algebras over a very general class of ternary algebras (triple systems). For N = 2 this construction leads to superconformal algebras corresponding to certain coset spaces of Lie groups with non-vanishing torsion and generalizes a previous construction over Jordan triple systems which are associated with Hermitian symmetric spaces. In general, a given Lie group admits more than one coset space of this type. We give examples for all simple Lie groups. In particular, the division algebras and their tensor products lead to N = 2 superconformal algebras associated with the groups of the Magic Square. For a very special class of ternary algebras, namely the Freudenthal triple (FT) systems, the N = 2 superconformal algebras can be extended to N = 4 superconformal algebras with the gauge group SU (2) × SU (2) × U (1). We give a complete list of the FT systems and the corresponding N = 4 models. They are associated with the unique quaternionic symmetric spaces of Lie groups.

scholarly journals Boundary maps and maximal representations on infinite-dimensional Hermitian symmetric spaces

2021 ◽  
pp. 1-50
Author(s):  
BRUNO DUCHESNE ◽  
JEAN LÉCUREUX ◽  
MARIA BEATRICE POZZETTI
Keyword(s):  
Symmetric Spaces ◽  
Hermitian Symmetric Spaces ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Boundary Map ◽  
Density Assumption ◽  
Maximal Representations ◽  
Tube Type ◽  
Maximal Representation ◽  
Hyperbolic Lattices

Abstract We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite-dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type, we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite-dimensional totally geodesic subspace on which the action is maximal. In the opposite direction, we construct examples of geometrically dense maximal representation in the infinite-dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, which we are able to construct in low ranks or under some suitable Zariski density assumption, circumventing the lack of local compactness in the infinite-dimensional setting.

Representations of simple anti-Jordan triple systems of m × n matrices

2017 ◽  
Vol 16 (05) ◽  
pp. 1750093 ◽  
Author(s):  
Hader A. Elgendy
Keyword(s):  
Triple System ◽  
Closed Field ◽  
Monomial Basis ◽  
Triple Systems ◽  
Infinite Dimensional ◽  
Matrix Algebras ◽  
Jordan Triple System ◽  
Finite Dimensional ◽  
Jordan Triple Systems

We show that the universal associative envelope of the simple anti-Jordan triple system of all [Formula: see text] ([Formula: see text] is even, [Formula: see text]) matrices over an algebraically closed field of characteristic 0 is finite-dimensional. The monomial basis and the center of the universal envelope are determined. The explicit decomposition of the universal envelope into matrix algebras is given. The classification of finite-dimensional irreducible representations of an anti-Jordan triple system is obtained. The semi-simplicity of the universal envelope is shown. We also show that the universal associative envelope of the simple polarized anti-Jordan triple system of [Formula: see text] matrices is infinite-dimensional.

scholarly journals ON MATRIX KP AND SUPER-KP HIERARCHIES IN THE HOMOGENEOUS GRADING

1996 ◽  
Vol 11 (18) ◽  
pp. 3257-3295 ◽  
Author(s):  
F. TOPPAN
Keyword(s):  
Power Series ◽  
Symmetric Spaces ◽  
Unexpected Result ◽  
Hermitian Symmetric Spaces ◽  
Regular Elements ◽  
Symmetry Properties ◽  
Alternating Sums ◽  
Spectral Parameter Power Series ◽  
Special Limit ◽  
Free Parameters

Constrained KP and super-KP hierarchies of integrable equations (generalized NLS hierarchies) are systematically produced through a Lie-algebraic AKS matrix framework associated with the homogeneous grading. The role played by different regular elements in defining the corresponding hierarchies is analyzed, as well as the symmetry properties under the Weyl group transformations. The coset structure of higher order Hamiltonian densities is proven. For a generic Lie algebra the hierarchies considered here are integrable and essentially dependent on continuous free parameters. The bosonic hierarchies studied in Refs. 1 and 2 are obtained as special limit restrictions on Hermitian symmetric spaces. In the supersymmetric case the homogeneous grading is introduced consistently by using alternating sums of bosons and fermions in the spectral parameter power series. The bosonic hierarchies obtained from [Formula: see text] and the supersymmetric ones derived from the N=1 affinization of sl (2), sl (3) and osp (1|2) are explicitly constructed. An unexpected result is found: only a restricted subclass of the sl (3) bosonic hierarchies can be supersymmetrically extended while preserving integrability.

scholarly journals Wave propagation on Riemannian symmetric spaces

1992 ◽  
Vol 107 (2) ◽  
pp. 270-278 ◽  
Author(s):  
G. 'Olafsson ◽  
H. Schlichtkrull
Keyword(s):  
Wave Propagation ◽  
Symmetric Spaces ◽  
Riemannian Symmetric Spaces
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