The KP hierarchy and infinite-dimensional Grassmann manifolds

1989 ◽  
pp. 51-66 ◽  
Author(s):  
Mikio Sato
Keyword(s):  
Grassmann Manifolds ◽  
Kp Hierarchy ◽  
Infinite Dimensional

scholarly journals Geodesics in infinite dimensional Stiefel and Grassmann manifolds

2012 ◽  
Vol 350 (15-16) ◽  
pp. 773-776 ◽  
Author(s):  
Philipp Harms ◽  
Andrea C.G. Mennucci
Keyword(s):  
Grassmann Manifolds ◽  
Infinite Dimensional

GEOMETRY OF UNIVERSAL GRASSMANN MANIFOLD FROM ALGEBRAIC POINT OF VIEW

1989 ◽  
Vol 01 (01) ◽  
pp. 1-46 ◽  
Author(s):  
KANEHISA TAKASAKI
Keyword(s):  
Lie Algebra ◽  
Grassmann Manifold ◽  
Point Of View ◽  
Simple Application ◽  
Grassmann Manifolds ◽  
Algebraic Point ◽  
Infinite Dimensional ◽  
Component Theory ◽  
Infinitesimal Transformations ◽  
Algebraic Formulation

An algebraic formulation of the geometry of the universal Grassmann manifold is presented along the line sketched by Sato and Sato [32]. General issues underlying the notion of infinite-dimensional manifolds are also discussed. A particular choice of affine coordinates on Grassmann manifolds, for both the finite- and infinite-dimensional case, turns out to be very useful for the understanding of geometric structures therein. The so-called “Kac-Peterson cocycle”, which is physically a kind of “commutator anomaly”, then arises as a cocycle of a Lie algebra of infinitesimal transformations on the universal Grassmann manifold. The action of group elements for that Lie algebra is also discussed. These ideas are extended to a multi-component theory. A simple application to a non-linear realization of current and Virasoro algebras is presented for illustration.

THE MODIFIED CLASSICAL YANG-BAXTER EQUATION AND SUPERSYMMETRIC GEL’FAND-DIKII BRACKETS

1993 ◽  
Vol 08 (02) ◽  
pp. 129-137 ◽  
Author(s):  
C.M. YUNG
Keyword(s):  
Pseudodifferential Operators ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Baxter Equation ◽  
Poisson Brackets ◽  
Trace Form ◽  
Kp Hierarchy ◽  
Infinite Dimensional

The classical Yang-Baxter equation as formulated by Semenov-Tyan-Shanskii is generalized to the case of Lie superalgebras [Formula: see text], for Grassmann even Yang-Baxter operators ℛ. When ℛ is “unitary” with respect to a super trace form defined on [Formula: see text], we prove the existence of two natural Poisson brackets on the dual [Formula: see text]*. If [Formula: see text] is the infinite-dimensional Lie superalgebra of N=1 super pseudodifferential operators, we recover the super Gel’fand-Dikii brackets underlying the N=1 super KP hierarchy and its reductions.

On Complex Infinite-Dimensional Grassmann Manifolds

2008 ◽  
Vol 3 (4) ◽  
pp. 739-758 ◽  
Author(s):  
Daniel Beltiţă ◽  
José E. Galé
Keyword(s):  
Grassmann Manifolds ◽  
Infinite Dimensional
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