scholarly journals ON THE EXACTNESS OF KOSTANT–KIRILLOV FORM AND THE SECOND COHOMOLOGY OF NILPOTENT ORBITS

2012 ◽  
Vol 23 (08) ◽  
pp. 1250086 ◽  
Author(s):  
INDRANIL BISWAS ◽  
PRALAY CHATTERJEE
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Nilpotent Orbits ◽  
Semisimple Lie Group ◽  
Simple Lie Algebra ◽  
Adjoint Orbits ◽  
Adjoint Orbit

We give a criterion for the Kostant–Kirillov form on an adjoint orbit in a real semisimple Lie group to be exact. We explicitly compute the second cohomology of all the nilpotent adjoint orbits in every complex simple Lie algebra.

scholarly journals Sur la structure transverse à une orbite nilpotente adjointe

2005 ◽  
Vol 57 (4) ◽  
pp. 750-770 ◽  
Author(s):  
Hervé Sabourin
Keyword(s):  
Lie Algebra ◽  
Poisson Structures ◽  
Nilpotent Orbits ◽  
Simple Lie Algebra ◽  
Adjoint Orbits

AbstractWe are interested in Poisson structures transverse to nilpotent adjoint orbits in a complex semi-simple Lie algebra, and we study their polynomial nature. Furthermore, in the case of sln, we construct some families of nilpotent orbits with quadratic transverse structures.

scholarly journals DRINFEL'D TWIST AND q-DEFORMING MAPS FOR LIE GROUP COVARIANT HEISENBERG ALGEBRAE

2000 ◽  
Vol 12 (03) ◽  
pp. 327-359 ◽  
Author(s):  
GAETANO FIORE
Keyword(s):  
Quantum Group ◽  
Clifford Algebra ◽  
Lie Algebra ◽  
Lie Group ◽  
Physical Systems ◽  
Covariant Quantum ◽  
Simple Lie Algebra ◽  
Systematic Procedure

Any deformation of a Weyl or Clifford algebra can be realized through a change of generators in the undeformed algebra. q-deformations of Weyl or Clifford algebrae that were covariant under the action of a simple Lie algebra g are characterized by their being covariant under the action of the quantum group Uhg, q:=eh. We present a systematic procedure for determining all possible corresponding changes of generators, together with the corresponding realizations of the Uhg-action. The intriguing relation between g-invariants and Uhg-invariants suggests that these changes of generators might be employed to simplify the dynamics of some g-covariant quantum physical systems.

On Some q-Analogs of a Theorem of Kostant-Rallis

2000 ◽  
Vol 52 (2) ◽  
pp. 438-448 ◽  
Author(s):  
N. R. Wallach ◽  
J. Willenbring
Keyword(s):  
Conjugacy Class ◽  
Quantum Computation ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Cartan Subgroup ◽  
Semisimple Lie Groups ◽  
Semisimple Lie Group ◽  
Real Group ◽  
Qubit States

AbstractIn the first part of this paper generalizationsof Hesselink’s q-analog of Kostant’smultiplicity formula for the action of a semisimple Lie group on the polynomials on its Lie algebra are given in the context of the Kostant-Rallis theorem. They correspond to the cases of real semisimple Lie groups with one conjugacy class of Cartan subgroup. In the second part of the paper a q-analog of the Kostant-Rallis theorem is given for the real group SL(4, ) (that is SO(4) acting on symmetric 4 × 4 matrices). This example plays two roles. First it contrasts with the examples of the first part. Second it has implications to the study of entanglement of mixed 2 qubit states in quantum computation.

scholarly journals Elements in Pointed Invariant Cones in Lie Algebras and Corresponding Affine Pairs

2021 ◽  
Author(s):  
Karl-Hermann Neeb ◽  
Daniel Oeh
Keyword(s):  
Convex Hull ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Closed Convex Hull ◽  
Finite Dimensional ◽  
Adjoint Orbits ◽  
Invariant Cones ◽  
Pointed Convex Cone ◽  
Adjoint Orbit

AbstractIn this note, we study in a finite dimensional Lie algebra $${\mathfrak g}$$ g the set of all those elements x for which the closed convex hull of the adjoint orbit contains no affine lines; this contains in particular elements whose adjoint orbits generates a pointed convex cone $$C_x$$ C x . Assuming that $${\mathfrak g}$$ g is admissible, i.e., contains a generating invariant convex subset not containing affine lines, we obtain a natural characterization of such elements, also for non-reductive Lie algebras. Motivated by the concept of standard (Borchers) pairs in QFT, we also study pairs (x, h) of Lie algebra elements satisfying $$[h,x]=x$$ [ h , x ] = x for which $$C_x$$ C x pointed. Given x, we show that such elements h can be constructed in such a way that $$\mathop {\mathrm{ad}}\nolimits h$$ ad h defines a 5-grading, and characterize the cases where we even get a 3-grading.

scholarly journals Approximation of nilpotent orbits for simple Lie groups

2021 ◽  
Vol 56 (2) ◽  
pp. 287-327
Author(s):  
Lucas Fresse ◽  
◽  
Salah Mehdi ◽  
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Finite Union ◽  
Nilpotent Orbits ◽  
Simple Lie Groups ◽  
Special Cases ◽  
Adjoint Orbits

We propose a systematic and topological study of limits \(\lim_{\nu\to 0^+}G_\mathbb{R}\cdot(\nu x)\) of continuous families of adjoint orbits for a non-compact simple real Lie group \(G_\mathbb{R}\). This limit is always a finite union of nilpotent orbits. We describe explicitly these nilpotent orbits in terms of Richardson orbits in the case of hyperbolic semisimple elements. We also show that one can approximate minimal nilpotent orbits or even nilpotent orbits by elliptic semisimple orbits. The special cases of \(\mathrm{SL}_n(\mathbb{R})\) and \(\mathrm{SU}(p,q)\) are computed in detail.

scholarly journals Slodowy slices and universal Poisson deformations

2011 ◽  
Vol 148 (1) ◽  
pp. 121-144 ◽  
Author(s):  
M. Lehn ◽  
Y. Namikawa ◽  
Ch. Sorger
Keyword(s):  
Lie Algebra ◽  
Nilpotent Orbits ◽  
Simple Lie Algebra

AbstractWe classify the nilpotent orbits in a simple Lie algebra for which the restriction of the adjoint quotient map to a Slodowy slice is the universal Poisson deformation of its central fibre. This generalises work of Brieskorn and Slodowy on subregular orbits. In particular, we find in this way new singular symplectic hypersurfaces of dimension four and six.

A LIMIT FORMULA FOR EVEN NILPOTENT ORBITS

2008 ◽  
Vol 19 (02) ◽  
pp. 223-236
Author(s):  
MLADEN BOŽIČEVIĆ
Keyword(s):  
Lie Group ◽  
Coadjoint Orbits ◽  
Nilpotent Orbits ◽  
Semisimple Lie Group ◽  
Real Form ◽  
Parabolic Subalgebra ◽  
Limit Formula ◽  
Canonical Measure ◽  
Complex Semisimple

Let Gℝ be a real form of a complex, semisimple Lie group G. Assume [Formula: see text] is an even nilpotent coadjoint Gℝ-orbit. We prove a limit formula, expressing the canonical measure on [Formula: see text] as a limit of canonical measures on semisimple coadjoint orbits, where the parameter of orbits varies over the negative chamber defined by the parabolic subalgebra associated with [Formula: see text].

On the Unitarized Adjoint Representation of a Semisimple Lie Group II

1977 ◽  
Vol 29 (6) ◽  
pp. 1217-1222
Author(s):  
Ronald L. Lipsman
Keyword(s):  
Lebesgue Measure ◽  
Lie Algebra ◽  
Unitary Representation ◽  
Lie Group ◽  
Adjoint Action ◽  
Adjoint Representation ◽  
Semisimple Lie Group ◽  
Group Ii ◽  
Image Position

Let G be a connected semisimple Lie group with Lie algebra . Lebesgue measure on is invariant under the adjoint action of G; and so there is a natural unitary representation TG of G on L2 given by

ε‎-Invariance

2018 ◽  
Author(s):  
Ercüment H. Ortaçgil
Keyword(s):  
Lie Algebra ◽  
Lie Group

The discussions up to Chapter 4 have been concerned with the Lie group. In this chapter, the Lie algebra is constructed by defining the operators ∇ and ∇̃.

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