Dirac structures and dynamical r -matrices

Zhang-Ju Liu[1]; Ping Xu[2]

  • [1] Peking University, Department of Mathematics, Beijing 100871 (Rép. Pop. Chine)
  • [2] Pennsylvania State University, Department of Mathematics, University Park PA 16802 (USA)

Annales de l’institut Fourier (2001)

  • Volume: 51, Issue: 3, page 835-859
  • ISSN: 0373-0956

Abstract

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The purpose of this paper is to establish a connection between various objects such as dynamical r -matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies on the theory of Dirac structures and Courant algebroids. In particular, we give a new method of classifying dynamical r -matrices of simple Lie algebras 𝔤 , and prove that dynamical r -matrices are in one-one correspondence with certain Lagrangian subalgebras of 𝔤 ⊕ 𝔤 .

How to cite

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Liu, Zhang-Ju, and Xu, Ping. "Dirac structures and dynamical $r$-matrices." Annales de l’institut Fourier 51.3 (2001): 835-859. <http://eudml.org/doc/115931>.

@article{Liu2001,
abstract = {The purpose of this paper is to establish a connection between various objects such as dynamical $r$-matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies on the theory of Dirac structures and Courant algebroids. In particular, we give a new method of classifying dynamical $r$-matrices of simple Lie algebras $\{\mathfrak \{g\}\}$, and prove that dynamical $r$-matrices are in one-one correspondence with certain Lagrangian subalgebras of $\{\mathfrak \{g\}\}\oplus \{\mathfrak \{g\}\}$.},
affiliation = {Peking University, Department of Mathematics, Beijing 100871 (Rép. Pop. Chine); Pennsylvania State University, Department of Mathematics, University Park PA 16802 (USA)},
author = {Liu, Zhang-Ju, Xu, Ping},
journal = {Annales de l’institut Fourier},
keywords = {dynamical $r$-matrices; Dirac structures; Lie bialgebroid; Courant algebroid; lagrangian subalgebra; dynamical r-matrices; Lagrangian subalgebra},
language = {eng},
number = {3},
pages = {835-859},
publisher = {Association des Annales de l'Institut Fourier},
title = {Dirac structures and dynamical $r$-matrices},
url = {http://eudml.org/doc/115931},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Liu, Zhang-Ju
AU - Xu, Ping
TI - Dirac structures and dynamical $r$-matrices
JO - Annales de l’institut Fourier
PY - 2001
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 3
SP - 835
EP - 859
AB - The purpose of this paper is to establish a connection between various objects such as dynamical $r$-matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies on the theory of Dirac structures and Courant algebroids. In particular, we give a new method of classifying dynamical $r$-matrices of simple Lie algebras ${\mathfrak {g}}$, and prove that dynamical $r$-matrices are in one-one correspondence with certain Lagrangian subalgebras of ${\mathfrak {g}}\oplus {\mathfrak {g}}$.
LA - eng
KW - dynamical $r$-matrices; Dirac structures; Lie bialgebroid; Courant algebroid; lagrangian subalgebra; dynamical r-matrices; Lagrangian subalgebra
UR - http://eudml.org/doc/115931
ER -

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