## Quantized Algebra and Physics: Proceedings of the International Workshop on Quantized Algebra and Physics, Tianjin, China, 23-26 July 2009The book aims to survey recent developments in quantum algebras and related topics. Quantum groups were introduced by Drinfeld and Jimbo in 1985 in their work on YangOCoBaxter equations. The subject from the very beginning has been an interesting one for both mathematics and theoretical physics. For example, Yangian is a special example of quantum group, corresponding to rational solution of YangOCoBaxter equation. Viewed as a generalization of the symmetric group, Yangians also have close connections to algebraic combinatorics. This is the proceeding for the International Workshop on Quantized Algebra and Physics. The workshop aims to gather experts and young investigators from China and abroad to discuss research problems in integrable systems, conformal field theory, string theory, Lie theory, quantum groups including Yangians and their representations. |

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### Contents

A Note on BrauerSchur Functions Kazuya Aokage Hiroshi Mizukawa and HiroFumi Yamada | 1 |

OOperators on Associative Algebras Associative YangBaxter Equations and Dendriform Algebras Chengming Bai Li Guo and Xiang Ni | 10 |

Irreducible Wakimotolike Modules for the Affine Lie Algebra gln Yun Gao and Ziting Zeng | 52 |

Verma Modules over Generic ExpPolynomial Lie Algebras Xiangqian Guo Xuewen Liu and Kaiming Zhao | 68 |

A Formal Infinite Dimensional Cauchy Problem and its Relation to Integrable Hierarchies G F Helminck E A Panasenko and A O Sergeeva | 89 |

Partially Harmonic Tensors and Quantized SchurWeyl Duality Jun Hu and Zhankui Xiao | 109 |

Quantum Entanglement and Approximation by Positive Matrices Xiaofen Huang and Naihuan Jing | 138 |

2Partitions of Root Systems Bin Li William Wong and Hechun Zhang | 147 |

A Survey on Weak Hopf Algebras Fang Li and Qinxiu Sun | 166 |

The Equitable Presentation for the Quantum Algebra Uqfk Yan Pan Meiling Zhu and Libin Li | 187 |

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2-partition A-bimodule k-algebra affine Lie algebra antisymmetric infinitesimal bialgebra bialgebra bimodule called China coefficients commutative Corollary define Definition dendriform dialgebra Dynkin diagram element exp-polynomial function exp-polynomial Lie algebras expº extended O-operators finite dimensional finite k-dimension GAYBE Hence Hermitian Hermitian matrix highest weight homomorphism I-good integer invertible isomorphism k-algebra k-algebra and let k-vector space Kac-Moody Kac-Moody algebra Lemma linear map Math Mathematics matrix monoid nonzero operator of weight Özi partition Phys polynomial positive semi-definite Proof Proposition quantum algebra quantum double quantum groups QYBE representation resp result root system Rota-Baxter operators satisfies Schur functions Section semilattice skew-symmetric solution solution of AYBE subset Suppose symmetric Theorem 2.1 total order trialgebra University Verma modules weak antipode weak Hopf algebra weight vector weight zero Yang-Baxter equation