Aug 2011, Volume 6 Issue 4

    

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• EDITORIAL

  Lie Algebras and Related Topics

  Yucai SU, Shaobin TAN, Hechun ZHANG

  2011, 6(4): 565-566. https://doi.org/10.1007/s11464-011-0145-6

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• SURVEY ARTICLE

  Parafermion vertex operator algebras

  Chongying DONG, Qing WANG

  2011, 6(4): 567-579. https://doi.org/10.1007/s11464-011-0138-5

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  This paper reviews some recent results on the parafermion vertex operator algebra associated to the integrable highest weight module ? (k, 0) of positive integer level k for any affine Kac-Moody Lie algebra g^, where g is a finite dimensional simple Lie algebra. In particular, the generators and the C₂-cofiniteness of the parafermion vertex operator algebras are discussed. A proof of the well-known fact that the parafermion vertex operator algebra can be realized as the commutant of a lattice vertex operator algebra in ? (k, 0) is also given.

• RESEARCH ARTICLE

  On classification of n-Lie algebras

  Ruipu BAI, Guojie SONG, Yaozhong ZHANG

  2011, 6(4): 581-606. https://doi.org/10.1007/s11464-011-0107-z

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  In this paper, we prove the isomorphic criterion theorem for (n+2)- dimensional n-Lie algebras, and give a complete classification of (n + 2)- dimensional n-Lie algebras over an algebraically closed field of characteristic zero.

• RESEARCH ARTICLE

  Twisted fermionic and bosonic representations for a class of BC-graded Lie algebras

  Fulin CHEN, Shaobin TAN

  2011, 6(4): 607-628. https://doi.org/10.1007/s11464-011-0147-4

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  In this paper, we study the fermionic and bosonic representations for a class of BC-graded Lie algebras coordinatized by skew Laurent polynomial rings. This generalizes the fermionic and bosonic constructions for the affine Kac-Moody algebras of type AN(2).

• RESEARCH ARTICLE

  Complex Lie algebras corresponding to weighted projective lines

  Rujing DOU, Jie SHENG, Jie XIAO

  2011, 6(4): 629-639. https://doi.org/10.1007/s11464-010-0070-0

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  The aim of this paper is to give an alternative proof of Kac’s theorem for weighted projective lines over the complex field. The geometric realization of complex Lie algebras arising from derived categories is essentially used.

• RESEARCH ARTICLE

  Schr?dinger-Virasoro type Lie bialgebra: a twisted case

  Huanxia FA, Yanjie LI, Junbo LI

  2011, 6(4): 641-657. https://doi.org/10.1007/s11464-011-0105-1

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  In this paper, we investigate Lie bialgebra structures on a twisted Schr?dinger-Virasoro type algebra L. All Lie bialgebra structures on L are triangular coboundary, which is different from the relative result on the original Schr?dinger-Virasoro type Lie algebra. In particular, we find for this Lie algebra that there are more hidden inner derivations from itself to L?L and we develop one method to search them.

• RESEARCH ARTICLE

  Generating index of finite-dimensional Lie algebras

  Fang FANG, Fuhai ZHU

  2011, 6(4): 659-670. https://doi.org/10.1007/s11464-011-0106-0

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  The notion of generating index of Lie algebras is introduced. We characterize some Lie algebras with full generating index and classify the Lie algebras with generating index 2 and 3. As a corollary, we give a characterization of 2-step nilpotent Lie algebras.

• RESEARCH ARTICLE

  First cohomology group of rank two Witt algebra to its Larsson modules

  Jinglian JIANG, Xiaoli KONG

  2011, 6(4): 671-688. https://doi.org/10.1007/s11464-011-0143-8

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  Let U=?2, Γ=?2, and let ?[x1±1,x2±1] be the ring of Laurent polynomials. The Witt algebra ? is the Lie algebra of derivations over ?[x1±1,x2±1], which is spanned by elements of the form D(u, r) = x^r(u₁d₁ + u₂d₂), u=(u1,u2)∈U, r∈Γ, where d₁ and d₂ are the degree derivations of ?[x1±1,x2±1]. The image of gl₂-module V under Larsson functor Fα, denoted by W=Fα(V), gives a class of ?-modules, often called the Larsson-modules of ?. In this paper, we study the derivations from the Witt algebra ? to its Larsson-modules W, and we determine the first cohomology group H¹(?, W).

• RESEARCH ARTICLE

  Path realization of crystal B(∞)

  Bin LI, Hechun ZHANG

  2011, 6(4): 689-706. https://doi.org/10.1007/s11464-010-0073-x

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  A class of piecewise linear paths, as a generalization of Littelmann’s paths, are introduced, and some operators, acting on the above paths with fixed parametrization, are defined. These operators induce the ordinary Littelmann’s root operators’ action on the equivalence classes of paths. With these induced operators, an explicit realization of B(∞) is given in terms of equivalence classes of paths, where B(∞) is the crystal base of the negative part of a quantum group Uq(g). Furthermore, we conjecture that there is a complete set of representatives for the above model by fixing a parametrization, and we prove the case when g is of finite type.

• RESEARCH ARTICLE

  Associating quantum vertex algebras to deformed Heisenberg Lie algebras

  Haisheng LI

  2011, 6(4): 707-730. https://doi.org/10.1007/s11464-011-0144-7

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  We associate quantum vertex algebras and their ?-coordinated quasi modules to certain deformed Heisenberg algebras.

• RESEARCH ARTICLE

  Whittaker modules for a Lie algebra of Block type

  Bin WANG, Xinyun ZHU

  2011, 6(4): 731-744. https://doi.org/10.1007/s11464-011-0121-1

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  In this paper, we study Whittaker modules for a Lie algebra of Block type. We define Whittaker modules and under some conditions, obtain a bijective correspondence between the set of isomorphism classes of Whittaker modules over this algebra and the set of ideals of a polynomial ring, parallel to a result from the classical setting and the case of the Virasoro algebra.

• RESEARCH ARTICLE

  Second cohomology group of extended W-algebras

  Wei WANG, Yongping WU, Chunguang XIA

  2011, 6(4): 745-758. https://doi.org/10.1007/s11464-011-0101-5

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  Let F be a field of characteristic 0, and let G be an additive subgroup of F. We define a class of infinite-dimensional Lie algebras W with an F-basis {Lμ,Vμ,Wμ|μ∈G}, which are very closely related to W-algebras. In this paper, the second cohomology group of W is determined.

• RESEARCH ARTICLE

  Partial differential equation approach to F₄

  Xiaoping XU

  2011, 6(4): 759-774. https://doi.org/10.1007/s11464-011-0131-z

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  Singular vectors of a representation of a finite-dimensional simple Lie algebra are weight vectors in the underlying module that are nullified by positive root vectors. In this article, we use partial differential equations to explicitly find all the singular vectors of the polynomial representation of the simple Lie algebra of type F₄ over its 26-dimensional basic irreducible module, which also supplements a proof of the completeness of Brion’s abstractly described generators. Moreover, we show that the number of irreducible submodules contained in the space of homogeneous harmonic polynomials with degree k≥2 is greater than or equal to ?k/3?+?(k-2)/3?+2.

• RESEARCH ARTICLE

  Support varieties of semisimple-character representations for Cartan type Lie algebras

  Yufeng YAO, Bin SHU

  2011, 6(4): 775-788. https://doi.org/10.1007/s11464-011-0133-x

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  In this paper, we study the support varieties for Lie algebras L of Cartan type. We give some description for the support varieties of any finitedimensional L-module with character χ, whenever the height of the character χ is not too large. And a more concrete computation can be made for a class of modules with semisimple characters.

• RESEARCH ARTICLE

  Ideals and simplicity of unitary Lie algebras

  Yelong ZHENG, Zhihua CHANG, Yun GAO

  2011, 6(4): 789-820. https://doi.org/10.1007/s11464-011-0148-3

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  Double graded ideals and simplicity of elementary unitary Lie algebra eun(R,-,γ) and Steinberg unitary Lie algebra stun(R,-,γ) are characterized, where R is a unital involutory associative algebra over a field F of characteristic zero, n≥5.