仿射Nappi-Witten李代数及无穷维李代数（？）（α，β）的表示
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摘要
李代数H4及W(α,β)来源于物理学,如今数学上对它们的研究也日趋增多,并且其逐渐成为李代数的很多方面的研究对象,例如VO代数,Virasoro代数,K-M李代数等等.因此研究它们的表示理论在物理背景下是一件有意义的工作,在数学中可以使李代数H4及W(α,β)的结构及表示理论更加完善.本篇论文重点考察了H4的扭李代数及李代数W(α,β)的表示理论.
Part1考察了无扭李代数H4的多项式表示及不可约非零水平的quasifi-nite模的分类.首先我们给出了无扭李代数H4在多项式环C[x0,xi,j,yi,j|i=1,2; j=1,2,...]上的表示.接着,我们分类了不可约非零水平的quasifi-nite H4-模,当k以非零元素作用时,我们得到无扭李代数H4上的不可约quasifinite模是Highest Weight Module (HW模)或Lowest Weight Module (LW模).
Part2考察了扭李代数H4[τ1]的表示.首先我们得到扭李代数H4[τ1]的Verma模MH4[τ1](k,l)是不可约模(?)k≠0,另外我们给出在可约条件下MH4[τ1](k,l)的不可约商模及singular元.其次,我们给出了扭李代数H4[τ1]的VO构造.最后,我们分类了李代数H4[τ1]的非零水平的不可约quasifinite模,即HW模或LW模.
Part3考察了扭李代数H4[τ2]的表示.我们有扭李代数H4[τ2]的Verma模MH4[τ2](c,d,l)是不可约模(?)l≠0.c(?)(2Z+1)l,另外我们给出在可约条件下MH4[τ2](c,d,l)的所有LI singular元.其次,我们分别构造了李代数H4和H4[τ2]的VO表示.最后,我们得到李代数H4[τ2]的非零水平的不可约quasifinite模是HW模或LW模.
Part4主要考察了李代数W(α,β)的表示.我们分类了当ai=0,bi(?){-1,0,1},i=1,2时,W(α,β)上的MIS,我们得到W(α,β)上的MIS与Virasoro代数上的MIS同构.进而,当ai=0,bi≠1,i=1,2时,我们分类了李代数W(α,β)上的不可约H-C模,我们得到W(α,β)的不可约H-C模是HW模,LW模或者UBM.另外,当ai=0,-1≤bi≠1如果bi∈Z,i=1,2时,我们分类了W(α,β)的有FDWS的不可约WM,我们得到这样的模实质上是H-C模.最后,我们研究了W(α,β)的一个子代数上的Verma模M(c,h,hv)我们得到M(c,h,hv)是不可约模(?)hv≠0.
The Lie algebra H4and W(α,β) are in the profound physical backgrounds, which have close connections with many important areas of Lie theory, for instance, VOA, Virasoro algebra, K-M Lie algebra and so on. Therefore it is significant in physics to study their representation theory, meanwhile, can enrich structures and representations of the Lie algebra H4and W(α,β) in mathematics. In this thesis, the representation theories of the twisted Lie algebras and Lie algebra W(α,β) are mainly researched.
Part1investigates the polynomial representations of the non-twisted Lie al-gebra H4and the classification of irreducible non-zero level quasifinite H4-modules. Firstly, the polynomial representations of the non-twisted Lie algebra H4on C[x0, xi,j, yi,j|i=1,2; j=1,2,...] are given. Secondly, the irreducible non-zero level quasifi-nite H4-modules are classified. The irreducible non-zero level quasifinite H4-module is the HW module or LW module.
In part2, the representation theory of the twisted Lie algebra H4[τ1] is studied. Firstly, the Verma module MH4[τ1](k,l) of Lie algebra H4[τ1] is an irreducible mod-ule(?)k≠0. Besides, the irreducible quotient module and singular vectors of Lie algebra H4[τ1] are presented under the reducibility. Secondly, the vertex operator representations of Lie algebra H4[τ1] are constructed. Finally, the irreducible non-zero level quasifinite modules of Lie algebra H4[τ1] are determined. The irreducible non-zero level quasifinite H4[τ1]-module is the HW module or LW module.
Part3mainly investigates the representation theory of the twisted Lie alge-bra H4[τ2]. The Verma module MH4[τ2](c, d,l) of Lie algebra H4[τ2] is irreducible (?)l≠0and c(?)(2Z+1)l, and all the linearly independent singular vectors of Lie algebra H4[τ2] under the reducibility are given. Then vertex operator repre- sentations of Lie algebra H4and H4[τ2] are constructed respectively. Finally, the irreducible non-zero level quasifinite modules of Lie algebra H4[τ2] are classified. That is, the irreducible non-zero level quasifinite H4[τ2]-module is the HW module or LW module.
Part4is devoted to study the representation theory of the Lie algebra W(α,β). Firstly, MIS over W(α,β) for ai=0, bi(?){-1,0,1},i=1,2are determined, and one can see that the W(α,β)-module of the intermediate series is isomorphic to MIS over Virasoro algebra. Secondly, all irreducible H-C modules over W(α,β) for ai=0,bi≠1,i=1,2are classified, and the irreducible H-C module over W(α,β) is either the HW/LW module or UBM. Moreover, the irreducible weight modules of W(α,β) with at least one non-trivial finite dimensional weight space for ai=0,-1≤bi≠1if bi∈Z, i=1,2are classified. Finally, the Verma module M(c, h, hy) over a subalgebra of W(α,β) is irreducible (?)hV≠0.
Part1考察了无扭李代数H4的多项式表示及不可约非零水平的quasifi-nite模的分类.首先我们给出了无扭李代数H4在多项式环C[x0,xi,j,yi,j|i=1,2; j=1,2,...]上的表示.接着,我们分类了不可约非零水平的quasifi-nite H4-模,当k以非零元素作用时,我们得到无扭李代数H4上的不可约quasifinite模是Highest Weight Module (HW模)或Lowest Weight Module (LW模).
Part2考察了扭李代数H4[τ1]的表示.首先我们得到扭李代数H4[τ1]的Verma模MH4[τ1](k,l)是不可约模(?)k≠0,另外我们给出在可约条件下MH4[τ1](k,l)的不可约商模及singular元.其次,我们给出了扭李代数H4[τ1]的VO构造.最后,我们分类了李代数H4[τ1]的非零水平的不可约quasifinite模,即HW模或LW模.
Part3考察了扭李代数H4[τ2]的表示.我们有扭李代数H4[τ2]的Verma模MH4[τ2](c,d,l)是不可约模(?)l≠0.c(?)(2Z+1)l,另外我们给出在可约条件下MH4[τ2](c,d,l)的所有LI singular元.其次,我们分别构造了李代数H4和H4[τ2]的VO表示.最后,我们得到李代数H4[τ2]的非零水平的不可约quasifinite模是HW模或LW模.
Part4主要考察了李代数W(α,β)的表示.我们分类了当ai=0,bi(?){-1,0,1},i=1,2时,W(α,β)上的MIS,我们得到W(α,β)上的MIS与Virasoro代数上的MIS同构.进而,当ai=0,bi≠1,i=1,2时,我们分类了李代数W(α,β)上的不可约H-C模,我们得到W(α,β)的不可约H-C模是HW模,LW模或者UBM.另外,当ai=0,-1≤bi≠1如果bi∈Z,i=1,2时,我们分类了W(α,β)的有FDWS的不可约WM,我们得到这样的模实质上是H-C模.最后,我们研究了W(α,β)的一个子代数上的Verma模M(c,h,hv)我们得到M(c,h,hv)是不可约模(?)hv≠0.
The Lie algebra H4and W(α,β) are in the profound physical backgrounds, which have close connections with many important areas of Lie theory, for instance, VOA, Virasoro algebra, K-M Lie algebra and so on. Therefore it is significant in physics to study their representation theory, meanwhile, can enrich structures and representations of the Lie algebra H4and W(α,β) in mathematics. In this thesis, the representation theories of the twisted Lie algebras and Lie algebra W(α,β) are mainly researched.
Part1investigates the polynomial representations of the non-twisted Lie al-gebra H4and the classification of irreducible non-zero level quasifinite H4-modules. Firstly, the polynomial representations of the non-twisted Lie algebra H4on C[x0, xi,j, yi,j|i=1,2; j=1,2,...] are given. Secondly, the irreducible non-zero level quasifi-nite H4-modules are classified. The irreducible non-zero level quasifinite H4-module is the HW module or LW module.
In part2, the representation theory of the twisted Lie algebra H4[τ1] is studied. Firstly, the Verma module MH4[τ1](k,l) of Lie algebra H4[τ1] is an irreducible mod-ule(?)k≠0. Besides, the irreducible quotient module and singular vectors of Lie algebra H4[τ1] are presented under the reducibility. Secondly, the vertex operator representations of Lie algebra H4[τ1] are constructed. Finally, the irreducible non-zero level quasifinite modules of Lie algebra H4[τ1] are determined. The irreducible non-zero level quasifinite H4[τ1]-module is the HW module or LW module.
Part3mainly investigates the representation theory of the twisted Lie alge-bra H4[τ2]. The Verma module MH4[τ2](c, d,l) of Lie algebra H4[τ2] is irreducible (?)l≠0and c(?)(2Z+1)l, and all the linearly independent singular vectors of Lie algebra H4[τ2] under the reducibility are given. Then vertex operator repre- sentations of Lie algebra H4and H4[τ2] are constructed respectively. Finally, the irreducible non-zero level quasifinite modules of Lie algebra H4[τ2] are classified. That is, the irreducible non-zero level quasifinite H4[τ2]-module is the HW module or LW module.
Part4is devoted to study the representation theory of the Lie algebra W(α,β). Firstly, MIS over W(α,β) for ai=0, bi(?){-1,0,1},i=1,2are determined, and one can see that the W(α,β)-module of the intermediate series is isomorphic to MIS over Virasoro algebra. Secondly, all irreducible H-C modules over W(α,β) for ai=0,bi≠1,i=1,2are classified, and the irreducible H-C module over W(α,β) is either the HW/LW module or UBM. Moreover, the irreducible weight modules of W(α,β) with at least one non-trivial finite dimensional weight space for ai=0,-1≤bi≠1if bi∈Z, i=1,2are classified. Finally, the Verma module M(c, h, hy) over a subalgebra of W(α,β) is irreducible (?)hV≠0.
引文
[1] C. Ahn, K. Schoutens and A. Sevrin, The full structure of the super-W3algebra, Int.Journ. Mod. Phys. A6(1991)3467-3488.
[2] E. Arbarello, C. De Concini, V.G. Kac, C. Procesi, Moduli spaces of curves and repre-sentation theory, Comm. Math. Phys.117(1988)1-36.
[3] I. Bakas, The large-N limit of extended conformal systems, Phys. Lett. B228(1989)57-63.
[4] Yixin Bao, Cuipo Jiang and Yufeng Pei, Representations of afne Nappi-Witten algebras,J. Algebra342(2011)111-133.
[5] E. Bergshoef, A. Bilal and K.S. Stelle, W symmetries:gauging and geometry, Int. Journ.Mod. Phys. A6(1991)4959-4983.
[6] Y. Billing, Representations of the twisted Heisenberg-Virasoro algebra at level zero,Canad. Math. Bull.46(2003)529-537.
[7] J. Brundan, S.M. Goodwin, A. Kleshchev, Highest Weight Theory for Finite W-Algebras,International Mathematics Research Notices,2008.
[8] Y. Cheung, L. Freidel and K. Savvidy, Strings in gravimagnetic fields, Journal of HighEnergy Physics054(2004)1-48.
[9] G. DAppollonio, E. Kiritsis, String interactions in gravitational wave backgrounds, Nucl.Phys. B674(2003)80-170.
[10] G. DAppollonio, T. Quella, The abelian cosets of the Heisenberg group, Journal of HighEnergy Physics045(2007).
[11] G. DAppollonio, T. Quella, The diagonal cosets of the Heisenberg group, Journal of HighEnergy Physics060(2008).
[12] J. De Boer, T. Tjin, Quantization and Representation-Theory of Finite W-Algebras,Communications in Mathematical Physics1583(1993)485-516.
[13] A. De Sole, V.G. Kac, Finite vs afne W-algebras, Japanese Journal of Mathematics1(2006)137-261.
[14] J. Distler, talk at the Strings93Conference, Berkeley, May1993.
[15] V. Drinfeld and V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, J.Sov. Math.30(1985)1975-2036.
[16] P. Etingof, T. Schedler, Traces on Finite W-Algebras, Transformation Groups15(4)(2010)843-850.
[17] J.M. Figueroa-O’Farrill and S. Stanciu, Nonsemisimple Sugawara constructions, PhysicsLetters B327(1994)40-46.
[18] I.B. Frenkel, J. Lepowsky, A. Meurman, Vertex Operator Algebras and the Monster,Academic Press, Boston,1989.
[19] D. Friedan, E. Martinec and S. Shenker, Conformal invariance, supersymmetry and stringtheory, Nucl. Phys. B271(1986)93-165.
[20] V.M. Futorny, Irreducible non-dense A(1)1-modules, Pacific J. Math.172(1996)83-99.
[21] V.M. Futorny, Representations of afne Lie algebras, Queen’s papers in Pure and AppliedMathematics106, Queen’s University, Kingston,1997.
[22] S. Gao, C. Jiang, Y. Pei, Low dimensional cohomology groups of the Lie algebras W (a, b),Commun. Alg.39(2011)397-423.
[23] P. Goddard, Meromorphic conformal field theory, in ‘Infinite dimensional Lie algebras andLie groups’, ed. V. Kac, Proc. CIRM-Luminy conf.,1988(World Scientific, Singapore,1989).
[24] S.M. Goodwin, A note on Verma modules for finite W-algebras, Journal of Algebra324(8)(2010)2058-2063.
[25] P. Goddard, A. Kent and D. Olive, Virasoro algebras and coset space models, Phys. Lett.B152(1985)88-92.
[26] Yasuaki, Hikida, Boundary ststes in the Nappi-Witten model, Int. J. Mod. Phys. A20(2005)7645-7661.
[27] Q. Ho-Kim and H.B. Zheng, Twisted conformal field theories with Z3invariance, Phys.Lett. B212(1988)71-76.
[28] Q. Ho-Kim and H.B. Zheng, Twisted structures of extended Virasoro algebras, Phys.Lett. B223(1989)57-60.
[29] Q. Ho-Kim and H.B. Zheng, Twisted characters and partition functions in extendedVirasoro algebras, Mod. Phys. Lett. A5(1990)1181-1189.
[30] K. Hornfeck, The minimal supersymmetric extensions of W An1, Phys. Lett. B275(1992)355-360.
[31] K. Hornfeck and E. Ragoucy, A coset construction for the super-W3algebra, Nucl. Phys.B340(1990)225-244.
[32] C.M. Hull, Classical and quantum W-gravity, in Proc. of ‘Strings and Symmetries1991’,Stony Brook, May1991, ed. N. Berkovits et al.(World Scientific, Singapore,1992).
[33] T. Inami, Y. Matsuo and I. Yamanaka, Extended conformal algebras with N=1super-symmetry, Phys. Lett. B215(1988)701-705.
[34] C. Itzykson, H. Saleur and J.-B. Zuber eds., Conformal Invariance and Applications toStatistical Mechanics, World Scientific, Singapore,1988.
[35] Q. Jiang, C. Jiang, Representations of the twisted Heisenberg-Virasoro algebra and thefull toroidal Lie algebras, Alg. Colloqu.14(1)(2007)117-134.
[36] W. Jiang, Y. Pei, Verma modules over the W-algebra W (2,2), J. Math. Phys.51(2)(2010)0223038pp.
[37] Cuipo Jiang, Song Wang, Extension of vertex operator algebra VH4(l,0),arXiv:1104.4232v1.
[38] V.G. Kac, Infinite Dimensional Lie Algebras,3rd ed., Cambridge, UK: Cambridge Uni-versity Press,1990.
[39] V.G. Kac, A.K. Raina, Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, Advanced Series in Mathematical Physics2, World Scientific,1987.
[40] I. Kaplansky, L.J. Santharoubane, Harish-Chandra Modules Over the Virasoro Algebra,Mathematical Sciences Research Institute Publication, Springer, New York,4(1985) pp.217-231.
[41] E. Kiritsis, C. Kounnas, String propagation in gravitational wave backgrounds, Phys.Lett. B594(1994)368-374.
[42] J. Lepowsky, Haisheng Li, Introduction to Vertex Operator Algebras and Their Repre-sentations, Progress in Math.227, Birkha¨user, Boston,2003.
[43] B. Lian, On the classification of simple vertex operator algebras, Commun. Math. Phys.163(1994)307-357.
[44] D. Liu, Classification of Harish-Chandra modules over some Lie algebras and supercon-formal algebras related to the Virasoro algebra,(2011) arXiv:1011.3438v3.
[45] D. Liu, S. Gao, L. Zhu, Classification of irreducible weight modules over W-algebraW (2,2), J. Math. Phys.49(11)(2008)1135036pp.
[46] D. Liu, C. Jiang, Harish-Chandra modules over the twisted Heisenberg-Virasoro algebra,J. Math. Phys.49(1)(2008)01290114pp.
[47] D. Liu, L. Zhu, Classification of Harish-Chandra modules over the W-algebra W (2,2),(2008) arXiv:0801.2601v2.
[48] R. Lv, K. Zhao, Classification of irreducible weight modules over the twisted Heisenberg-Virasoro algebra, Commun. Contemp. Math.12(2010)183-205.
[49] O. Mathieu, Classification of Harish-Chandra modules over the Virasoro Lie algebra,Invent. Math.107(1992) no.2,225-234.
[50] V. Mazorchuk, K. Zhao, Classification of simple weight Virasoro modules with a finitedimensional weight space, J. Algebra307(2007)209-214.
[51] N. Mohammedi, On bosonic and supersymmetric current algebras for non-semisimplegroups, Phys. Lett. B325(1994)371-376.
[52] R.V. Moody and A. Pianzola, Lie Algebras with Triangular Decompositions, CanadianMathematical Society Series of Monographs and Advanced Texts. John Wiley&SonsInc., New York,1995.
[53] C. Nappi, E. Witten, Wess-Zumino-Witten model based on a nonsemisimple group, Phys.Rev. Lett.23(1993)3751-3753.
[54] S. Odake and T. Sano,W1+∞and super-W1+∞algebras with SU(N) symmetry, Phys.Lett. B258(1991)369-374.
[55] D.I. Olive, E. Rabinovici and A. Schwimmer, A class of string backgrounds as a semi-classical limit of WZW models, Phys. Lett. B321(1994)361-364.
[56] J.M. Osborn, K. Zhao, DoublyZ-graded Lie algebras containing a Virasoro algebra, J.Algebra219(1999)266-298.
[57] L. Page, A New Relativity Paper I. Fundamental Principles and Transformations BetweenAccelerated Systems, Phys. Rev.49(3)(1936)254-268.
[58] L. Page, N.I. Adams, A New Relativity Paper II. Transformations of the ElectromagneticField Between Accelerated Systems and the Force Equation, Phys. Rev.49(6)(1936)466-469.
[59] C.N. Pope, Lectures onWalgebras andWgravity, in Proc. of the1991Trieste SummerSchool in High-Energy Physics (World Scieific, Singapore,1992).
[60] C.N. Pope, L.J. Romans and X. Shen, The complete structure ofW∞, Phys. Lett. B236(1990)173-178.
[61] C.N. Pope, L.J. Romans and X. Shen,W∞and the Racah-Wigner algebra, Nucl. Phys.B339(1990)191-221.
[62] C.N. Pope, L.J. Romans and X. Shen, A new higher-spin algebra and the lone-starproduct, Phys. Lett. B242(1990)401-406.
[63] C.N. Pope, L.J. Romans and X. Shen, A brief history ofW∞, in Proc. of ‘Strings90’,Texas A M, ed. R. Arnowitt et al.(World Scieific, Singapore,1991).
[64] A. Premet, Commutative quotients of finite W-algebras, Advances in Mathematics225(1)(2010)269-306.
[65] Gor, Sarkissian, On D-branes in the Nappi-Witten and GMM gauged WZW models,Journal of High Energy Physics059(2003).
[66] K. Schoutens and A. Sevrin, Minimal superWNalgebras in coset conformal field theories,Phys. Lett. B258(1991)134-140.
[67] K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, Induced gauge theories andW-gravity, in Proc. of ‘Strings and Symmetries1991’, Stony, May1991, ed. N. Berkovits etal.(World Scieific, Singapore,1992).
[68] K. Sfetsos, Exact string backgrounds from WZW models based on non-semisimple group,Int. J. Mod. Phys. A09(1994).
[69] K. Sfetsos, Gauged WZW models and non-abelian duality, Phys. Rev. D50(1994)2784-2798.
[70] R. Shen, Y. Su, Classification of irreducible weight modules with a finite dimensionalweight space over twisted Heisenberg-Virasoro algebra, Acta Mathematica Sinica23(1)(2007)189-192.
[71] Yucai Su, Classification of quasifinite modules over the Lie algebras of Weyl type, Adv.Math.174(2003)57-68.
[72] Y. Su, Y. Xu, X. Yue, Indecomposable modules of the immediate series over W (a, b)algebras,(2011) arXiv:1103.3850.
[73] M. Walton, Afne Kac-Moody algebras and the Wess-Zumino-Witten Model,(1999)arXiv:hep-th/9911187v124.
[74] S. Wang, Vertex operator algebras and modules associated to the afne Nappi-Wittenalgebra H4, Phd thesis,(2011), Shanghai Jiaotong University.
[75] W. Wang, W1+∞Algebra, W3Algebra, and Friedan Martinec Shenker Bosonization,Comm. Math. Phys.195(1998)95-111.
[76] W. Wang, The structures and representations of some infinite-dimensional Lie algebrasrelated to the W-algebras, Phd thesis,(2011), University of Science and Technology ofChina.
[77] M. Willard, Lie Theory and Special Functions, Academic Press, Inc., New York,1968.
[78] E. Witten, Non-abelian bosonization in two-dimensions, Commun. Math. Phys.92(1984)455-472.
[79] A.B. Zamolodchikov, Infinite additional symmentries in two dimensional conformal quan-tum field theory, Theor. Math. Phys.65(1985)1205-1213.
[80] W. Zhang, C. Dong, W-algebra W (2,2) and the vertex operator algebra L(12,0) L(12,0),Commun. Math. Phys.285(2009)991-1004.
[2] E. Arbarello, C. De Concini, V.G. Kac, C. Procesi, Moduli spaces of curves and repre-sentation theory, Comm. Math. Phys.117(1988)1-36.
[3] I. Bakas, The large-N limit of extended conformal systems, Phys. Lett. B228(1989)57-63.
[4] Yixin Bao, Cuipo Jiang and Yufeng Pei, Representations of afne Nappi-Witten algebras,J. Algebra342(2011)111-133.
[5] E. Bergshoef, A. Bilal and K.S. Stelle, W symmetries:gauging and geometry, Int. Journ.Mod. Phys. A6(1991)4959-4983.
[6] Y. Billing, Representations of the twisted Heisenberg-Virasoro algebra at level zero,Canad. Math. Bull.46(2003)529-537.
[7] J. Brundan, S.M. Goodwin, A. Kleshchev, Highest Weight Theory for Finite W-Algebras,International Mathematics Research Notices,2008.
[8] Y. Cheung, L. Freidel and K. Savvidy, Strings in gravimagnetic fields, Journal of HighEnergy Physics054(2004)1-48.
[9] G. DAppollonio, E. Kiritsis, String interactions in gravitational wave backgrounds, Nucl.Phys. B674(2003)80-170.
[10] G. DAppollonio, T. Quella, The abelian cosets of the Heisenberg group, Journal of HighEnergy Physics045(2007).
[11] G. DAppollonio, T. Quella, The diagonal cosets of the Heisenberg group, Journal of HighEnergy Physics060(2008).
[12] J. De Boer, T. Tjin, Quantization and Representation-Theory of Finite W-Algebras,Communications in Mathematical Physics1583(1993)485-516.
[13] A. De Sole, V.G. Kac, Finite vs afne W-algebras, Japanese Journal of Mathematics1(2006)137-261.
[14] J. Distler, talk at the Strings93Conference, Berkeley, May1993.
[15] V. Drinfeld and V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, J.Sov. Math.30(1985)1975-2036.
[16] P. Etingof, T. Schedler, Traces on Finite W-Algebras, Transformation Groups15(4)(2010)843-850.
[17] J.M. Figueroa-O’Farrill and S. Stanciu, Nonsemisimple Sugawara constructions, PhysicsLetters B327(1994)40-46.
[18] I.B. Frenkel, J. Lepowsky, A. Meurman, Vertex Operator Algebras and the Monster,Academic Press, Boston,1989.
[19] D. Friedan, E. Martinec and S. Shenker, Conformal invariance, supersymmetry and stringtheory, Nucl. Phys. B271(1986)93-165.
[20] V.M. Futorny, Irreducible non-dense A(1)1-modules, Pacific J. Math.172(1996)83-99.
[21] V.M. Futorny, Representations of afne Lie algebras, Queen’s papers in Pure and AppliedMathematics106, Queen’s University, Kingston,1997.
[22] S. Gao, C. Jiang, Y. Pei, Low dimensional cohomology groups of the Lie algebras W (a, b),Commun. Alg.39(2011)397-423.
[23] P. Goddard, Meromorphic conformal field theory, in ‘Infinite dimensional Lie algebras andLie groups’, ed. V. Kac, Proc. CIRM-Luminy conf.,1988(World Scientific, Singapore,1989).
[24] S.M. Goodwin, A note on Verma modules for finite W-algebras, Journal of Algebra324(8)(2010)2058-2063.
[25] P. Goddard, A. Kent and D. Olive, Virasoro algebras and coset space models, Phys. Lett.B152(1985)88-92.
[26] Yasuaki, Hikida, Boundary ststes in the Nappi-Witten model, Int. J. Mod. Phys. A20(2005)7645-7661.
[27] Q. Ho-Kim and H.B. Zheng, Twisted conformal field theories with Z3invariance, Phys.Lett. B212(1988)71-76.
[28] Q. Ho-Kim and H.B. Zheng, Twisted structures of extended Virasoro algebras, Phys.Lett. B223(1989)57-60.
[29] Q. Ho-Kim and H.B. Zheng, Twisted characters and partition functions in extendedVirasoro algebras, Mod. Phys. Lett. A5(1990)1181-1189.
[30] K. Hornfeck, The minimal supersymmetric extensions of W An1, Phys. Lett. B275(1992)355-360.
[31] K. Hornfeck and E. Ragoucy, A coset construction for the super-W3algebra, Nucl. Phys.B340(1990)225-244.
[32] C.M. Hull, Classical and quantum W-gravity, in Proc. of ‘Strings and Symmetries1991’,Stony Brook, May1991, ed. N. Berkovits et al.(World Scientific, Singapore,1992).
[33] T. Inami, Y. Matsuo and I. Yamanaka, Extended conformal algebras with N=1super-symmetry, Phys. Lett. B215(1988)701-705.
[34] C. Itzykson, H. Saleur and J.-B. Zuber eds., Conformal Invariance and Applications toStatistical Mechanics, World Scientific, Singapore,1988.
[35] Q. Jiang, C. Jiang, Representations of the twisted Heisenberg-Virasoro algebra and thefull toroidal Lie algebras, Alg. Colloqu.14(1)(2007)117-134.
[36] W. Jiang, Y. Pei, Verma modules over the W-algebra W (2,2), J. Math. Phys.51(2)(2010)0223038pp.
[37] Cuipo Jiang, Song Wang, Extension of vertex operator algebra VH4(l,0),arXiv:1104.4232v1.
[38] V.G. Kac, Infinite Dimensional Lie Algebras,3rd ed., Cambridge, UK: Cambridge Uni-versity Press,1990.
[39] V.G. Kac, A.K. Raina, Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, Advanced Series in Mathematical Physics2, World Scientific,1987.
[40] I. Kaplansky, L.J. Santharoubane, Harish-Chandra Modules Over the Virasoro Algebra,Mathematical Sciences Research Institute Publication, Springer, New York,4(1985) pp.217-231.
[41] E. Kiritsis, C. Kounnas, String propagation in gravitational wave backgrounds, Phys.Lett. B594(1994)368-374.
[42] J. Lepowsky, Haisheng Li, Introduction to Vertex Operator Algebras and Their Repre-sentations, Progress in Math.227, Birkha¨user, Boston,2003.
[43] B. Lian, On the classification of simple vertex operator algebras, Commun. Math. Phys.163(1994)307-357.
[44] D. Liu, Classification of Harish-Chandra modules over some Lie algebras and supercon-formal algebras related to the Virasoro algebra,(2011) arXiv:1011.3438v3.
[45] D. Liu, S. Gao, L. Zhu, Classification of irreducible weight modules over W-algebraW (2,2), J. Math. Phys.49(11)(2008)1135036pp.
[46] D. Liu, C. Jiang, Harish-Chandra modules over the twisted Heisenberg-Virasoro algebra,J. Math. Phys.49(1)(2008)01290114pp.
[47] D. Liu, L. Zhu, Classification of Harish-Chandra modules over the W-algebra W (2,2),(2008) arXiv:0801.2601v2.
[48] R. Lv, K. Zhao, Classification of irreducible weight modules over the twisted Heisenberg-Virasoro algebra, Commun. Contemp. Math.12(2010)183-205.
[49] O. Mathieu, Classification of Harish-Chandra modules over the Virasoro Lie algebra,Invent. Math.107(1992) no.2,225-234.
[50] V. Mazorchuk, K. Zhao, Classification of simple weight Virasoro modules with a finitedimensional weight space, J. Algebra307(2007)209-214.
[51] N. Mohammedi, On bosonic and supersymmetric current algebras for non-semisimplegroups, Phys. Lett. B325(1994)371-376.
[52] R.V. Moody and A. Pianzola, Lie Algebras with Triangular Decompositions, CanadianMathematical Society Series of Monographs and Advanced Texts. John Wiley&SonsInc., New York,1995.
[53] C. Nappi, E. Witten, Wess-Zumino-Witten model based on a nonsemisimple group, Phys.Rev. Lett.23(1993)3751-3753.
[54] S. Odake and T. Sano,W1+∞and super-W1+∞algebras with SU(N) symmetry, Phys.Lett. B258(1991)369-374.
[55] D.I. Olive, E. Rabinovici and A. Schwimmer, A class of string backgrounds as a semi-classical limit of WZW models, Phys. Lett. B321(1994)361-364.
[56] J.M. Osborn, K. Zhao, DoublyZ-graded Lie algebras containing a Virasoro algebra, J.Algebra219(1999)266-298.
[57] L. Page, A New Relativity Paper I. Fundamental Principles and Transformations BetweenAccelerated Systems, Phys. Rev.49(3)(1936)254-268.
[58] L. Page, N.I. Adams, A New Relativity Paper II. Transformations of the ElectromagneticField Between Accelerated Systems and the Force Equation, Phys. Rev.49(6)(1936)466-469.
[59] C.N. Pope, Lectures onWalgebras andWgravity, in Proc. of the1991Trieste SummerSchool in High-Energy Physics (World Scieific, Singapore,1992).
[60] C.N. Pope, L.J. Romans and X. Shen, The complete structure ofW∞, Phys. Lett. B236(1990)173-178.
[61] C.N. Pope, L.J. Romans and X. Shen,W∞and the Racah-Wigner algebra, Nucl. Phys.B339(1990)191-221.
[62] C.N. Pope, L.J. Romans and X. Shen, A new higher-spin algebra and the lone-starproduct, Phys. Lett. B242(1990)401-406.
[63] C.N. Pope, L.J. Romans and X. Shen, A brief history ofW∞, in Proc. of ‘Strings90’,Texas A M, ed. R. Arnowitt et al.(World Scieific, Singapore,1991).
[64] A. Premet, Commutative quotients of finite W-algebras, Advances in Mathematics225(1)(2010)269-306.
[65] Gor, Sarkissian, On D-branes in the Nappi-Witten and GMM gauged WZW models,Journal of High Energy Physics059(2003).
[66] K. Schoutens and A. Sevrin, Minimal superWNalgebras in coset conformal field theories,Phys. Lett. B258(1991)134-140.
[67] K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, Induced gauge theories andW-gravity, in Proc. of ‘Strings and Symmetries1991’, Stony, May1991, ed. N. Berkovits etal.(World Scieific, Singapore,1992).
[68] K. Sfetsos, Exact string backgrounds from WZW models based on non-semisimple group,Int. J. Mod. Phys. A09(1994).
[69] K. Sfetsos, Gauged WZW models and non-abelian duality, Phys. Rev. D50(1994)2784-2798.
[70] R. Shen, Y. Su, Classification of irreducible weight modules with a finite dimensionalweight space over twisted Heisenberg-Virasoro algebra, Acta Mathematica Sinica23(1)(2007)189-192.
[71] Yucai Su, Classification of quasifinite modules over the Lie algebras of Weyl type, Adv.Math.174(2003)57-68.
[72] Y. Su, Y. Xu, X. Yue, Indecomposable modules of the immediate series over W (a, b)algebras,(2011) arXiv:1103.3850.
[73] M. Walton, Afne Kac-Moody algebras and the Wess-Zumino-Witten Model,(1999)arXiv:hep-th/9911187v124.
[74] S. Wang, Vertex operator algebras and modules associated to the afne Nappi-Wittenalgebra H4, Phd thesis,(2011), Shanghai Jiaotong University.
[75] W. Wang, W1+∞Algebra, W3Algebra, and Friedan Martinec Shenker Bosonization,Comm. Math. Phys.195(1998)95-111.
[76] W. Wang, The structures and representations of some infinite-dimensional Lie algebrasrelated to the W-algebras, Phd thesis,(2011), University of Science and Technology ofChina.
[77] M. Willard, Lie Theory and Special Functions, Academic Press, Inc., New York,1968.
[78] E. Witten, Non-abelian bosonization in two-dimensions, Commun. Math. Phys.92(1984)455-472.
[79] A.B. Zamolodchikov, Infinite additional symmentries in two dimensional conformal quan-tum field theory, Theor. Math. Phys.65(1985)1205-1213.
[80] W. Zhang, C. Dong, W-algebra W (2,2) and the vertex operator algebra L(12,0) L(12,0),Commun. Math. Phys.285(2009)991-1004.