On quantum Galois theory.(English)Zbl 0890.17031
The authors initiate a program to systematically study the vertex operator subalgebra \(V^G\) of \(G\)-invariants for a given simple vertex operator algebra \(V\) with a finite and faithful group \(G\) of automorphisms of \(V\), a structure that a physicist might refer to as the “operator content of orbifold models” [cf.R. Dijkgraaf, C. Vafa, E. Verlinde andH. Verlinde, Commun. Math. Phys. 123, 485-526 (1989;Zbl 0674.46051)]. The authors show that \(V^G\) is also simple and that, under the assumption that \(G\) is either abelian or dihedral, the map \(H\mapsto V^H\) is a bijection between the subgroups of \(G\) and the vertex operator subalgebras of \(V\) which contain \(V^G\). This Galois correspondence is a consequence of results concerning a direct sum decomposition \(V=\oplus_{\chi\in\text{Irr}(G)} V^\chi\), where \( V^\chi\) is a graded subspace on which \(G\) acts according to the simple character \(\chi\). It is shown that each \(V^\chi\) is nonzero and, under the assumption that \(G\) is solvable, the main result is a description \(V^\chi=M_\chi\otimes V_\chi\) in terms of simple \(G\)-modules \(M_\chi\) and simple \(V^G\)-modules \(V_\chi\) which are contained in \(V\), with \(M_\chi\mapsto V_\chi\) being a bijection.
Reviewer: Mirko Primc (Zagreb)
MSC:
| 17B69 | Vertex operators; vertex operator algebras and related structures |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
Keywords:
vertex operator subalgebra;\(G\)-invariants;operator content;orbifold models;Galois correspondence;direct sum decompositionCitations:
Zbl 0674.46051Cite
References:
| [1] | R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster , Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 10, 3068-3071. JSTOR: ·Zbl 0613.17012 ·doi:10.1073/pnas.83.10.3068 |
| [2] | C. Curtis and I. Reiner, Methods of Representation Theory, with Applications to Finite Groups and Orders, Vol. 1 , Pure Appl. Math., A Wiley-Interscience Publication, John & Wiley Sons, New York, 1981. ·Zbl 0469.20001 |
| [3] | R. Dijkgraaf, C. Vafa, E. Verlinde, and H. Verlinde, The operator algebra of orbifold models , Comm. Math. Phys. 123 (1989), no. 3, 485-526. ·Zbl 0674.46051 ·doi:10.1007/BF01238812 |
| [4] | C. Dong and J. Lepowsky, Generalized Vertex Algebras and Relative Vertex Operators , Progr. Math., vol. 112, Birkhäuser, Boston, 1993. ·Zbl 0803.17009 |
| [5] | C. Dong and G. Mason, Nonabelian orbifolds and the boson-fermion correspondence , Comm. Math. Phys. 163 (1994), no. 3, 523-559. ·Zbl 0808.17019 ·doi:10.1007/BF02101462 |
| [6] | I. B. Frenkel, Y.-Z. Huang, and J. Lepowsky, On Axiomatic Approaches to Vertex Operator Algebras and Modules , Mem. Amer. Math. Soc. 104 (1993), no. 494, viii+64. ·Zbl 0789.17022 |
| [7] | I. Frenkel, J. Lepowsky, and A. Meurman, Vertex Operator Algebras and the Monster , Pure Appl. Math., vol. 134, Academic Press, Boston, 1988. ·Zbl 0674.17001 |
| [8] | N. Iiyori and H. Yamada, Automorphism groups of vertex operators algebras , |
| [9] | V. Jones, Subfactors and Knots , Regional Conf. Ser. in Math., vol. 80, Amer. Math. Soc., Providence, 1991. ·Zbl 0743.46058 |
| [10] | H. Li, An approach to tensor product theory for representations of a vertex operator algebra , Ph.D. thesis, Rutgers University, 1994. |
| [11] | G. Mason, The quantum double of a finite group and its role in conformal field theory , Groups ’93 Galway/St. Andrews, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 212, Cambridge Univ. Press, Cambridge, 1995, pp. 405-417. ·Zbl 0856.20005 |
| [12] | S. Montgomery, Hopf Algebras and Their Actions on Rings , Regional Conf. Ser. Math., vol. 82, Amer. Math. Soc., Providence, 1993. ·Zbl 0793.16029 |
| [13] | Y. Zhu, Modular invariance of characters of vertex operator algebras , J. Amer. Math. Soc. 9 (1996), no. 1, 237-302. JSTOR: ·Zbl 0854.17034 ·doi:10.1090/S0894-0347-96-00182-8 |
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