Tensor products of strongly graded vertex algebras and their modules. (English) Zbl 1281.17032
Summary: We study strongly graded vertex algebras and their strongly graded modules, which are conformal vertex algebras and their modules with a second, compatible grading by an abelian group satisfying certain grading restriction conditions. We consider a tensor product of strongly graded vertex algebras and its tensor product strongly graded modules. We prove that a tensor product of strongly graded irreducible modules for a tensor product of strongly graded vertex algebras is irreducible, and that such irreducible modules, up to equivalence, exhaust certain naturally defined strongly graded irreducible modules for a tensor product of strongly graded vertex algebras. We also prove that certain naturally defined strongly graded modules for the tensor product strongly graded vertex algebra are completely reducible if and only if every strongly graded module for each of the tensor product factors is completely reducible. These results generalize the corresponding known results for vertex operator algebras and their modules.
MSC:
| 17B69 | Vertex operators; vertex operator algebras and related structures |
| 17B70 | Graded Lie (super)algebras |
References:
| [1] | Borcherds, R. E., Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA, 83, 3068-3071 (1986) · Zbl 0613.17012 |
| [2] | Conway, J. H.; Norton, S. P., Monstrous moonshine, Bull. Lond. Math. Soc, 11, 308-339 (1979) · Zbl 0424.20010 |
| [3] | Dong, C., Vertex algebras associated with even lattices, J. Algebra, 161, 245-265 (1993) · Zbl 0807.17022 |
| [4] | Dong, C., Representations of the moonshine module vertex operator algebra, (Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups. Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups, South Hadley, MA, 1992. Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups. Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups, South Hadley, MA, 1992, Contemporary Math., vol. 175 (1994), Amer. Math. Soc: Amer. Math. Soc Providence, RI), 27-36 · Zbl 0808.17013 |
| [5] | Dong, C.; Li, H.; Mason, G., Regularity of rational vertex operator algebras, Adv. Math., 132, 148-166 (1997) · Zbl 0902.17014 |
| [6] | Dong, C.; Mason, G.; Zhu, Y., Discrete series of the Virasoro algebra and the moonshine module, (Algebraic Groups and their Generalizations: Quantum and Infinite-dimensional Methods. Algebraic Groups and their Generalizations: Quantum and Infinite-dimensional Methods, University Park, PA, 1991. Algebraic Groups and their Generalizations: Quantum and Infinite-dimensional Methods. Algebraic Groups and their Generalizations: Quantum and Infinite-dimensional Methods, University Park, PA, 1991, Proc. Sympos. Pure Math., vol. 56 (1994), Amer. Math. Soc: Amer. Math. Soc Providence, RI), 295-316 · Zbl 0813.17019 |
| [7] | Frenkel, I. B.; Huang, Y.-Z.; Lepowsky, J., On axiomatic approaches to vertex operator algebras and modules, (Mem. Amer. Math. Soc., vol. 104 (1993), Amer. Math. Soc: Amer. Math. Soc Providence), no. 494 (preprint, 1989) |
| [8] | Frenkel, I. B.; Lepowsky, J.; Meurman, A., (Vertex Operator Algebras and the Monster. Vertex Operator Algebras and the Monster, Pure and Appl. Math., vol. 134 (1988), Academic Press: Academic Press Boston) · Zbl 0674.17001 |
| [9] | Huang, Y.-Z., A theory of tensor products for module categories for a vertex operator algebra, IV, J. Pure Appl. Algebra, 100, 173-216 (1995) · Zbl 0841.17015 |
| [10] | Huang, Y.-Z.; Lepowsky, J., Tensor products of modules for a vertex operator algebras and vertex tensor categories, (Brylinski, R.; Brylinski, J.-L.; Guillemin, V.; Kac, V., Lie Theory and Geometry, in honor of Bertram Kostant (1994), Birkhäuser: Birkhäuser Boston), 349-383 · Zbl 0848.17031 |
| [11] | Huang, Y.-Z.; Lepowsky, J., A theory of tensor products for module categories for a vertex operator algebra, I, Selecta Mathematica (New Series), 1, 699-756 (1995) · Zbl 0854.17032 |
| [12] | Huang, Y.-Z.; Lepowsky, J., A theory of tensor products for module categories for a vertex operator algebra, II, Selecta Mathematica (New Series), 1, 757-786 (1995) · Zbl 0854.17033 |
| [13] | Huang, Y.-Z.; Lepowsky, J., A theory of tensor products for module categories for a vertex operator algebra, III, J. Pure Appl. Algebra, 100, 141-171 (1995) · Zbl 0841.17014 |
| [14] | Y.-Z. Huang, J. Lepowsky, L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, Parts I-VIII, arXiv:1012.4193, arXiv:1012.4196, arXiv:1012.4197, arXiv:1012.4198, arXiv:1012.4199, arXiv:1012.4202, arXiv:1110.1929, arXiv:1110.1931. · Zbl 1345.81112 |
| [15] | Jacobson, N., (Structure of Rings. Structure of Rings, American Math. Soc. Colloquium. Publ., vol. 37 (1964), Amer. Math. Soc: Amer. Math. Soc Providence, RI) |
| [16] | Lepowsky, J., Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc., 176, 1-44 (1973) · Zbl 0264.22012 |
| [17] | Lepowsky, J.; Li, H., (Introduction to Vertex Operator Algebras and Their Representations. Introduction to Vertex Operator Algebras and Their Representations, Progress in Math., vol. 227 (2003), Birkhäuser: Birkhäuser Boston) |
| [18] | Wallach, N. R., Real Reductive Groups, Vol. 1 (1988), Academic Press · Zbl 0666.22002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
