Gradings of Lie algebras, magical spin geometries and matrix factorizations. (English) Zbl 1479.17032
Summary: We describe a remarkable rank 14 matrix factorization of the octic \(\operatorname{Spin}_{14} \)-invariant polynomial on either of its half-spin representations. We observe that this representation can be, in a suitable sense, identified with a tensor product of two octonion algebras. Moreover the matrix factorisation can be deduced from a particular \(\mathbb{Z} \)-grading of \(\mathfrak{e}_8\). Intriguingly, the whole story can in fact be extended to the whole Freudenthal-Tits magic square and yields matrix factorizations on other spin representations, as well as for the degree seven invariant on the space of three-forms in several variables. As an application of our results on \(\operatorname{Spin}_{14} \), we construct a special rank seven vector bundle on a double-octic threefold, that we conjecture to be spherical.
MSC:
| 17B45 | Lie algebras of linear algebraic groups |
| 13A50 | Actions of groups on commutative rings; invariant theory |
| 15A66 | Clifford algebras, spinors |
| 17B70 | Graded Lie (super)algebras |
Software:
LiEReferences:
| [1] | Abuaf, Roland, On quartic double fivefolds and the matrix factorizations of exceptional quaternionic representations, Ann. Inst. Fourier (Grenoble), 70, 4, 1403-1430 (2020) · Zbl 1473.14035 |
| [2] | Agricola, Ilka, Old and new on the exceptional group \(G_2\), Notices Amer. Math. Soc., 55, 8, 922-929 (2008) · Zbl 1194.22023 |
| [3] | Allison, B. N., Models of isotropic simple Lie algebras, Comm. Algebra, 7, 17, 1835-1875 (1979) · Zbl 0422.17006 · doi:10.1080/00927877908822432 |
| [4] | Allison, B. N.; Faulkner, J. R., Norms on structurable algebras, Comm. Algebra, 20, 1, 155-188 (1992) · Zbl 0753.17004 · doi:10.1080/00927879208824337 |
| [5] | Baez, John C., The octonions, Bull. Amer. Math. Soc. (N.S.), 39, 2, 145-205 (2002) · Zbl 1026.17001 · doi:10.1090/S0273-0979-01-00934-X |
| [6] | Beauville, Arnaud, Determinantal hypersurfaces, Michigan Math. J., 48, 39-64 (2000) · Zbl 1076.14534 · doi:10.1307/mmj/1030132707 |
| [7] | Bertin, Jos\'{e}, Clifford algebras and matrix factorizations, Adv. Appl. Clifford Algebr., 18, 3-4, 417-430 (2008) · Zbl 1177.15026 · doi:10.1007/s00006-008-0079-6 |
| [8] | Buchweitz, Ragnar-Olaf; Eisenbud, David; Herzog, J\"{u}rgen, Cohen-Macaulay modules on quadrics. Singularities, representation of algebras, and vector bundles, Lambrecht, 1985, Lecture Notes in Math. 1273, 58-116 (1987), Springer, Berlin · Zbl 0633.13008 · doi:10.1007/BFb0078838 |
| [9] | Candelas, P.; Derrick, E.; Parkes, L., Generalized Calabi-Yau manifolds and the mirror of a rigid manifold, Nuclear Phys. B, 407, 1, 115-154 (1993) · Zbl 0899.32011 · doi:10.1016/0550-3213(93)90276-U |
| [10] | Chevalley, Claude, The algebraic theory of spinors and Clifford algebras, xiv+214 pp. (1997), Springer-Verlag, Berlin · Zbl 0899.01032 |
| [11] | Clerc, Jean-Louis, Special prehomogeneous vector spaces associated to \(F_4,\ E_6,\ E_7,\ E_8\) and simple Jordan algebras of rank 3, J. Algebra, 264, 1, 98-128 (2003) · Zbl 1034.17006 · doi:10.1016/S0021-8693(03)00115-7 |
| [12] | Deligne, Pierre; Gross, Benedict H., On the exceptional series, and its descendants, C. R. Math. Acad. Sci. Paris, 335, 11, 877-881 (2002) · Zbl 1017.22008 · doi:10.1016/S1631-073X(02)02590-6 |
| [13] | Eisenbud, David, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc., 260, 1, 35-64 (1980) · Zbl 0444.13006 · doi:10.2307/1999875 |
| [14] | Garibaldi, Skip, Cohomological invariants: exceptional groups and spin groups, Mem. Amer. Math. Soc., 200, 937, xii+81 pp. (2009) · Zbl 1191.11009 · doi:10.1090/memo/0937 |
| [15] | Gruson, Laurent; Sam, Steven V.; Weyman, Jerzy, Moduli of abelian varieties, Vinberg \(\theta \)-groups, and free resolutions. Commutative algebra, 419-469 (2013), Springer, New York · Zbl 1274.13027 · doi:10.1007/978-1-4614-5292-8\_13 |
| [16] | Gyoja, Akihiko, Construction of invariants, Tsukuba J. Math., 14, 2, 437-457 (1990) · Zbl 0746.20025 · doi:10.21099/tkbjm/1496161465 |
| [17] | Kac, V. G., Some remarks on nilpotent orbits, J. Algebra, 64, 1, 190-213 (1980) · Zbl 0431.17007 · doi:10.1016/0021-8693(80)90141-6 |
| [18] | Igusa, Jun-ichi, A classification of spinors up to dimension twelve, Amer. J. Math., 92, 997-1028 (1970) · Zbl 0217.36203 · doi:10.2307/2373406 |
| [19] | Iliev, Atanas; Manivel, Laurent, Fano manifolds of Calabi-Yau Hodge type, J. Pure Appl. Algebra, 219, 6, 2225-2244 (2015) · Zbl 1314.14085 · doi:10.1016/j.jpaa.2014.07.033 |
| [20] | Gatti, V.; Viniberghi, E., Spinors of \(13\)-dimensional space, Adv. in Math., 30, 2, 137-155 (1978) · Zbl 0429.20043 · doi:10.1016/0001-8708(78)90034-8 |
| [21] | Hitchin, Nigel, \(SL(2)\) over the octonions, Math. Proc. R. Ir. Acad., 118A, 1, 21-38 (2018) · Zbl 1473.53076 · doi:10.3318/pria.2018.118.04 |
| [22] | Kimura, Tatsuo, Remark on some combinatorial construction of relative invariants, Tsukuba J. Math., 5, 1, 101-115 (1981) · Zbl 0492.14006 · doi:10.21099/tkbjm/1496159322 |
| [23] | Kimura, Tatsuo, Introduction to prehomogeneous vector spaces, Translations of Mathematical Monographs 215, xxii+288 pp. (2003), American Mathematical Society, Providence, RI · Zbl 1035.11060 · doi:10.1090/mmono/215 |
| [24] | Sato, M.; Kimura, T., A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., 65, 1-155 (1977) · Zbl 0321.14030 |
| [25] | kwE7 W. Kraskiewicz, J. Weyman, Geometry of orbit closures for the representations associated to gradings of Lie algebras of type \(E_7, 1301.0720\). · Zbl 1350.17009 |
| [26] | kwE8 W. Kraskiewicz, J. Weyman , Geometry of orbit closures for the representations associated to gradings of Lie algebras of type \(E_8\), preprint. · Zbl 1350.17009 |
| [27] | Landsberg, J. M.; Manivel, L., The projective geometry of Freudenthal’s magic square, J. Algebra, 239, 2, 477-512 (2001) · Zbl 1064.14053 · doi:10.1006/jabr.2000.8697 |
| [28] | Lie LiE, A computer algebra package for Lie group computations, available online at http://wwwmathlabo.univ-poitiers.fr/ maavl/LiE/ |
| [29] | Manivel, Laurent, The Cayley Grassmannian, J. Algebra, 503, 277-298 (2018) · Zbl 1423.14293 · doi:10.1016/j.jalgebra.2018.02.010 |
| [30] | Manivel, Laurent, Double spinor Calabi-Yau varieties, \'{E}pijournal G\'{e}om. Alg\'{e}brique, 3, Art. 2, 14 pp. (2019) · Zbl 1443.14042 · doi:10.46298/epiga.2019.volume3.3965 |
| [31] | Manivel, Laurent, Ulrich and aCM bundles from invariant theory, Comm. Algebra, 47, 2, 706-718 (2019) · Zbl 1436.14082 · doi:10.1080/00927872.2018.1495222 |
| [32] | Popov, V. L., Classification of the spinors of dimension fourteen, Uspehi Mat. Nauk, 32, 1(193), 199-200 (1977) · Zbl 0327.15028 |
| [33] | Rosenfeld, Boris, Geometry of Lie groups, Mathematics and its Applications 393, xviii+393 pp. (1997), Kluwer Academic Publishers Group, Dordrecht · Zbl 0867.53002 · doi:10.1007/978-1-4757-5325-7 |
| [34] | Schimmrigk, Rolf, Mirror symmetry and string vacua from a special class of Fano varieties, Internat. J. Modern Phys. A, 11, 17, 3049-3096 (1996) · Zbl 1044.32504 · doi:10.1142/S0217751X96001486 |
| [35] | Seidel, Paul; Thomas, Richard, Braid group actions on derived categories of coherent sheaves, Duke Math. J., 108, 1, 37-108 (2001) · Zbl 1092.14025 · doi:10.1215/S0012-7094-01-10812-0 |
| [36] | Vinberg, \`E. B., Classification of homogeneous nilpotent elements of a semisimple graded Lie algebra, Trudy Sem. Vektor. Tenzor. Anal., 19, 155-177 (1979) · Zbl 0431.17006 |
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.
