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On central extensions and simply laced Lie algebras. (English) Zbl 1468.17017

The main aim of this paper is to construct a \(\mathbb Z_d\)-graded semisimple Lie algebra attached to each input datum. The author was lead to this purely algebraic problem when studying some families of algebraic curves, which provide naturally such an input datum. Note that this work is part of a program trying to use representation theory to extend results in arithmetic statistics, although precisely in this occasion the work is more focused on the algebraic part than on the number-theoretic applications, which are relegated.
For \(k\) a field of characteristic zero containing a primitive \(d\)th root of the unit \(\zeta\), the input datum is a quadruple \((\Lambda, w,\mathcal H, \varepsilon)\) with
A simply laced root lattice \(\Lambda\), that is, a finitely generated free \(\mathbb Z\)-module with a symmetric positive-definite bilinear form \((\cdot,\cdot)\colon\Lambda\times \Lambda\to\mathbb Z\) generated by \(\{\alpha\in\Lambda:(\alpha,\alpha)=2\}\);
\(w\colon \Lambda\to\Lambda\) a lattice automorphism of order \(d\) which is elliptic, that is, the group of coinvariants \(\Lambda_w:=\Lambda/(1-w)\Lambda\) is finite;
A central extension \(1\to\langle\zeta\rangle\to\mathcal H\to \Lambda_w\to 1\) with \(\langle\cdot,\cdot\rangle\colon\Lambda_w\times \Lambda_w\to \langle\zeta\rangle\) its commutator pairing;
A bilinear map \(\varepsilon\colon \Lambda\times \Lambda\to k^\times\) (that is, \(\langle \alpha,\beta+\gamma \rangle=\langle \alpha,\beta \rangle\langle \alpha,\gamma \rangle\) and similarly for the other variable), satisfying \(\varepsilon(w\alpha, w\beta)=\varepsilon(\alpha, \beta)\) and \(\varepsilon(\alpha, \beta)/\varepsilon( \beta,\alpha)=-\langle \beta,\alpha \rangle\) whenever \((\alpha, \beta)=-1\).
For each quadruple as above, the author constructs a Lie algebra with Dynkin type given by \(\Lambda\), endowed with a grading produced by an order \(d\) automorphism which is a lifting of \(w\). The author claims that the input data are easy to find: first, if one has \((\Lambda, w)\), then it is always possible to give a group \(\mathcal H\) and a bilinear map \(\varepsilon\) such that \((\Lambda, w,\mathcal H, \varepsilon)\) is a suitable quadruple as required above. Also, and this is important for the applications, input data arise naturally from some algebraic curves. For instance, an interesting collection of examples coming from a simple singularity of type \(E_8\) is described when \(k\) is algebraically closed. From a semiuniversal deformation of the surface \(X^5+Y^3+Z^2=0\), the Picard group of each smooth fiber is a root lattice of type \(E_8\), and this lattice has elliptic automorphisms of orders \(2\), \(3\) and \(5\).
Note that, for some usual choice of \(\varepsilon\), representation theory of \(\mathcal H\) is closely related with representation theory of the Lie subalgebra fixed by the automorphism.
Finally, new proofs of some well-known results on lifts of automorphisms on simply laced Lie algebras are given as a consequence of the construction here developed.

MSC:

17B20 Simple, semisimple, reductive (super)algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
14H40 Jacobians, Prym varieties
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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[1] Adams, Jeffrey; He, Xuhua, Lifting of elements of Weyl groups, J. Algebra, 485, 142-165 (2017), MR3659328 · Zbl 1395.20024
[2] Bourbaki, Nicolas, Lie Groups and Lie Algebras, Elements of Mathematics (Berlin) (2002), Springer-Verlag: Springer-Verlag Berlin, translated from the 1968 French original by Andrew Pressley, MR1890629 · Zbl 0983.17001
[3] Bhargava, Manjul; Shankar, Arul, The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1, preprint, available at · Zbl 1307.11071
[4] Beyl, F. Rudolf; Tappe, Jürgen, Group Extensions, Representations, and the Schur Multiplicator, Lecture Notes in Mathematics, vol. 958 (1982), Springer-Verlag: Springer-Verlag Berlin-New York, MR681287 · Zbl 0544.20001
[5] Carter, R. W., Conjugacy classes in the Weyl group, Compos. Math., 25, 1-59 (1972), MR318337 · Zbl 0254.17005
[6] Cremona, J. E.; Fisher, T. A.; O’Neil, C.; Simon, D.; Stoll, M., Explicit n-descent on elliptic curves. I. Algebra, J. Reine Angew. Math., 615, 121-155 (2008), MR2384334 · Zbl 1242.11039
[7] Gross, Benedict H., On Bhargava’s representation and Vinberg’s invariant theory, (Frontiers of Mathematical Sciences (2011)), 317-321, MR3050830 · Zbl 1523.20079
[8] Kulkarni, Avinash, An arithmetic invariant theory of curves from \(E_8\) · Zbl 1422.11222
[9] Lepowsky, J., Calculus of twisted vertex operators, Proc. Natl. Acad. Sci. USA, 82, 24, 8295-8299 (1985), MR820716 · Zbl 0579.17010
[10] Lurie, Jacob, On simply laced Lie algebras and their minuscule representations, Comment. Math. Helv., 76, 3, 515-575 (2001), MR1854697 · Zbl 1017.17011
[11] Prasad, Dipendra, Notes on central extensions · Zbl 0969.22008
[12] Reeder, Mark, Torsion automorphisms of simple Lie algebras, Enseign. Math. (2), 56, 1-2, 3-47 (2010), MR2674853 · Zbl 1223.17020
[13] Reeder, Mark, Elliptic centralizers in Weyl groups and their coinvariant representations, Represent. Theory, 15, 63-111 (2011), MR2765477 · Zbl 1251.20042
[14] Reeder, Mark; Levy, Paul; Yu, Jiu-Kang; Gross, Benedict H., Gradings of positive rank on simple Lie algebras, Transform. Groups, 17, 4, 1123-1190 (2012), MR3000483 · Zbl 1310.17017
[15] Rains, Eric M.; Sam, Steven V., Invariant theory of \(\wedge^3(9)\) and genus-2 curves, Algebra Number Theory, 12, 4, 935-957 (2018), MR3830207 · Zbl 1405.15035
[16] Romano, Beth; Thorne, Jack A., On the arithmetic of simple singularities of type E, Res. Number Theory, 4, 2 (2018) · Zbl 1441.14080
[17] Romano, Beth; Thorne, Jack, \( E_8\) and the and the average size of the 3-Selmer group of the Jacobian of a pointed genus-2 curve · Zbl 1441.14080
[18] Shioda, Tetsuji, Mordell-Weil lattices and Galois representation. II, III, Proc. Jpn. Acad., Ser. A, Math. Sci., 65, 8, 296-303 (1989), MR1030204 · Zbl 0715.14017
[19] Shioda, Tetsuji, Mordell-Weil lattices of type \(E_8\) and deformation of singularities, (Prospects in Complex Geometry. Prospects in Complex Geometry, Katata and Kyoto, 1989 (1991)), 177-202, MR1123543 · Zbl 0751.14006
[20] Slodowy, Peter, Simple Singularities and Simple Algebraic Groups, Lecture Notes in Mathematics, vol. 815 (1980), Springer: Springer Berlin, MR584445 · Zbl 0441.14002
[21] Schütt, Matthias; Shioda, Tetsuji, Elliptic surfaces, (Algebraic Geometry in East Asia—Seoul 2008 (2010)), 51-160, MR2732092 · Zbl 1216.14036
[22] Schütt, Matthias; Shioda, Tetsuji, Mordell-Weil Lattices, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 70 (2019), Springer: Springer Singapore, MR3970314 · Zbl 1433.14002
[23] Steinberg, Robert, Endomorphisms of Linear Algebraic Groups, Memoirs of the American Mathematical Society, vol. 80 (1968), American Mathematical Society: American Mathematical Society Providence, R.I., MR0230728 · Zbl 0164.02902
[24] Thorne, Jack A., Vinberg’s representations and arithmetic invariant theory, Algebra Number Theory, 7, 9, 2331-2368 (2013), MR3152016 · Zbl 1321.11045
[25] Thorne, Jack A., Arithmetic invariant theory and 2-descent for plane quartic curves, Algebra Number Theory, 10, 7, 1373-1413 (2016), with an appendix by Tasho Kaletha, MR3554236 · Zbl 1416.11044
[26] Zarhin, Yu. G., Del Pezzo surfaces of degree 1 and Jacobians, Math. Ann., 340, 2, 407-435 (2008), MR2368986 · Zbl 1222.14084
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