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(Co)homology of Lie algebras via algebraic Morse theory. (English) Zbl 1402.17025

Summary: The fundamental theorem of cancellation AMT [M. Jöllenbeck, Algebraic discrete Morse theory and applications to commutative algebra. Diss. Univ. Marburg (2005; Zbl 1334.13001)] and [E. Sköldberg [Trans. Am. Math. Soc. 358, No. 1, 115–129 (2006; Zbl 1150.16008)], which is the algebraic generalization of discrete Morse theory [R. Forman, Adv. Math. 134, No. 1, 90–145 (1998; Zbl 0896.57023) for simplicial complexes and smooth Morse theory [M. Morse, Trans. Am. Math. Soc. 27, 345–396 (1925; JFM 51.0451.01)] for differentiable manifolds, is discussed in the context of general chain complexes of free modules.
The Chevalley (co)homology table of a Lie algebra is often a tremendous beast. Using AMT, we compute the homology of the Lie algebra of all triangular matrices \(\mathfrak{sol}_n\) over \(\mathbb{Q}\) or \(\mathbb{Z}_p\) for large enough primes \(p\). We determine the column and row in the table of \(H_k(\mathfrak{sol}_n; \mathbb{Z})\) where the \(p\)-torsion first appears. Module \(H_k(\mathfrak{sol}_n; \mathbb{Z}_p)\) is expressed by the homology of a chain subcomplex for the Lie algebra of all strictly triangular matrices \(\mathfrak{nil}_n\), using the Künneth formula. All conclusions are accompanied by computer experiments.
Then we generalize some results to Lie algebras of (strictly) triangular matrices \(\mathfrak{gl}_n^\prec\) and \(\mathfrak{gl}_n^\preceq\) with respect to any partial ordering \(\preceq\) on \([n]\). We determine the multiplicative structure of \(H^\ast(\mathfrak{gl}_n^\preceq)\) w.r.t. the cup product over fields of zero or sufficiently large characteristic, the result being the exterior algebra.
Matchings used here can be analogously defined for other Lie algebra families and in other (co)homology theories; we collectively call them normalization matchings. They are useful for theoretical as well as computational purposes.

MSC:

17B56 Cohomology of Lie (super)algebras
13P20 Computational homological algebra
17B30 Solvable, nilpotent (super)algebras
17B50 Modular Lie (super)algebras
18G35 Chain complexes (category-theoretic aspects), dg categories
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
68W30 Symbolic computation and algebraic computation

Software:

Homology
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References:

[1] Dumas, J. G.; Heckenbach, F.; Saunders, D.; Welker, V., Computing simplicial homology based on efficient Smith normal form algorithms, (Algebra, Geometry, and Software Systems (2003), Springer: Springer Berlin), 177-206 · Zbl 1026.55010
[2] Forman, R., Morse theory for cell complexes, Adv. Math., 134, 1, 90-145 (1998) · Zbl 0896.57023
[3] Hozo, I., Inclusion of poset homology into Lie algebra homology, J. Pure Appl. Algebra, 111, 169-180 (1996) · Zbl 0869.17020
[4] Jöllenbeck, M., Algebraic Discrete Morse Theory and Applications to Commutative Algebra (2005), Thesis · Zbl 1334.13001
[5] Jöllenbeck, M.; Welker, V., Minimal resolutions via algebraic discrete Morse theory, Mem. Amer. Math. Soc., 197, 923 (2009) · Zbl 1160.13007
[6] Jöllenbeck, M.; Welker, V., Resolution of the residue class field via algebraic discrete Morse theory (2005)
[7] Lampret, L.; Vavpetič, A., (Co)homology of poset Lie algebras · Zbl 1433.17026
[9] Loday, J. L., Cyclic Homology, Grundlehren der Mathematischen Wissenschaften, vol. 301 (1998), Springer-Verlag: Springer-Verlag Berlin · Zbl 0885.18007
[10] Morse, M., Relations between the critical points of a real function of \(n\) independent variables, Trans. Amer. Math. Soc., 27, 3, 345-396 (1925) · JFM 51.0451.01
[11] Sköldberg, E., Morse theory from an algebraic viewpoint, Trans. Amer. Math. Soc., 358, 1, 115-129 (2006) · Zbl 1150.16008
[12] Sköldberg, E., The homology of Heisenberg Lie algebras over fields of characteristic two, Math. Proc. R. Ir. Acad., 105A, 2, 47-49 (2005) · Zbl 1095.17008
[13] Tauvel, P.; Yu, R. W.T., Lie Algebras and Algebraic Groups, Springer Monographs in Mathematics (2005), Springer-Verlag: Springer-Verlag Berlin · Zbl 1068.17001
[14] Weibel, C. A., An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, vol. 38 (1994), Cambridge University Press · Zbl 0797.18001
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