Lampret, Leon; Vavpetič, Aleš (Co)homology of Lie algebras via algebraic Morse theory. (English) Zbl 1402.17025 J. Algebra 463, 254-277 (2016). 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. Cited in 1 ReviewCited in 4 Documents 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 Keywords:algebraic/discrete Morse theory; homological algebra; chain complex; acyclic matching; solvable Lie algebra; triangular matrices; torsion table; algebraic combinatorics Citations:Zbl 1334.13001; Zbl 1150.16008; Zbl 0896.57023; JFM 51.0451.01 Software:Homology PDFBibTeX XMLCite \textit{L. Lampret} and \textit{A. Vavpetič}, J. Algebra 463, 254--277 (2016; Zbl 1402.17025) Full Text: DOI arXiv 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 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.