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| nonassociative algebra | |
|---|---|
| Name | Nonassociative algebra |
| Type | Algebraic structure |
| Related | Lie algebra; Jordan algebra; Alternative algebra; Malcev algebra; Octonions |
nonassociative algebra
Nonassociative algebra studies algebraic systems in which the associative law need not hold, focusing on identities, structural decomposition, and representation theory; researchers in this area include Sophus Lie, Camille Jordan, Arthur Cayley, Élie Cartan, and John von Neumann, while institutions such as the Institute for Advanced Study, Mathematical Association of America, and Royal Society have supported related work. Connections run to classical figures and organizations like Hermann Weyl, Emmy Noether, Niels Henrik Abel, Bernhard Riemann, David Hilbert, Felix Klein, Isaac Newton, James Clerk Maxwell, Albert Einstein, Paul Dirac, Évariste Galois, Hermann Grassmann, William Rowan Hamilton, Augustin-Louis Cauchy, Niels Bohr, Murray Gell-Mann, Roger Penrose, Alonzo Church, Alan Turing, Harish-Chandra, Israel Gelfand, Jean-Pierre Serre, Shing-Tung Yau, Maxwell Sachs, Felix Kleinert, Michael Atiyah, Isadore Singer, Raoul Bott.
A nonassociative algebra over a field is a vector space equipped with a bilinear product that may violate associativity; foundational contributors include Camille Jordan, Sophus Lie, Arthur Cayley, Élie Cartan, and Hermann Weyl, and standard examples and counterexamples appear in the work of William Rowan Hamilton, John Graves, Augustin-Louis Cauchy, Évariste Galois, and James Joseph Sylvester. Core notions such as identities, commutators, associators, and isotopes are developed alongside concepts introduced by Emmy Noether, Niels Henrik Abel, Bernhard Riemann, and David Hilbert in their broader algebraic investigations. Early classification efforts involved mathematicians connected to Felix Klein, Isaac Newton, Murray Gell-Mann, Paul Dirac, Albert Einstein, and Roger Penrose who influenced geometric and physical interpretations. Modern programmatic study has been carried forward at places like Institute for Advanced Study, Princeton University, University of Cambridge, École Normale Supérieure, University of Göttingen, and University of Chicago by researchers connected to Jean-Pierre Serre, Michael Atiyah, Isadore Singer, Raoul Bott, and Harish-Chandra.
Prominent classes include Lie algebras developed by Sophus Lie and elaborated by Élie Cartan, Harish-Chandra, Claude Chevalley, Jean-Pierre Serre, Nathan Jacobson, and Élie Cartan; Jordan algebras introduced by Camille Jordan and used by Pascual Jordan, John von Neumann, Eugene Wigner, Richard Brauer, and Nathan Jacobson; alternative algebras studied by Arthur Cayley, John Graves (leading to the Cayley–Dickson construction associated with William Rowan Hamilton), and Ruth Moufang who discovered Moufang loops and identities bearing her name. Malcev algebras trace to work by Anatoly Malcev and interact with research by A.I. Malcev collaborators and contemporaries such as Nikolai Ivanovich Lobachevsky-era schools and later analysts at Moscow State University and Steklov Institute. Other notable examples include the octonions tied to John T. Graves and Arthur Cayley, sedenions from iterated doubling influenced by Cayley–Dickson construction, flexible algebras linked to Emmy Noether-era algebraists, and power-associative algebras considered by Richard Brauer and Nathan Jacobson. Researchers like Israel Gelfand, Jean-Louis Loday, Murray Gerstenhaber, Michel Saeki, Gerhard Hochschild, Daniel Quillen, Hyman Bass, and Maxim Kontsevich have extended classes and homological perspectives.
Structure theory parallels classical work by Niels Henrik Abel-era algebraists and modern developments by Nathan Jacobson, Claude Chevalley, Jean-Pierre Serre, Emmy Noether, and David Hilbert with analogues of ideals, solvable and nilpotent radicals, Levi decompositions, and simple components developed by contributors at Princeton University, Harvard University, University of Chicago, Moscow State University, and École Normale Supérieure. Radicals named after Kurosh and results analogous to the Wedderburn–Artin theorem were investigated by Lev Kurosh, Issai Schur, Richard Brauer, Jacobson, and Claude Chevalley; simplicity and semisimplicity criteria echo themes in work by Élie Cartan, Harish-Chandra, Cartan-related classification, and the Killing form investigations influenced by Wilhelm Killing and Élie Cartan. Classification of finite-dimensional simple algebras in special classes involved contributors like I.M. Gelfand, Nathan Jacobson, Seligman, Pascual Jordan, and later researchers in algebraic groups such as Armand Borel and Jacques Tits.
Representation theory for nonassociative algebras extends module concepts developed by Emmy Noether, Claude Chevalley, Jean-Pierre Serre, Israel Gelfand, Michael Atiyah, Isadore Singer, and Harish-Chandra; key approaches adapt enveloping algebra methods of Poincaré-type and categorical viewpoints influenced by Alexander Grothendieck, Daniel Quillen, Maxim Kontsevich, Jean-Louis Loday, and Gerhard Hochschild. Modules, cohomology, and extension theories build on work by Hochschild, Gerstenhaber, Murray Gerstenhaber, Pierre Deligne, Alexander Beilinson, and Vladimir Drinfeld with applications of homological algebra from Samuel Eilenberg, Samuel Mac Lane, and Saunders Mac Lane. Representations of Lie algebras studied by Harish-Chandra and Nicolas Bourbaki-associated authors inform analogous theories for Jordan and alternative algebras developed by Nathan Jacobson, Richard Schafer, Ruth Moufang, and Anatoly Malcev.
Constructions include identities such as the Jacobi identity introduced by Sophus Lie, the Jordan identity from Camille Jordan, and alternativity linked to Arthur Cayley and John Graves; derivation algebras and inner derivations were studied by Nathan Jacobson, Richard Schafer, Élie Cartan, Harish-Chandra, and Israel Gelfand. Universal enveloping constructions for Lie-type nonassociative structures parallel those by Élie Cartan and Poincaré and have been developed further by J.-P. Serre-influenced schools, Gerhard Hochschild, A.I. Kostrikin, Seligman, Victor Kac, and James Lepowsky. Cohomology theories, deformation theory, and operadic formalisms draw on work by Murray Gerstenhaber, Maxim Kontsevich, Jean-Louis Loday, Vladimir Drinfeld, Pierre Deligne, Alexander Grothendieck, and Daniel Quillen.
Applications tie to differential geometry explored by Élie Cartan, Bernhard Riemann, Felix Klein, Hermann Weyl, Shiing-Shen Chern, Shing-Tung Yau, and Roger Penrose; theoretical physics contributions come from Albert Einstein, Paul Dirac, Murray Gell-Mann, Pascual Jordan, John von Neumann, Eugene Wigner, Niels Bohr, Max Born, and Richard Feynman using Jordan and Lie structures in quantum theory and symmetry. Further connections appear in coding and design theory developed by Richard Hamming, Claude Shannon, E. T. Parker, Elwyn Berlekamp, Ronald Graham, J. H. Conway, Neil J. A. Sloane, Vladimir Levenshtein, André Weil, and Hermann Weyl; cryptography and information theory draw from algebraic techniques used by Alan Turing, Alonzo Church, Claude Shannon, and Whitfield Diffie. Interdisciplinary work links to string theory and exceptional structures investigated by Edward Witten, Michael Green, John H. Schwarz, Piotr W. Michor, Pierre Ramond, Gabriele Veneziano, and John Baez, while computational and algorithmic aspects have been advanced at Los Alamos National Laboratory, Bell Labs, IBM Research, Microsoft Research, and by researchers like Donald Knuth and Stephen Cook.