Jordan, David A.; Oh, Sei-Qwon Poisson spectra in polynomial algebras. (English) Zbl 1351.17023 J. Algebra 400, 56-71 (2014). Summary: A significant class of Poisson brackets on the polynomial algebra \(\mathbb C[x_1,\dots ,x_n]\) is studied and, for this class of Poisson brackets, the Poisson prime ideals, Poisson primitive ideals and symplectic cores are determined. Moreover it is established that these Poisson algebras satisfy the Poisson Dixmier-Moeglin equivalence. Cited in 3 Documents MSC: 17B63 Poisson algebras 16S36 Ordinary and skew polynomial rings and semigroup rings Keywords:Poisson algebra; Poisson prime ideal; polynomial algebra Software:Macaulay2 PDF BibTeX XML Cite \textit{D. A. Jordan} and \textit{S.-Q. Oh}, J. Algebra 400, 56--71 (2014; Zbl 1351.17023) Full Text: DOI arXiv OpenURL References: [1] Brown, K. A.; Goodearl, K. R., Lectures on algebraic quantum groups, Adv. Courses Math. CRM Barcelona, (2002), Birkhäuser Basel, Boston, Berlin · Zbl 1027.17010 [2] Brown, K. A.; Gordon, I., Poisson orders, symplectic reflection algebras and representation theory, J. Reine Angew. Math., 559, 193-216, (2003) · Zbl 1025.17007 [3] Erdmann, K.; Wildon, M. J., Introduction to Lie algebras, (2006), Springer London · Zbl 1139.17001 [4] Gelfand, I. M.; Kapranov, M.; Zelevinsky, A., Discriminants, resultants, and multidimensional determinants, (1994), Birkhäuser Boston · Zbl 0827.14036 [5] Goodearl, K. R., Semiclassical limits of quantized coordinate rings, (Huynh, D. V.; Lopez-Permouth, S., Advances in Ring Theory, (2009), Birkhäuser Basel), 165-204 · Zbl 1202.16027 [6] Goodearl, K. R., A Dixmier-Moeglin equivalence for Poisson algebras with torus actions, (Huynh, D. V.; Jain, S. K.; Lopez-Permouth, S. R., Algebra and Its Applications, Contemp. Math., vol. 419, (2006)), 131-154 · Zbl 1147.17017 [7] Grayson, D. R.; Stillman, M. E., Macaulay2, a software system for research in algebraic geometry, available at [8] Hodge, W. V.D.; Pedoe, D., Methods of algebraic geometry, vol. 1, Cambridge Math. Lib., (1994), Cambridge University Press Cambridge, (reissue) · Zbl 0796.14003 [9] Grabowski, J.; Marmo, G.; Perelomov, A. M., Poisson structures: towards a classification, Modern Phys. Lett. A, 8, 18, 1719-1733, (1993) · Zbl 1020.37529 [10] Jordan, D. A.; Oh, Sei-Qwon, Poisson brackets and Poisson spectra in polynomial algebras, Contemp. Math., 562, 169-187, (2012) · Zbl 1273.17027 [11] Jordan, D. A., Ore extensions and Poisson algebras, Glasg. Math. J., (13 August 2013), Firstview Articles, published online [12] Oh, Sei-Qwon, Symplectic ideals of Poisson algebras and the Poisson structure associated to quantum matrices, Comm. Algebra, 27, 2163-2180, (1999) · Zbl 0936.16041 [13] Oh, Sei-Qwon, Symplectic submanifolds and symplectic ideals, J. Lie Theory, 16, 1, 131-138, (2006) · Zbl 1118.53053 [14] Oh, Sei-Qwon, Poisson prime ideals of Poisson polynomial rings, Comm. Algebra, 35, 3007-3012, (2007) · Zbl 1160.17015 [15] Oh, Sei-Qwon, Quantum and Poisson structures of multi-parameter symplectic and Euclidean spaces, J. Algebra, 319, 4485-4535, (2008) · Zbl 1142.17011 [16] Panov, A. N., n-Poisson and n-Sklyanin brackets, J. Math. Sci., 110, 2322-2329, (2002) · Zbl 1048.17002 [17] Przybysz, R., On one class of exact Poisson structures, J. Math. Phys., 42, 1913-1920, (2001) · Zbl 1016.53062 [18] Rowen, L. H., Graduate algebra: commutative view, (2006), American Mathematical Society Providence RI · Zbl 1116.13001 [19] Sharp, R. Y., Steps in commutative algebra, London Math. Soc. Stud. Texts, vol. 51, (2000), Cambridge University Press · Zbl 0969.13001 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.