×

zbMATH — the first resource for mathematics

Multilinear forms and graded algebras. (English) Zbl 1141.17010
In the paper under review, the author studies connected graded algebras which are finitely generated in degree \(1\), finitely presented with relations in degree \(\geq 2\), of finite global dimension \(D\) and Gorenstein, plus the condition of being homogeneous and Koszul in the case where \(D \geq 4\). For \(D = 2\), it is shown that the algebras are associated to a nondegenerate bilinear form and correspond to the natural quantum spaces for the action of the quantum groups associated to that form in the paper [Phys. Lett., B 245, No. 2, 175–177 (1990; Zbl 1119.16307)], by G. Launer and the author. It is also shown that all homogeneous Koszul-Gorenstein algebras of finite global dimension are associated with multilinear forms which are in particular preregular, a concept introduced in this paper. Homogeneous algebras associated to a multilinear form are introduced and semi-crossed product is investigated for this class of algebras. Generalizing the above mentioned paper with Launer, the author also introduces quantum groups preserving multilinear forms which act on the quantum spaces associated with these forms. The paper analyzes several examples, including the Yang-Mills algebra and the extended \(4\)-dimensional Sklyanin algebra.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W50 Graded rings and modules (associative rings and algebras)
16S37 Quadratic and Koszul algebras
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Artin, M.; Schelter, W.F., Graded algebras of global dimension 3, Adv. math., 66, 171-216, (1987) · Zbl 0633.16001
[2] Artin, M.; Tate, J.; Van den Bergh, M., Some algebras associated to automorphisms of elliptic curves, (), 33-85 · Zbl 0744.14024
[3] Artin, M.; Tate, J.; Van den Bergh, M., Modules over regular algebras of dimension 3, Invent. math., 106, 335-388, (1991) · Zbl 0763.14001
[4] Berger, R., Koszulity for nonquadratic algebras, J. algebra, 239, 705-734, (2001) · Zbl 1035.16023
[5] Berger, R., Dimension de Hochschild des algèbres graduées, C. R. acad. sci. Paris Sér. I, 341, 597-600, (2005) · Zbl 1082.16009
[6] Berger, R.; Dubois-Violette, M.; Wambst, M., Homogeneous algebras, J. algebra, 261, 172-185, (2003) · Zbl 1061.16034
[7] Berger, R.; Marconnet, N., Koszul and Gorenstein properties for homogeneous algebras, Algebr. represent. theory, 9, 67-97, (2006) · Zbl 1125.16017
[8] Bichon, J., Cosovereign Hopf algebras, J. pure appl. algebra, 157, 121-133, (2001) · Zbl 0976.16027
[9] Bichon, J., The representation category of the quantum group of a non-degenerate bilinear form, Comm. algebra, 31, 4831-4851, (2003) · Zbl 1034.16042
[10] Bondal, A.I.; Polishchuk, A.E., Homological properties of associative algebras: the method of helices, Russian acad. sci. izv. math., 42, 219-260, (1994) · Zbl 0847.16010
[11] Cartan, H.; Eilenberg, S., Homological algebra, (1973), Princeton University Press
[12] Connes, A., Non-commutative differential geometry, Publ. math. inst. hautes études sci., 62, 257-360, (1986)
[13] Connes, A., Non-commutative geometry, (1994), Academic Press · Zbl 0933.46069
[14] Connes, A.; Dubois-Violette, M., Noncommutative finite-dimensional manifolds. I. spherical manifolds and related examples, Comm. math. phys., 230, 539-579, (2002) · Zbl 1026.58005
[15] Connes, A.; Dubois-Violette, M., Yang – mills algebra, Lett. math. phys., 61, 149-158, (2002) · Zbl 1028.53025
[16] Connes, A.; Dubois-Violette, M., Moduli space and structure of noncommutative 3-spheres, Lett. math. phys., 66, 91-121, (2003) · Zbl 1052.58012
[17] Connes, A.; Dubois-Violette, M., Yang – mills and some related algebras, (2004), in: Rigorous Quantum Field Theory
[18] Connes, A.; Dubois-Violette, M., Noncommutative finite-dimensional manifolds. II. moduli space and structure of noncommutative 3-spheres, (2005)
[19] Dubois-Violette, M., \(d^N = 0\): generalized homology, K-theory, 14, 371-404, (1998) · Zbl 0918.18008
[20] Dubois-Violette, M., Graded algebras and multilinear forms, C. R. acad. sci. Paris Sér. I, 341, 719-724, (2005) · Zbl 1105.16020
[21] Dubois-Violette, M.; Launer, G., The quantum group of a non-degenerated bilinear form, Phys. lett. B, 245, 175-177, (1990) · Zbl 1119.16307
[22] Dubois-Violette, M.; Popov, T., Homogeneous algebras, statistics and combinatorics, Lett. math. phys., 61, 159-170, (2002) · Zbl 1020.16007
[23] Ewen, H.; Ogievetsky, O., Classification of the \(\mathit{GL}(3)\) quantum matrix groups
[24] Irving, R.S., Prime ideals of ore extensions, J. algebra, 58, 399-423, (1979) · Zbl 0411.16025
[25] J.L. Loday, Notes on Koszul duality for associative algebras, 1999
[26] Lu, D.M.; Palmieri, J.H.; Wu, Q.S.; Zhang, J.J., Regular algebras of dimension 4 and their \(A_\infty\)-ext-algebras · Zbl 1193.16014
[27] Manin, Yu.I., Quantum groups and non-commutative geometry, (1988), CRM Université de Montréal · Zbl 0724.17006
[28] Manin, Yu.I., Some remarks on Koszul algebras and quantum groups, Ann. inst. Fourier (Grenoble), 37, 191-205, (1987) · Zbl 0625.58040
[29] Nekrasov, N., Lectures on open strings and noncommutative gauge theories, (), 477-495
[30] Popov, T., Automorphisms of regular algebras, () · Zbl 1342.16032
[31] Pottier, A., Stabilité de la propriété de Koszul pour LES algèbres homogènes vis-à-vis du produit semi-croisé, C. R. acad. sci. Paris Sér. I, 343, 161-164, (2006) · Zbl 1150.16021
[32] Priddy, S.B., Koszul resolutions, Trans. amer. math. soc., 152, 39-60, (1970) · Zbl 0261.18016
[33] Sklyanin, E.K., Some algebraic structures connected with the yang – baxter equation, Funct. anal. appl., 16, 263-270, (1982) · Zbl 0513.58028
[34] Smith, S.P.; Stafford, J.T., Regularity of the four-dimensional Sklyanin algebra, Compos. math., 83, 259-289, (1992) · Zbl 0758.16001
[35] Stafford, J.T., Noncommutative projective geometry, Icm, III, 1-3, (2002) · Zbl 1057.14004
[36] Van den Bergh, M., A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. amer. math. soc., 126, 1345-1348, (1998) · Zbl 0894.16005
[37] Van den Bergh, M., Erratum, Proc. amer. math. soc., 130, 2809-2810, (2002)
[38] Woronowicz, S.L., Tannaka – krein duality for compact matrix pseudogroups. twisted \(\mathit{SU}(N)\) groups, Invent. math., 93, 35-76, (1988) · Zbl 0664.58044
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.