Dimension and enumeration of primitive ideals in quantum algebras
J. Bell
, S. Launois
and N. Nguyen3
3S. Launois ( ) Institute of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, Kent, CT2 7NF, UK
DOI: 10.1007/s10801-008-0132-5
Abstract
In this paper, we study the primitive ideals of quantum algebras supporting a rational torus action. We first prove a quantum analogue of a Theorem of Dixmier; namely, we show that the Gelfand-Kirillov dimension of primitive factors of various quantum algebras is always even. Next we give a combinatorial criterion for a prime ideal that is invariant under the torus action to be primitive. We use this criterion to obtain a formula for the number of primitive ideals in the algebra of 2\times n quantum matrices that are invariant under the action of the torus. Roughly speaking, this can be thought of as giving an enumeration of the points that are invariant under the induced action of the torus in the “variety of 2\times n quantum matrices”.
Pages: 269–294
Keywords: keywords primitive ideals; quantum matrices; quantised enveloping algebras; cauchon diagrams; perfect matchings; pfaffians
Full Text: PDF
References
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29. North-Holland, Amsterdam (1986) J Algebr Comb (2009) 29: 269-294
2. Brown, K.A., Goodearl, K.R.: Lectures on Algebraic Quantum Groups. Advanced Courses in Mathematics CRM Barcelona. Birkhäuser, Basel (2002)
3. Cauchon, G.: Effacement des dérivations et spectres premiers des algèbres quantiques. J. Algebra 260, 476-518 (2003)
4. Cauchon, G.: Effacement des dérivations et quotients premiers de U w q (g). Preprint, University of Reims
5. Cauchon, G.: Spectre premier de Oq (Mn(k)), image canonique et séparation normale. J. Algebra 260, 519-569 (2003)
6. Goodearl, K.R., Letzter, E.S.: Prime factor algebras of the coordinate ring of quantum matrices. Proc. Am. Math. Soc. 121, 1017-1025 (1994)
7. Goodearl, K.R., Letzter, E.S.: Prime and primitive spectra of multiparameter quantum affine spaces. In: Trends in Ring Theory, CMS Conf Proc, (Miskolc, 1996) vol. 22, pp. 39-58. Am. Math. Soc., Providence (1998)
8. Goodearl, K.R., Letzter, E.S.: The Dixmier-Moeglin equivalence in quantum matrices and quantum Weyl algebras. Trans. Am. Math. Soc. 352(3), 1381-1403 (2000)
9. Hodges, T.J., Levasseur, T.: Primitive ideals of Cq [SL(3)]. Comm. Math. Phys. 156, 581-605 (1993)
10. Hodges, T.J., Levasseur, T.: Primitive ideals of Cq [SL(n)]. J. Algebra 168, 455-468 (1994)
11. Joseph, A.: On the prime and primitive spectrum of the algebra of functions on a quantum group. J. Algebra 169, 441-511 (1994)
12. Joseph, A.: Sur les idéaux génériques de l'algèbre des fonctions sur un groupe quantique. C.R. Acad. Sci. Paris 321, 135-140 (1995)
13. Joseph, A.: Quantum Groups and Their Primitive Ideals. Ergebnisse der Math. (3), vol.
29. Springer, Berlin (1995)
14. Launois, S.: Combinatorics of H-primes in quantum matrices. J. Algebra 309, 139-167 (2007)
15. Launois, S.: Primitive ideals and automorphism group of + Uq (B2). J. Algebra Appl. 6(1), 21-47 (2007)
16. Launois, S., Lenagan, T.H.: Primitive ideals and automorphisms of quantum matrices. Algebr. Represent. Theory 10, 339-365 (2007)
17. Launois, S., Lenagan, T.H., Rigal, L.: Quantum unique factorisation domains. J. London Math. Soc. (2) 74(2), 321-340 (2006)
18. Lovász, L., Plummer, M.D.: Matching Theory. Ann. Discrete Math., vol.
29. North-Holland, Amsterdam (1986) J Algebr Comb (2009) 29: 269-294