zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Imaginary vectors in the dual canonical basis of $U_q({\germ n})$. (English) Zbl 1044.17009
If $\frak{g}$ is a simple Lie algebra over $\Bbb C$, $\germ{n}$ a maximal nilpotent subalgebra of $\frak{g}$, let $\bold B$ be the canonical basis of $U_q(\frak{n})$ and ${\bold B}^{\ast}$ the dual basis with respect to the natural scalar product in $U_q(\frak{n})$. Berenstein and Zelevinsky had conjectured that the product $b_1b_2$ is of the form $q^mb$, for $b_1,b_2,b\in{\bold B}^{\ast}$ if and only if $b_1$ and $b_2$ $q$-commute. This would imply that $b_1^2$ is always of the form $q^mb$, for $b_1 \in {\bold B}^{\ast}$. Such vectors are called {\it real}, otherwise they are called {\it imaginary}. The paper shows that there are imaginary vectors except when $\frak{g}$ if of type $A_1,A_2,A_3,A_4,B_2$. The author uses this to exhibit an explicit irreducible representation $V$ for $U_q(\hat{sl}_N)$ such that $V\otimes V$ is not irreducible.

17B37Quantum groups and related deformations
Full Text: DOI
[1] S. Ariki, On the decomposition numbers of the Heeke algebra of G(n, 1, m), J. Math. Kyoto Univ. 36 (1996), 789--808. · Zbl 0888.20011
[2] A. Berenstein, A. Zelevinsky, String bases for quantum groups of typ e Ar, Adv. Soviet Math. 16 (1993), 51--89. · Zbl 0794.17007
[3] P. Caldero, Adapted algebras for the Berenstein -- Zelevinsky conjecture, Transformation Groups 8 (2003), No. 1, 37--50. · Zbl 1044.17007 · doi:10.1007/s00031-003-1121-3
[4] P. Caldero, A multiplicative property of quantum flag minors, preprint, 2001, math. RT/0112205. · Zbl 1030.17009
[5] V. Chari, A. Pressley, Quantum affine algebras and affine Heeke algebras, Pacific J. Math. 174 (1996), 295--326. · Zbl 0881.17011
[6] I. V. Cherednik, A new interpretation of Gelfand-Tzetlin bases, Duke Math. J. 54 (1987), 563--577. · Zbl 0645.17006 · doi:10.1215/S0012-7094-87-05423-8
[7] S. Fomin, A. Zelevinsky, Cluster algebras 1: Foundations, J. Amer. Math. Soc. 15 (2002), 497--529. · Zbl 1021.16017 · doi:10.1090/S0894-0347-01-00385-X
[8] V. Ginzburg, N. Yu. Reshetikhin, E. Vasserot, Quantum groups and flag varieties, A.M.S. Contemp. Math. 175 (1994), 101--130. · Zbl 0818.17018 · doi:10.1090/conm/175/01840
[9] J. A. Green, Quantum groups, Hall algebras and quantum shuffles, in: Finite Reductive Groups, Related Structures and Representations, Ed. M. Cabanes, Prog. Math., Vol. Prog. Math., Birkhiiuser, 1996, 273--290.
[10] M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465--516. · Zbl 0739.17005 · doi:10.1215/S0012-7094-91-06321-0
[11] M. Kashiwara, Global bases of quantum groups, Duke Math. J. 69 (1993), 455--485. · Zbl 0774.17018 · doi:10.1215/S0012-7094-93-06920-7
[12] M. Kashiwara, Y. Saito, Geometric construction of crystal bases, Duke Math. J. 89 (1997), 9--6. · Zbl 0901.17006 · doi:10.1215/S0012-7094-97-08902-X
[13] P. Lalonde, A. Ram, Standard Lyndon bases of Lie algebras and enveloping algebras, Trans. Amer. Math. Soc. 347 (1995), 1821--1830. · Zbl 0833.17003 · doi:10.1090/S0002-9947-1995-1273505-4
[14] B. Leclerc, Dual canonical bases, quantum shuffles and q-characters, preprint 2002, math. QA/0209133.
[15] B. Leclerc, M. Nazarov, J.- Y. Thibon, Induced representativns of affine Heeke algebras and canonical bases of quantum groups, in: Studies in Memory of Issai Schur, Ed. A. Joseph, A. Melnikov, R. Rentschler, Prog. Math., Vol. 210, Birkhiiuser, 2003, 115--153. · Zbl 1085.17010
[16] M. Lothaire, Combinatorics on Words, Readings, Massachusetts, 1983. · Zbl 0514.20045
[17] G. Lusztig, Introduction to Quantum Groups, Prog. Math., Vol. 110, Birkhiiuser, 1993. · Zbl 0788.17010
[18] G. Lusztig, Quivers, perverse sheaves and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), 365--421. · Zbl 0738.17011 · doi:10.1090/S0894-0347-1991-1088333-2
[19] M. Reineke, Multi plicative properties of dual canonical bases of quantum groups, J. Algebra 211 (1999), 134--149. · Zbl 0917.17008 · doi:10.1006/jabr.1998.7570
[20] C. Reutenauer, Free Lie Algebras, Oxford University Press, 1993. · Zbl 0798.17001
[21] M. Rosso, Groupes quantiques et algebres de battage quantiques, C. R. Acad. Sci. Paris 320 (1995), 145--148.
[22] M. Rosso, Quantum groups and quantum shuffles, Invent. Math. 133 (1998), 399--416. · Zbl 0912.17005 · doi:10.1007/s002220050249
[23] M. Rosso, Lyndon bases and the multiplicative formula for R-matrices, preprint, 2002.
[24] A. Zelevinsky, Connected components of real double Bruhat cells, Intern. Math. Res. Notices 21 2000, 1131--1153. · Zbl 0978.20021 · doi:10.1155/S1073792800000568
[25] A. Zelevin sky, P ersonal communication.
[26] A. Zelevinsky, From Littlewood- Richardson coefficients to cluster algebms in three lectures, in: Symmetric Functions 2001: Surveys of Developments and Perspectives, Ed. S. Fomin, NATO Science Series II, Vol. 74, Kluwer, 2002, 253--273. · Zbl 1155.17303
[27] A. Zelevinsky, Induced representations of reductive p-adic groups, II, Ann. Sci. E.N.S. 13 (1980), 165--210. · Zbl 0441.22014
[28] A. B. {\cyr Z}e{\cyr l}e{\cyr v}{\cyr i}{\cyr s}{\cyr k}{\cyr i0}, B. C. Petax, O{\cyr s}{\cyr n}oe{\cyr n}oe a{\cyr f}{\cyr f}u’qn{\cyr n}oe npo{\cyr s}mpa{\cyr n}{\cyr s}{\cyr m}eo u {\cyr k}a{\cyr n}o{\cyr n}u{\cyr ch}e{\cyr s}{\cyr k}u{\cyr e1} {\cyr b}a{\cyr z}u{\cyr s}{\cyr y} e {\cyr n}enpueo{\cyr m}{\cyr y}x npe{\cyr m}a{\cyr v}{\cyr l}e{\cyr n}u{\cyr ya}x pynn{\cyr y} Sp(4), {\cyr D}AH CCP 300 (1998), No. 1, 31--35. Engl. transl.: A.V. Zelevinskii, V. S. Retakh, The base affine space and canonical bases in irreducible representations of the group SP(4), Soviet Math. Dokl. 37 (1988), No. 3, 618--622.