Transactions of the American Mathematical Society

Author(s): Arkady Berenstein; Sebastian Zwicknagl
Journal: Trans. Amer. Math. Soc. 360 (2008), 3429-3472.
MSC (2000): Primary 17B37; Secondary 17B63
Posted: February 13, 2008
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Abstract: The goal of the paper is to introduce and study symmetric and exterior algebras in certain braided monoidal categories such as the category $\mathcal{O}$ for quantum groups. We relate our braided symmetric algebras and braided exterior algebras with their classical counterparts.

1.: P. Akueson, Geometry of the tangent space on a quantum hyperboloid (French), Cahiers Topologie Géom. Différentielle Catég. 42 (2001), no. 1, 2-50. MR 1820764 (2002c:17019)
2.: N. Andruskiewitsch, H.-J. Schneider, Finite Quantum Groups and Cartan Matrices. Adv. Math. 154 (2000), 1-45. MR 1780094 (2001g:16070)
3.: A. Berenstein and D. Kazhdan, Geometric and unipotent crystals, Geom. Funct. Anal. 2000, Special Volume, Part I, 188-236. MR 1826254 (2003b:17013)
4.: A. Berenstein and D. Kazhdan, Geometric and unipotent crystals II: from geometric crystals to crystal bases, submitted to Contemporary Mathematics.
5.: A. Berenstein, D.Kazhdan, Lecture notes on geometric crystals and their combinatorial analogues, to appear in Proceedings of the workshop on Combinatorial Aspect of Integrable Systems, RIMS August 2004.
6.: A. Berenstein, D. Kazhdan, Algebro-geometric distributions on reductive groups and geometric crystals, in preparation.
7.: A. Berenstein, A. Zelevinsky, String Bases for Quantum Groups of Type $A_{r}$ , Advances in Soviet Math., 16, Part 1 (1993), 51-89. MR 1237826 (94g:17019)
8.: A. Berenstein, A. Zelevinsky, Canonical bases for the quantum group of type $A_{r}$ , and piecewise-linear combinatorics, Duke Math. J. 82 (1996), no. 3, 473-502. MR 1387682 (97g:17007)
9.: A. Berenstein, A. Zelevinsky, Quantum cluster algebras, Advances in Mathematics, vol. 195, 2 (2005), pp. 405-455. MR 2146350 (2006a:20092)
10.: K. Brown and K. Goodearl, Lectures on algebraic quantum groups, Birkhäuser, 2002. MR 1898492 (2003f:16067)
11.: J. Donin, Double quantization on the coadjoint representation of ${\rm sl}(n)$ , Quantum groups and integrable systems, Part I (Prague, 1997). Czechoslovak J. Phys. 47 (1997), no. 11, 1115-1122. MR 1615893 (99h:17019)
12.: M. Durdevic, Z. Oziewicz, Clifford Algebras and Spinors for Arbitrary Braids, Differential geometric methods in theoretical physics (Ixtapa-Zihuatanejo, 1993). Adv. Appl. Clifford Algebras 4 (1994), Suppl. 1, 461-467. MR 1337730 (96f:15024)
13.: V. Drinfel'd, Quasi-Hopf Algebras, Leningrad Math. Journal, 1 (1990), no. 6. MR 1047964 (91b:17016)
14.: V. Drinfel'd, Commutation Relations in the Quasi-Classical Case, Selecta Mathematica Sovietica, 11, no.4 (1992) MR 1206296
15.: G. Greuel, G. Pfister, and H. Schönemann.
SINGULAR 2.0. A Computer Algebra System for Polynomial Computations.
Centre for Computer Algebra, University of Kaiserslautern (2001).
http://www.singular.uni-kl.de.
16.: A. Joseph, Quantum groups and their primitive ideals, Ergebnisse der Math. (3) 29, Springer-Verlag, Berlin, 1995. MR 1315966 (96d:17015)
17.: G. Lusztig, Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc. 3 (1990), no. 2, 447-498. MR 1035415 (90m:17023)
18.: S. Majid, Free braided differential calculus, braided binomial theorem, and the braided exponential map, J. Math. Phys. 34 (1993), no. 10, 4843-4856. MR 1235979 (94i:58013)
19.: O. Rossi-Doria, A -representation with no quantum symmetric algebra. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 (1999), no. 1, 5-9. MR 1768515 (2001f:17029)
20.: S. Woronowicz, Differential Calculus on Matrix Pseudogroups (Quantum Groups), Comm. in Math. Phys. 122 125-170, 1989. MR 994499 (90g:58010)
21.: R. Zhang, Howe Duality and the Quantum General Linear Group, Proc. Amer. Math. Soc., Amer. Math. Soc., Providence, RI, 131, no. 9, 2681-1692. MR 1974323 (2004b:17036)
22.: S. Zwicknagl, Flat deformations of modules over semisimple Lie algebras, Ph.D. thesis, in preparation.

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 17B37, 17B63

Arkady Berenstein
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: arkadiy@math.uoregon.edu

Sebastian Zwicknagl
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Address at time of publication: Department of Mathematics, University of California, Riverside, California 92521
Email: zwicki@noether.uoregon.edu, zwick@math.ucr.edu

DOI: 10.1090/S0002-9947-08-04373-0
PII: S 0002-9947(08)04373-0
Received by editor(s): November 9, 2005
Posted: February 13, 2008
Additional Notes: This research was supported in part by NSF grants \#DMS-0102382 and \#DMS-0501103
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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