Journal of the Mathematical Society of Japan

Diagram automorphisms and quantum groups

Toshiaki SHOJI and Zhiping ZHOU

Advance publication

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Abstract

Let $\mathbf{U}^-_q = \mathbf{U}^-_q(\mathfrak{g})$ be the negative part of the quantum group associated to a finite dimensional simple Lie algebra $\mathfrak{g}$, and $\sigma : \mathfrak{g} \to \mathfrak{g}$ be the automorphism obtained from the diagram automorphism. Let $\mathfrak{g}^{\sigma}$ be the fixed point subalgebra of $\mathfrak{g}$, and put $\underline{\mathbf{U}}^-_q = \mathbf{U}^-_q(\mathfrak{g}^{\sigma})$. Let $\mathbf{B}$ be the canonical basis of $\mathbf{U}_q^-$ and $\underline{\mathbf{B}}$ the canonical basis of $\underline{\mathbf{U}}_q^-$. $\sigma$ induces a natural action on $\mathbf{B}$, and we denote by $\mathbf{B}^{\sigma}$ the set of $\sigma$-fixed elements in $\mathbf{B}$. Lusztig proved that there exists a canonical bijection $\mathbf{B}^{\sigma} \simeq \underline{\mathbf{B}}$ by using geometric considerations. In this paper, we construct such a bijection in an elementary way. We also consider such a bijection in the case of certain affine quantum groups, by making use of PBW-bases constructed by Beck and Nakajima.

Article information

Source
J. Math. Soc. Japan, Advance publication (2019), 33 pages.

Dates
Received: 17 October 2018
First available in Project Euclid: 2 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1572660116

Digital Object Identifier
doi:10.2969/jmsj/81488148

Subjects
Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]
Secondary: 20G42: Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50] 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37]

Keywords
quantum groups canonical bases PBW-bases

Citation

SHOJI, Toshiaki; ZHOU, Zhiping. Diagram automorphisms and quantum groups. J. Math. Soc. Japan, advance publication, 2 November 2019. doi:10.2969/jmsj/81488148. https://projecteuclid.org/euclid.jmsj/1572660116


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References

  • [B] J. Beck, Braid group action and quantum affine algebras, Comm. Math. Phys., 165 (1994), 555–568.
  • [BCP] J. Beck, V. Chari and A. Pressley, An algebraic characterization of the affine canonical basis, Duke Math. J., 99 (1999), 455–487.
  • [BN] J. Beck and H. Nakajima, Crystal bases and two-sided cells of quantum affine algebras, Duke Math. J., 123 (2004), 335–402.
  • [Ka] V. G. Kac, Infinite Dimensional Lie Algebras, third edition, Cambridge Univ. Press, 1990.
  • [K1] M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J., 63 (1991), 465–516.
  • [K2] M. Kashiwara, On level-zero representation of quantized affine algebras, Duke Math. J., 112 (2002), 117–175.
  • [L1] G. Lusztig, Quantum groups at roots of 1, Geom. Dedicata, 35 (1990), 89–113.
  • [L2] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc., 3 (1990), 447–498.
  • [L3] G. Lusztig, Introduction to Quantum Groups, Progr. Math., 110, Birkhäuser, Boston, 1993.
  • [L4] G. Lusztig, Piecewise linear parametrization of canonical bases, Pure Appl. Math. Q., 7 (2011), 783–796.
  • [LS] S. Levendorskii and Y. Soibelman, Some applications of the quantum Weyl groups, J. Geom. Phys., 7 (1990), 241–254.
  • [MNZ] Y. Ma, D. Niu and Z. Zhou, Diagram automorphisms and canonical bases—some examples in low rank cases—, in preparation.
  • [N] H. Nakajima, Crystal bases of quantum affine algebras, Lecture note on the lectures at Sophia Univ., 2006 (in Japanese).
  • [NS] S. Naito and D. Sagaki, Crystal base elements of an extremal weight module fixed by a diagram automorphism, Algebr. Represent. Theory, 8 (2005), 689–707.
  • [S] A. Savage, A geometric construction of crystal graphs using quiver varieties: extension to the non-simply laced case, Contemp. Math., 392 (2005), 133–154.
  • [X1] N. Xi, On the PBW bases of the quantum group $U_v(G_2)$, Algebra Colloq., 2 (1995), 355–362.
  • [X2] N. Xi, Canonical basis for type $B_2$, J. Algebra, 214 (1999), 8–21.