Journal of the Mathematical Society of Japan

Diagram automorphisms and quantum groups

Toshiaki SHOJI and Zhiping ZHOU

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Permanent link to this document

Digital Object Identifier

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]

quantum groups canonical bases PBW-bases


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

Export citation


  • [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.