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- Title
Crystal Base Elements of an Extremal Weight Module Fixed by a Diagram Automorphism.
- Authors
Satoshi Naito; Daisuke Sagaki
- Abstract
In our previous work, we introduced a bijection between the elements of the crystal base of the negative (resp. positive) part of the quantized universal enveloping algebra <img src="/fulltext-image.asp?format=htmlnonpaginated&src=73R3462255008218_html\10468_2005_Article_20234_TeX2GIFIEq1.gif" border="0" alt="$U_{q}(\mathfrak{g})$" /> of a Kac–Moody algebra <img src="/fulltext-image.asp?format=htmlnonpaginated&src=73R3462255008218_html\10468_2005_Article_20234_TeX2GIFIEq2.gif" border="0" alt="$\mathfrak{g}$" /> that are fixed by a diagram automorphism and the elements of the crystal base of the negative (resp. positive) part of the quantized universal enveloping algebra <img src="/fulltext-image.asp?format=htmlnonpaginated&src=73R3462255008218_html\10468_2005_Article_20234_TeX2GIFIEq3.gif" border="0" alt="$U_{q}(\breve {\mathfrak{g}})$" /> of the orbit Lie algebra <img src="/fulltext-image.asp?format=htmlnonpaginated&src=73R3462255008218_html\10468_2005_Article_20234_TeX2GIFIEq4.gif" border="0" alt="$\breve {\mathfrak{g}}$" /> of <img src="/fulltext-image.asp?format=htmlnonpaginated&src=73R3462255008218_html\10468_2005_Article_20234_TeX2GIFIEq5.gif" border="0" alt="$\mathfrak{g}$" /> . In this paper, we prove that this bijection commutes with the *-operation. As an application of this result we show that there exists a canonical bijection between the elements ℬ0(λ) of the crystal base ℬ(λ) of an extremal weight module of extremal weight λ over <img src="/fulltext-image.asp?format=htmlnonpaginated&src=73R3462255008218_html\10468_2005_Article_20234_TeX2GIFIEq6.gif" border="0" alt="$U_{q}(\mathfrak{g})$" /> that are fixed by a diagram automorphism and the elements of the crystal base <img src="/fulltext-image.asp?format=htmlnonpaginated&src=73R3462255008218_html\10468_2005_Article_20234_TeX2GIFIEq7.gif" border="0" alt="$\breve {\mathcal{B}}(\widehat {\lambda})$" /> of an extremal weight module of extremal weight <img src="/fulltext-image.asp?format=htmlnonpaginated&src=73R3462255008218_html\10468_2005_Article_20234_TeX2GIFIEq8.gif" border="0" alt="$\widehat {\lambda}$" /> over <img src="/fulltext-image.asp?format=htmlnonpaginated&src=73R3462255008218_html\10468_2005_Article_20234_TeX2GIFIEq9.gif" border="0" alt="$U_{q}(\breve {\mathfrak{g}})$" /> , if the crystal graph of <img src="/fulltext-image.asp?format=htmlnonpaginated&src=73R3462255008218_html\10468_2005_Article_20234_TeX2GIFIEq10.gif" border="0" alt="$\breve {\mathcal{B}}(\widehat {\lambda})$" /> is connected.
- Publication
Algebras & Representation Theory, 2005, Vol 8, Issue 5, p689
- ISSN
1386-923X
- Publication type
Article
- DOI
10.1007/s10468-005-0234-x