1######################### BEGIN COPYRIGHT MESSAGE #########################
2# GBNP - computing Gröbner bases of noncommutative polynomials
3# Copyright 2001-2010 by Arjeh M. Cohen, Dié A.H. Gijsbers, Jan Willem
4# Knopper, Chris Krook. Address: Discrete Algebra and Geometry (DAM) group
5# at the Department of Mathematics and Computer Science of Eindhoven
6# University of Technology.
7#
8# For acknowledgements see the manual. The manual can be found in several
9# formats in the doc subdirectory of the GBNP distribution. The
10# acknowledgements formatted as text can be found in the file chap0.txt.
11#
12# GBNP is free software; you can redistribute it and/or modify it under
13# the terms of the Lesser GNU General Public License as published by the
14# Free Software Foundation (FSF); either version 2.1 of the License, or
15# (at your option) any later version. For details, see the file 'LGPL' in
16# the doc subdirectory of the GBNP distribution or see the FSF's own site:
17# http://www.gnu.org/licenses/lgpl.html
18########################## END COPYRIGHT MESSAGE ##########################
19
20### filename = "exampleNoah.g"
21### authors Cohen & Wales
22
23### THIS IS A GAP PACKAGE FOR COMPUTING NON-COMMUTATIVE GROBNER BASES
24
25### Last change: Aug 12 2008, amc.
26### dahg
27## [A.M. Cohen, D.A.H. Gijsbers  D.B. Wales, BMW Algebras of simply laced type, J. Algebra, 286 (2005) 107--153].
28
29# <#GAPDoc Label="Example07">
30# <Section Label="Example07"><Heading>The Birman-Murakami-Wenzl algebra of type A<M>_3</M></Heading>
31# We study the Birman-Murakami-Wenzl algebra of type A<M>_3</M> as an algebra
32# given by generators and relations.
33# A reference for the relations used is <Cite Key="MR2124811"/>.
34# <P/>
35
36# First load the package and set the standard infolevel <Ref
37# InfoClass="InfoGBNP" Style="Text"/> to 1 and the time infolevel <Ref
38# Func="InfoGBNPTime" Style="Text"/> to 1 (for more information about the info
39# level, see Chapter <Ref Chap="Info"/>).
40
41# <L>
42LoadPackage("GBNP","0",false);;
43SetInfoLevel(InfoGBNP,1);
44SetInfoLevel(InfoGBNPTime,1);
45# </L>
46
47# The variables are <M>g_1</M>, <M>g_2</M>, <M>g_3</M>,
48# <M>e_1</M>, <M>e_2</M>, <M>e_3</M>, in this order.
49# In order to have the results printed out with these symbols, we
50# invoke
51# <Ref Func="GBNP.ConfigPrint" Style="Text"/>
52# <L>
53GBNP.ConfigPrint("g1","g2","g3","e1","e2","e3");
54# </L>
55
56
57# Now enter the relations. This will be done in NP form (see <Ref Sect="NP"/>).
58# The inderminates <M>m</M> and <M>l</M>
59# in the coefficient ring of the Birman-Murakami-Wenzl algebra
60# are specialized to 7 and 11 in order to make the computations more efficient.
61# <L>
62m:= 7;;
63l:= 11;;
64
65#relations Theorem 1.1
66k1 := [[[4],[1,1],[1],[]],[1,-l/m,-l,l/m]];;
67k2 := [[[5],[2,2],[2],[]],[1,-l/m,-l,l/m]];;
68k3 := [[[6],[3,3],[3],[]],[1,-l/m,-l,l/m]];;
69
70#relations B1
71#empty set here
72
73#relations B2:
74k4 := [[[1,2,1],[2,1,2]],[1,-1]];;
75k5 := [[[2,3,2],[3,2,3]],[1,-1]];;
76k6 := [[[1,3],[3,1]],[1,-1]];;
77
78#relations R1
79kr1 := [[[1,4],[4]],[1,-1/l]];;
80kr2 := [[[2,5],[5]],[1,-1/l]];;
81kr3 := [[[3,6],[6]],[1,-1/l]];;
82
83#relations R2:
84kr4 := [[[4,2,4],[4]],[1,-l]];;
85kr5 := [[[5,1,5],[5]],[1,-l]];;
86kr6 := [[[5,3,5],[5]],[1,-l]];;
87kr7 := [[[6,2,6],[6]],[1,-l]];;
88
89#relations R2'
90km1 := [[[4,5,4],[4]],[1,-1]];;
91km2 := [[[5,4,5],[5]],[1,-1]];;
92km3 := [[[5,6,5],[5]],[1,-1]];;
93km4 := [[[6,5,6],[6]],[1,-1]];;
94
95KI := [k1,k2,k3,k4,k5,k6,kr1,kr2,kr3,kr4,kr5,kr6,kr7,km1,km2,km3,km4];;
96# </L>
97
98# Now print the relations with <Ref Func="PrintNPList" Style="Text"/>:
99
100# <L>
101PrintNPList(KI);
102Length(KI);
103# </L>
104
105# Now calculate the Gröbner basis with <Ref Func="SGrobner" Style="Text"/>:
106
107# <L>
108GB := SGrobner(KI);;
109PrintNPList(GB);
110# </L>
111
112# Now calculate the dimension of the quotient algebra with <Ref Func="DimQA"
113# Style="Text"/> (the second argument is the number of symbols):
114
115# <L>
116DimQA(GB,6);
117# </L>
118
119# The conclusion is that the BMW algebra of type A3 has dimension 105.
120# </Section>
121# <#/GAPDoc>
122