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