For your convenience, SBStrips comes bundled with \(5\) SB algebras built in. We detail these algebras in this appendix. They may be obtained by calling SBStripsExampleAlgebra (A.1-1).
‣ SBStripsExampleAlgebra( n ) | ( function ) |
Arguments: n, an integer between 1 and 5 inclusive
Returns: a SB algebra
Calling this function with argument 1, 2, 3, 4 or 5 respectively returns the algebras described in subsections A.2-1, A.2-2, A.2-3, A.2-4 or A.2-5.
Each algebra is of the form \(KQ/\langle \rho \rangle\), where \(K\) is the field Rationals in GAP and where \(Q\) and \(\rho\) are respectively a quiver and a set of relations. These change from example to example.
The LaTeX version of this documentation provides pictures of each quiver.
The quiver and relations of this algebra are specified to QPA as follows.
gap> quiv := Quiver( > 2, > [ [ 1, 1, "a" ], [ 1, 2, "b" ], [ 2, 1, "c" ], [ 2, 2, "d" ] ] > ); <quiver with 2 vertices and 4 arrows> pa := PathAlgebra( Rationals, quiv ); <Rationals[<quiver with 2 vertices and 4 arrows>]> gap> rels := [ > pa.a * pa.a, pa.b * pa.d, pa.c * pa.b, pa.d * pa.c, > pa.c * pa.a * pa.b, (pa.d)^4, > pa.a * pa.b * pa.c - pa.b * pa.c * pa.a > ]; [ (1)*a^2, (1)*b*d, (1)*c*b, (1)*d*c, (1)*c*a*b, (1)*d^4, (1)*a*b*c+(-1)*b*c*a ]
The relations of this algebra are chosen so that the nonzero paths of length \(2\) are: a*b, b*c, c*a, d*d.
The simple module associated to vertex v2 has infinite syzygy type.
The quiver and relations of this algebra are specified to QPA as follows.
gap> quiv := Quiver( > 3, > [ [ 1, 2, "a" ], [ 2, 3, "b" ], [ 3, 1, "c" ] ] > ); <quiver with 3 vertices and 3 arrows> gap> pa := PathAlgebra( Rationals, quiv ); <Rationals[<quiver with 3 vertices and 3 arrows>]> gap> rels := NthPowerOfArrowIdeal( pa, 4 ); [ (1)*a*b*c*a, (1)*b*c*a*b, (1)*c*a*b*c ]
(In other words, this quiver is the \(3\)-cycle quiver, and the relations are the paths of length \(4\).) The nonzero paths of length \(2\) are: a*b, b*c, c*a.
This algebra is a Nakayama algebra, and so has finite representation type. A fortiori, it is syzygy-finite.
The quiver and relations of this algebra are specified to QPA as follows.
gap> quiv := Quiver( > 4, > [ [1,2,"a"], [2,3,"b"], [3,4,"c"], [4,1,"d"], [4,4,"e"], [1,2,"f"], > [2,3,"g"], [3,1,"h"] ] > ); <quiver with 4 vertices and 8 arrows> gap> pa := PathAlgebra( Rationals, quiv ); <Rationals[<quiver with 4 vertices and 8 arrows>]> gap> rels := [ > pa.a * pa.g, pa.b * pa.h, pa.c * pa.e, pa.d * pa.f, > pa.e * pa.d, pa.f * pa.b, pa.g * pa.c, pa.h * pa.a, > pa.a * pa.b * pa.c * pa.d * pa.a - ( pa.f * pa.g * pa.h )^2 * pa.f, > pa.d * pa.a * pa.b * pa.c - ( pa.e )^3, > pa.c * pa.d * pa.a * pa.b * pa.c, > ( pa.h * pa.f * pa.g )^2 * pa.h > ]; [ (1)*a*g, (1)*b*h, (1)*c*e, (1)*d*f, (1)*e*d, (1)*f*b, (1)*g*c, (1)*h*a, (1)*a*b*c*d*a+(-1)*f*g*h*f*g*h*f, (-1)*e^3+(1)*d*a*b*c, (1)*c*d*a*b*c, (1)*h*f*g*h*f*g*h ]
The relations of this algebra are chosen so that the nonzero paths of length \(2\) are: a*b, b*c, c*d, d*a, e*e, f*g, g*h and h*f.
The quiver and relations of this algebra are specified to QPA as follows.
gap> quiv := Quiver( > 8, > [ [ 1, 1, "a" ], [ 1, 2, "b" ], [ 2, 2, "c" ], [ 2, 3, "d" ], > [ 3, 4, "e" ], [ 4, 3, "f" ], [ 3, 4, "g" ], [ 4, 5, "h" ], > [ 5, 6, "i" ], [ 6, 5, "j" ], [ 5, 7, "k" ], [ 7, 6, "l" ], > [ 6, 7, "m" ], [ 7, 8, "n" ], [ 8, 8, "o" ], [ 8, 1, "p" ] ] > ); <quiver with 8 vertices and 16 arrows> gap> pa := PathAlgebra( Rationals, quiv ); <Rationals[<quiver with 8 vertices and 16 arrows>]> gap> rels := [ > pa.a * pa.a, pa.b * pa.d, pa.c * pa.c, pa.d * pa.g, pa.e * pa.h, > pa.f * pa.e, pa.g * pa.f, pa.h * pa.k, pa.i * pa.m, pa.j * pa.i, > pa.k * pa.n, pa.l * pa.j, > pa.m * pa.l, pa.n * pa.p, pa.o * pa.o, pa.p * pa.b, > pa.a * pa.b * pa.c * pa.d, > pa.e * pa.f * pa.g * pa.h, > pa.g * pa.h * pa.i * pa.j * pa.k, > pa.c * pa.d * pa.e - pa.d * pa.e * pa.f * pa.g, > pa.f * pa.g * pa.h * pa.i - pa.h * pa.i * pa.j * pa.k * pa.l, > pa.j * pa.k * pa.l * pa.m * pa.n - pa.m * pa.n * pa.o, > pa.o * pa.p * pa.a * pa.b - pa.p * pa.a * pa.b * pa.c > ];
The relations of this algebra are chosen so that the nonzero paths of length \(2\) are: a*b, b*c, c*d, d*e, e*f, f*g, g*h, h*i, i*j, j*k, k*l, l*m, m*n, n*o, o*p and p*a.
The quiver and relations of this algebra are specified to QPA as follows.
gap> quiv := Quiver( > 4, > [ [ 1, 2, "a" ], [ 2, 3, "b" ], [ 3, 4, "c" ], [ 4, 1, "d" ], > [ 1, 2, "e" ], [ 2, 3, "f" ], [ 3, 1, "g" ], [ 4, 4, "h" ] ] > ); <quiver with 4 vertices and 8 arrows> gap> pa := PathAlgebra( Rationals, quiv5 ); <Rationals[<quiver with 4 vertices and 8 arrows>]> gap> rels := [ > pa.a * pa.f, pa.b * pa.g, pa.c * pa.h, pa.d * pa.e, pa.e * pa.b, > pa.f * pa.c, pa.g * pa.a, pa.h * pa.d, > pa.b * pa.c * pa.d * pa.a * pa.b * pa.c, > pa.d * pa.a * pa.b * pa.c * pa.d * pa.a, > ( pa.h )^6, > pa.a * pa.b * pa.c * pa.d * pa.a * pa.b - > pa.e * pa.f * pa.g * pa.e * pa.f * pa.g * pa.e * pa.f, > pa.c * pa.d * pa.a * pa.b * pa.c * pa.d - > pa.g * pa.e * pa.f * pa.g * pa.e * pa.f * pa.g > ]; [ (1)*a*f, (1)*b*g, (1)*c*h, (1)*d*e, (1)*e*b, (1)*f*c, (1)*g*a, (1)*h*d, (1)*b*c*d*a*b*c, (1)*d*a*b*c*d*a, (1)*h^6, (1)*a*b*c*d*a*b+(-1)*e*f*g*e*f*g*e*f, (1)*c*d*a*b*c*d+(-1)*g*e*f*g*e*f*g ]
The relations of this algebra are chosen so that the nonzero paths of length \(2\) are: a*b, b*c, c*d, d*a, e*f, f*g, g*e, h*h.
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