Q-Systems#
AUTHORS:
Travis Scrimshaw (2013-10-08): Initial version
Travis Scrimshaw (2017-12-08): Added twisted Q-systems
- class sage.algebras.q_system.QSystem(base_ring, cartan_type, level, twisted)[source]#
Bases:
CombinatorialFreeModule
A Q-system.
Let \(\mathfrak{g}\) be a tamely-laced symmetrizable Kac-Moody algebra with index set \(I\) and Cartan matrix \((C_{ab})_{a,b \in I}\) over a field \(k\). Follow the presentation given in [HKOTY1999], an unrestricted Q-system is a \(k\)-algebra in infinitely many variables \(Q^{(a)}_m\), where \(a \in I\) and \(m \in \ZZ_{>0}\), that satisfies the relations
\[\left(Q^{(a)}_m\right)^2 = Q^{(a)}_{m+1} Q^{(a)}_{m-1} + \prod_{b \sim a} \prod_{k=0}^{-C_{ab} - 1} Q^{(b)}_{\left\lfloor \frac{m C_{ba} - k}{C_{ab}} \right\rfloor},\]with \(Q^{(a)}_0 := 1\). Q-systems can be considered as T-systems where we forget the spectral parameter \(u\) and for \(\mathfrak{g}\) of finite type, have a solution given by the characters of Kirillov-Reshetikhin modules (again without the spectral parameter) for an affine Kac-Moody algebra \(\widehat{\mathfrak{g}}\) with \(\mathfrak{g}\) as its classical subalgebra. See [KNS2011] for more information.
Q-systems have a natural bases given by polynomials of the fundamental representations \(Q^{(a)}_1\), for \(a \in I\). As such, we consider the Q-system as generated by \(\{ Q^{(a)}_1 \}_{a \in I}\).
There is also a level \(\ell\) restricted Q-system (with unit boundary condition) given by setting \(Q_{d_a \ell}^{(a)} = 1\), where \(d_a\) are the entries of the symmetrizing matrix for the dual type of \(\mathfrak{g}\).
Similarly, for twisted affine types (we omit type \(A_{2n}^{(2)}\)), we can define the twisted Q-system by using the relation:
\[(Q^{(a)}_{m})^2 = Q^{(a)}_{m+1} Q^{(a)}_{m-1} + \prod_{b \neq a} (Q^{(b)}_{m})^{-C_{ba}}.\]See [Wil2013] for more information.
EXAMPLES:
We begin by constructing a Q-system and doing some basic computations in type \(A_4\):
sage: Q = QSystem(QQ, ['A', 4]) sage: Q.Q(3,1) Q^(3)[1] sage: Q.Q(1,2) Q^(1)[1]^2 - Q^(2)[1] sage: Q.Q(3,3) -Q^(1)[1]*Q^(3)[1] + Q^(1)[1]*Q^(4)[1]^2 + Q^(2)[1]^2 - 2*Q^(2)[1]*Q^(3)[1]*Q^(4)[1] + Q^(3)[1]^3 sage: x = Q.Q(1,1) + Q.Q(2,1); x Q^(1)[1] + Q^(2)[1] sage: x * x Q^(1)[1]^2 + 2*Q^(1)[1]*Q^(2)[1] + Q^(2)[1]^2
>>> from sage.all import * >>> Q = QSystem(QQ, ['A', Integer(4)]) >>> Q.Q(Integer(3),Integer(1)) Q^(3)[1] >>> Q.Q(Integer(1),Integer(2)) Q^(1)[1]^2 - Q^(2)[1] >>> Q.Q(Integer(3),Integer(3)) -Q^(1)[1]*Q^(3)[1] + Q^(1)[1]*Q^(4)[1]^2 + Q^(2)[1]^2 - 2*Q^(2)[1]*Q^(3)[1]*Q^(4)[1] + Q^(3)[1]^3 >>> x = Q.Q(Integer(1),Integer(1)) + Q.Q(Integer(2),Integer(1)); x Q^(1)[1] + Q^(2)[1] >>> x * x Q^(1)[1]^2 + 2*Q^(1)[1]*Q^(2)[1] + Q^(2)[1]^2
Next we do some basic computations in type \(C_4\):
sage: Q = QSystem(QQ, ['C', 4]) sage: Q.Q(4,1) Q^(4)[1] sage: Q.Q(1,2) Q^(1)[1]^2 - Q^(2)[1] sage: Q.Q(2,3) Q^(1)[1]^2*Q^(4)[1] - 2*Q^(1)[1]*Q^(2)[1]*Q^(3)[1] + Q^(2)[1]^3 - Q^(2)[1]*Q^(4)[1] + Q^(3)[1]^2 sage: Q.Q(3,3) Q^(1)[1]*Q^(4)[1]^2 - 2*Q^(2)[1]*Q^(3)[1]*Q^(4)[1] + Q^(3)[1]^3
>>> from sage.all import * >>> Q = QSystem(QQ, ['C', Integer(4)]) >>> Q.Q(Integer(4),Integer(1)) Q^(4)[1] >>> Q.Q(Integer(1),Integer(2)) Q^(1)[1]^2 - Q^(2)[1] >>> Q.Q(Integer(2),Integer(3)) Q^(1)[1]^2*Q^(4)[1] - 2*Q^(1)[1]*Q^(2)[1]*Q^(3)[1] + Q^(2)[1]^3 - Q^(2)[1]*Q^(4)[1] + Q^(3)[1]^2 >>> Q.Q(Integer(3),Integer(3)) Q^(1)[1]*Q^(4)[1]^2 - 2*Q^(2)[1]*Q^(3)[1]*Q^(4)[1] + Q^(3)[1]^3
We compare that with the twisted Q-system of type \(A_7^{(2)}\):
sage: Q = QSystem(QQ, ['A',7,2], twisted=True) sage: Q.Q(4,1) Q^(4)[1] sage: Q.Q(1,2) Q^(1)[1]^2 - Q^(2)[1] sage: Q.Q(2,3) Q^(1)[1]^2*Q^(4)[1] - 2*Q^(1)[1]*Q^(2)[1]*Q^(3)[1] + Q^(2)[1]^3 - Q^(2)[1]*Q^(4)[1] + Q^(3)[1]^2 sage: Q.Q(3,3) -Q^(1)[1]*Q^(3)[1]^2 + Q^(1)[1]*Q^(4)[1]^2 + Q^(2)[1]^2*Q^(3)[1] - 2*Q^(2)[1]*Q^(3)[1]*Q^(4)[1] + Q^(3)[1]^3
>>> from sage.all import * >>> Q = QSystem(QQ, ['A',Integer(7),Integer(2)], twisted=True) >>> Q.Q(Integer(4),Integer(1)) Q^(4)[1] >>> Q.Q(Integer(1),Integer(2)) Q^(1)[1]^2 - Q^(2)[1] >>> Q.Q(Integer(2),Integer(3)) Q^(1)[1]^2*Q^(4)[1] - 2*Q^(1)[1]*Q^(2)[1]*Q^(3)[1] + Q^(2)[1]^3 - Q^(2)[1]*Q^(4)[1] + Q^(3)[1]^2 >>> Q.Q(Integer(3),Integer(3)) -Q^(1)[1]*Q^(3)[1]^2 + Q^(1)[1]*Q^(4)[1]^2 + Q^(2)[1]^2*Q^(3)[1] - 2*Q^(2)[1]*Q^(3)[1]*Q^(4)[1] + Q^(3)[1]^3
REFERENCES:
- class Element[source]#
Bases:
IndexedFreeModuleElement
An element of a Q-system.
- Q(a, m)[source]#
Return the generator \(Q^{(a)}_m\) of
self
.EXAMPLES:
sage: Q = QSystem(QQ, ['A', 8]) sage: Q.Q(2, 1) Q^(2)[1] sage: Q.Q(6, 2) -Q^(5)[1]*Q^(7)[1] + Q^(6)[1]^2 sage: Q.Q(7, 3) -Q^(5)[1]*Q^(7)[1] + Q^(5)[1]*Q^(8)[1]^2 + Q^(6)[1]^2 - 2*Q^(6)[1]*Q^(7)[1]*Q^(8)[1] + Q^(7)[1]^3 sage: Q.Q(1, 0) 1
>>> from sage.all import * >>> Q = QSystem(QQ, ['A', Integer(8)]) >>> Q.Q(Integer(2), Integer(1)) Q^(2)[1] >>> Q.Q(Integer(6), Integer(2)) -Q^(5)[1]*Q^(7)[1] + Q^(6)[1]^2 >>> Q.Q(Integer(7), Integer(3)) -Q^(5)[1]*Q^(7)[1] + Q^(5)[1]*Q^(8)[1]^2 + Q^(6)[1]^2 - 2*Q^(6)[1]*Q^(7)[1]*Q^(8)[1] + Q^(7)[1]^3 >>> Q.Q(Integer(1), Integer(0)) 1
Twisted Q-system:
sage: Q = QSystem(QQ, ['D',4,3], twisted=True) sage: Q.Q(1,2) Q^(1)[1]^2 - Q^(2)[1] sage: Q.Q(2,2) -Q^(1)[1]^3 + Q^(2)[1]^2 sage: Q.Q(2,3) 3*Q^(1)[1]^4 - 2*Q^(1)[1]^3*Q^(2)[1] - 3*Q^(1)[1]^2*Q^(2)[1] + Q^(2)[1]^2 + Q^(2)[1]^3 sage: Q.Q(1,4) -2*Q^(1)[1]^2 + 2*Q^(1)[1]^3 + Q^(1)[1]^4 - 3*Q^(1)[1]^2*Q^(2)[1] + Q^(2)[1] + Q^(2)[1]^2
>>> from sage.all import * >>> Q = QSystem(QQ, ['D',Integer(4),Integer(3)], twisted=True) >>> Q.Q(Integer(1),Integer(2)) Q^(1)[1]^2 - Q^(2)[1] >>> Q.Q(Integer(2),Integer(2)) -Q^(1)[1]^3 + Q^(2)[1]^2 >>> Q.Q(Integer(2),Integer(3)) 3*Q^(1)[1]^4 - 2*Q^(1)[1]^3*Q^(2)[1] - 3*Q^(1)[1]^2*Q^(2)[1] + Q^(2)[1]^2 + Q^(2)[1]^3 >>> Q.Q(Integer(1),Integer(4)) -2*Q^(1)[1]^2 + 2*Q^(1)[1]^3 + Q^(1)[1]^4 - 3*Q^(1)[1]^2*Q^(2)[1] + Q^(2)[1] + Q^(2)[1]^2
- algebra_generators()[source]#
Return the algebra generators of
self
.EXAMPLES:
sage: Q = QSystem(QQ, ['A',4]) sage: Q.algebra_generators() Finite family {1: Q^(1)[1], 2: Q^(2)[1], 3: Q^(3)[1], 4: Q^(4)[1]} sage: Q = QSystem(QQ, ['D',4,3], twisted=True) sage: Q.algebra_generators() Finite family {1: Q^(1)[1], 2: Q^(2)[1]}
>>> from sage.all import * >>> Q = QSystem(QQ, ['A',Integer(4)]) >>> Q.algebra_generators() Finite family {1: Q^(1)[1], 2: Q^(2)[1], 3: Q^(3)[1], 4: Q^(4)[1]} >>> Q = QSystem(QQ, ['D',Integer(4),Integer(3)], twisted=True) >>> Q.algebra_generators() Finite family {1: Q^(1)[1], 2: Q^(2)[1]}
- cartan_type()[source]#
Return the Cartan type of
self
.EXAMPLES:
sage: Q = QSystem(QQ, ['A',4]) sage: Q.cartan_type() ['A', 4] sage: Q = QSystem(QQ, ['D',4,3], twisted=True) sage: Q.cartan_type() ['G', 2, 1]^* relabelled by {0: 0, 1: 2, 2: 1}
>>> from sage.all import * >>> Q = QSystem(QQ, ['A',Integer(4)]) >>> Q.cartan_type() ['A', 4] >>> Q = QSystem(QQ, ['D',Integer(4),Integer(3)], twisted=True) >>> Q.cartan_type() ['G', 2, 1]^* relabelled by {0: 0, 1: 2, 2: 1}
- dimension()[source]#
Return the dimension of
self
, which is \(\infty\).EXAMPLES:
sage: F = QSystem(QQ, ['A',4]) sage: F.dimension() +Infinity
>>> from sage.all import * >>> F = QSystem(QQ, ['A',Integer(4)]) >>> F.dimension() +Infinity
- gens()[source]#
Return the generators of
self
.EXAMPLES:
sage: Q = QSystem(QQ, ['A',4]) sage: Q.gens() (Q^(1)[1], Q^(2)[1], Q^(3)[1], Q^(4)[1])
>>> from sage.all import * >>> Q = QSystem(QQ, ['A',Integer(4)]) >>> Q.gens() (Q^(1)[1], Q^(2)[1], Q^(3)[1], Q^(4)[1])
- index_set()[source]#
Return the index set of
self
.EXAMPLES:
sage: Q = QSystem(QQ, ['A',4]) sage: Q.index_set() (1, 2, 3, 4) sage: Q = QSystem(QQ, ['D',4,3], twisted=True) sage: Q.index_set() (1, 2)
>>> from sage.all import * >>> Q = QSystem(QQ, ['A',Integer(4)]) >>> Q.index_set() (1, 2, 3, 4) >>> Q = QSystem(QQ, ['D',Integer(4),Integer(3)], twisted=True) >>> Q.index_set() (1, 2)
- level()[source]#
Return the restriction level of
self
orNone
if the system is unrestricted.EXAMPLES:
sage: Q = QSystem(QQ, ['A',4]) sage: Q.level() sage: Q = QSystem(QQ, ['A',4], 5) sage: Q.level() 5
>>> from sage.all import * >>> Q = QSystem(QQ, ['A',Integer(4)]) >>> Q.level() >>> Q = QSystem(QQ, ['A',Integer(4)], Integer(5)) >>> Q.level() 5
- one_basis()[source]#
Return the basis element indexing \(1\).
EXAMPLES:
sage: Q = QSystem(QQ, ['A',4]) sage: Q.one_basis() 1 sage: Q.one_basis().parent() is Q._indices True
>>> from sage.all import * >>> Q = QSystem(QQ, ['A',Integer(4)]) >>> Q.one_basis() 1 >>> Q.one_basis().parent() is Q._indices True
- sage.algebras.q_system.is_tamely_laced(ct)[source]#
Check if the Cartan type
ct
is tamely-laced.A (symmetrizable) Cartan type with index set \(I\) is tamely-laced if \(A_{ij} < -1\) implies \(d_i = -A_{ji} = 1\) for all \(i,j \in I\), where \((d_i)_{i \in I}\) is the diagonal matrix symmetrizing the Cartan matrix \((A_{ij})_{i,j \in I}\).
EXAMPLES:
sage: from sage.algebras.q_system import is_tamely_laced sage: all(is_tamely_laced(ct) ....: for ct in CartanType.samples(crystallographic=True, finite=True)) True sage: for ct in CartanType.samples(crystallographic=True, affine=True): ....: if not is_tamely_laced(ct): ....: print(ct) ['A', 1, 1] ['BC', 1, 2] ['BC', 5, 2] ['BC', 1, 2]^* ['BC', 5, 2]^* sage: cm = CartanMatrix([[2,-1,0,0],[-3,2,-2,-2],[0,-1,2,-1],[0,-1,-1,2]]) sage: is_tamely_laced(cm) True
>>> from sage.all import * >>> from sage.algebras.q_system import is_tamely_laced >>> all(is_tamely_laced(ct) ... for ct in CartanType.samples(crystallographic=True, finite=True)) True >>> for ct in CartanType.samples(crystallographic=True, affine=True): ... if not is_tamely_laced(ct): ... print(ct) ['A', 1, 1] ['BC', 1, 2] ['BC', 5, 2] ['BC', 1, 2]^* ['BC', 5, 2]^* >>> cm = CartanMatrix([[Integer(2),-Integer(1),Integer(0),Integer(0)],[-Integer(3),Integer(2),-Integer(2),-Integer(2)],[Integer(0),-Integer(1),Integer(2),-Integer(1)],[Integer(0),-Integer(1),-Integer(1),Integer(2)]]) >>> is_tamely_laced(cm) True