Loop Crystals¶
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sage.categories.loop_crystals.
KirillovReshetikhinCrystals
¶ Category of Kirillov-Reshetikhin crystals.
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class
sage.categories.loop_crystals.
LocalEnergyFunction
(B, Bp, normalization=0)¶ Bases:
sage.categories.map.Map
The local energy function.
Let \(B\) and \(B'\) be Kirillov-Reshetikhin crystals with maximal vectors \(u_B\) and \(u_{B'}\) respectively. The local energy function \(H : B \otimes B' \to \ZZ\) is the function which satisfies
\[\begin{split}H(e_0(b \otimes b')) = H(b \otimes b') + \begin{cases} 1 & \text{if } i = 0 \text{ and LL}, \\ -1 & \text{if } i = 0 \text{ and RR}, \\ 0 & \text{otherwise,} \end{cases}\end{split}\]where LL (resp. RR) denote \(e_0\) acts on the left (resp. right) on both \(b \otimes b'\) and \(R(b \otimes b')\), and normalized by \(H(u_B \otimes u_{B'}) = 0\).
INPUT:
B
– a Kirillov-Reshetikhin crystalBp
– a Kirillov-Reshetikhin crystalnormalization
– (default: 0) the normalization value
EXAMPLES:
sage: K = crystals.KirillovReshetikhin(['C',2,1], 1,2) sage: K2 = crystals.KirillovReshetikhin(['C',2,1], 2,1) sage: H = K.local_energy_function(K2) sage: T = tensor([K, K2]) sage: hw = T.classically_highest_weight_vectors() sage: for b in hw: ....: b, H(b) ([[], [[1], [2]]], 1) ([[[1, 1]], [[1], [2]]], 0) ([[[2, -2]], [[1], [2]]], 1) ([[[1, -2]], [[1], [2]]], 1)
REFERENCES:
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sage.categories.loop_crystals.
LoopCrystals
¶ The category of \(U_q'(\mathfrak{g})\)-crystals, where \(\mathfrak{g}\) is of affine type.
The category is called loop crystals as we can also consider them as crystals corresponding to the loop algebra \(\mathfrak{g}_0[t]\), where \(\mathfrak{g}_0\) is the corresponding classical type.
EXAMPLES:
sage: from sage.categories.loop_crystals import LoopCrystals sage: C = LoopCrystals() sage: C Category of loop crystals sage: C.super_categories() [Category of crystals] sage: C.example() Kirillov-Reshetikhin crystal of type ['A', 3, 1] with (r,s)=(1,1)
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sage.categories.loop_crystals.
RegularLoopCrystals
¶ The category of regular \(U_q'(\mathfrak{g})\)-crystals, where \(\mathfrak{g}\) is of affine type.