gh-35387: The Fusion Ring of the Drinfeld Double of a Finite Group
    
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This code is very similar to the existing FusionRing class. Both
FusionRing and the new FusionDouble classes implement Fusion Rings of
Modular Tensor Categories, important in conformal field theory and
related quantum math. The FusionDouble is the fusion ring of the
Drinfeld double of a finite group. It has methods corresponding to
fusion, S-matrix, twists of simple elements, quantum dimension. An
interface is provide to the F-matrix code, but due to the fact that the
number of simple objects is fairly large even for fairly small groups,
the F-matrix code is only confirmed to work for very small group such as
Z_2 and S_3. Apart from the F-matrix code, the code works for Drinfeld
doubles of moderate size, tested for example on groups of order 16, and
the simple groups A5 and GL(3,2) of orders 60 and 168.

A prototype of this code by Wenqi Li is here:
https://github.com/enqiLi/drinfeld_doubles

Work issues:

*The code would benefit from refactoring with the existing FusionRing
code. It is proposed that we skip this since the existing code works
well.
*The Drinfeld double can be twisted by a 3-cocycle for H^3(G). This may
be implemented in a later PR.

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URL: #35387
Reported by: Daniel Bump
Reviewer(s): Daniel Bump, Matthias Köppe, Travis Scrimshaw

Author: Release Manager

src/doc/en/reference/algebras/fusion_rings.rst

@@ -6,6 +6,7 @@ Fusion Rings
 
    sage/algebras/fusion_rings/fusion_ring
    sage/algebras/fusion_rings/f_matrix
+   sage/algebras/fusion_rings/fusion_double
 
 F-Matrix Backend
 ----------------

src/doc/en/reference/references/index.rst

@@ -43,6 +43,10 @@ REFERENCES:
             University of New South Wales,
             1995
 
+.. [Ab2022] Willie Aboumrad,
+        *Quantum compution with anyons: an F-matrix and braid calculator*
+        (2022). https://arxiv.org/abs/2212.00831
+
 .. [AB2007] \M. Aschenbrenner, C. Hillar,
             *Finite generation of symmetric ideals*.
             Trans. Amer. Math. Soc. 359 (2007), no. 11, 5171--5192.
@@ -1888,7 +1892,7 @@ REFERENCES:
 .. [CW2005] \J. E. Cremona and M. Watkins. Computing isogenies of elliptic
             curves. preprint, 2005.
 
-.. [CHW2015] Shawn X.; Hong, Seung-Moon; Wang, Zhenghan Universal quantum computation 
+.. [CHW2015] Cui, Shawn X.; Hong, Seung-Moon; Wang, Zhenghan Universal quantum computation 
              with weakly integral anyons. Quantum Inf. Process. 14 (2015),
              no. 8, 2687-2727. 
 
@@ -2703,6 +2707,14 @@ REFERENCES:
 .. [GJ2007] \A. Glen, J. Justin, Episturmian words: a survey, Preprint,
             2007, :arxiv:`0801.1655`.
 
+.. [Goff1999] Christopher Goff. *Isomorphic fusion algebras of twisted quantum
+              doubles of finite groups*. PhD Thesis,
+              University of California, Santa Cruz. 1999.
+
+.. [GoMa2010] Christopher Goff and Geoffrey Mason, 
+            *Generalized twisted quantum doubles and the McKay correspondence*,
+            J. Algebra 324 (2010), no. 11, 3007–3016. 
+
 .. [GJ2016] Muddappa Seetharama Gowda and Juyoung Jeong.
             Spectral cones in Euclidean Jordan algebras.
             Linear Algebra and its Applications, 509:286-305, 2016.
@@ -4432,6 +4444,10 @@ REFERENCES:
 .. [Mat2002] Jiří Matousek, "Lectures on Discrete Geometry", Springer,
              2002
 
+.. [Mas1995] Mason, Geoffrey. *The quantum double of a finite group and its role
+             in conformal field theory*. Groups '93 Galway/St. Andrews, Vol. 2,
+             405-417, London Math. Soc. Lecture Note Ser., 212, Cambridge, 1995.
+
 .. [Ma2009] Sarah Mason, An Explicit Construction of Type A Demazure
             Atoms, Journal of Algebraic Combinatorics, Vol. 29,
             (2009), No. 3, p.295-313. :arxiv:`0707.4267`

src/sage/algebras/fusion_rings/all.py

@@ -14,5 +14,4 @@
 from sage.misc.lazy_import import lazy_import
 
 lazy_import('sage.algebras.fusion_rings.fusion_ring', ['FusionRing'])
-
-del lazy_import
+lazy_import('sage.algebras.fusion_rings.fusion_double', ['FusionDouble'])

src/sage/algebras/fusion_rings/f_matrix.py

@@ -130,7 +130,7 @@ class FMatrix(SageObject):
 
     See [TTWL2009]_ for an introduction to this topic,
     [EGNO2015]_ Section 4.9 for a precise mathematical
-    definition, and [Bond2007]_ Section 2.5 for a discussion
+    definition, and [Bond2007]_ Section 2.5 and [Ab2022]_ for discussions
     of how to compute the F-matrix. In addition to
     [Bond2007]_, worked out F-matrices may be found in
     [RoStWa2009]_ and [CHW2015]_.
@@ -2217,7 +2217,7 @@ def _substitute_degree_one(self, eqns=None):
         r"""
         Substitute known value from linear univariate polynomial and
         solve, following [Bond2007]_ p.37, for two-term linear equation
-        for one of the variables.
+        for one of the variables. See also [Ab2022]_.
 
         EXAMPLES::