add read me

venv/lib/python3.12/site-packages/sympy/liealgebras/tests/test_cartan_matrix.py

@@ -0,0 +1,10 @@
+from sympy.liealgebras.cartan_matrix import CartanMatrix
+from sympy.matrices import Matrix
+
+def test_CartanMatrix():
+    c = CartanMatrix("A3")
+    m = Matrix(3, 3, [2, -1, 0, -1, 2, -1, 0, -1, 2])
+    assert c == m
+    a = CartanMatrix(["G",2])
+    mt = Matrix(2, 2, [2, -1, -3, 2])
+    assert a == mt

venv/lib/python3.12/site-packages/sympy/liealgebras/tests/test_cartan_type.py

@@ -0,0 +1,12 @@
+from sympy.liealgebras.cartan_type import CartanType, Standard_Cartan
+
+def test_Standard_Cartan():
+    c = CartanType("A4")
+    assert c.rank() == 4
+    assert c.series == "A"
+    m = Standard_Cartan("A", 2)
+    assert m.rank() == 2
+    assert m.series == "A"
+    b = CartanType("B12")
+    assert b.rank() == 12
+    assert b.series == "B"

venv/lib/python3.12/site-packages/sympy/liealgebras/tests/test_dynkin_diagram.py

@@ -0,0 +1,9 @@
+from sympy.liealgebras.dynkin_diagram import DynkinDiagram
+
+def test_DynkinDiagram():
+    c = DynkinDiagram("A3")
+    diag = "0---0---0\n1   2   3"
+    assert c == diag
+    ct = DynkinDiagram(["B", 3])
+    diag2 = "0---0=>=0\n1   2   3"
+    assert ct == diag2

venv/lib/python3.12/site-packages/sympy/liealgebras/tests/test_root_system.py

@@ -0,0 +1,18 @@
+from sympy.liealgebras.root_system import RootSystem
+from sympy.liealgebras.type_a import TypeA
+from sympy.matrices import Matrix
+
+def test_root_system():
+    c = RootSystem("A3")
+    assert c.cartan_type == TypeA(3)
+    assert c.simple_roots() ==  {1: [1, -1, 0, 0], 2: [0, 1, -1, 0], 3: [0, 0, 1, -1]}
+    assert c.root_space() == "alpha[1] + alpha[2] + alpha[3]"
+    assert c.cartan_matrix() == Matrix([[ 2, -1,  0], [-1,  2, -1], [ 0, -1,  2]])
+    assert c.dynkin_diagram() == "0---0---0\n1   2   3"
+    assert c.add_simple_roots(1, 2) == [1, 0, -1, 0]
+    assert c.all_roots() == {1: [1, -1, 0, 0], 2: [1, 0, -1, 0],
+            3: [1, 0, 0, -1], 4: [0, 1, -1, 0], 5: [0, 1, 0, -1],
+            6: [0, 0, 1, -1], 7: [-1, 1, 0, 0], 8: [-1, 0, 1, 0],
+            9: [-1, 0, 0, 1], 10: [0, -1, 1, 0],
+            11: [0, -1, 0, 1], 12: [0, 0, -1, 1]}
+    assert c.add_as_roots([1, 0, -1, 0], [0, 0, 1, -1]) == [1, 0, 0, -1]

venv/lib/python3.12/site-packages/sympy/liealgebras/tests/test_type_A.py

@@ -0,0 +1,17 @@
+from sympy.liealgebras.cartan_type import CartanType
+from sympy.matrices import Matrix
+
+def test_type_A():
+    c = CartanType("A3")
+    m = Matrix(3, 3, [2, -1, 0, -1, 2, -1, 0, -1, 2])
+    assert m == c.cartan_matrix()
+    assert c.basis() == 8
+    assert c.roots() == 12
+    assert c.dimension() == 4
+    assert c.simple_root(1) == [1, -1, 0, 0]
+    assert c.highest_root() == [1, 0, 0, -1]
+    assert c.lie_algebra() == "su(4)"
+    diag = "0---0---0\n1   2   3"
+    assert c.dynkin_diagram() == diag
+    assert c.positive_roots() == {1: [1, -1, 0, 0], 2: [1, 0, -1, 0],
+            3: [1, 0, 0, -1], 4: [0, 1, -1, 0], 5: [0, 1, 0, -1], 6: [0, 0, 1, -1]}

venv/lib/python3.12/site-packages/sympy/liealgebras/tests/test_type_B.py

@@ -0,0 +1,17 @@
+from sympy.liealgebras.cartan_type import CartanType
+from sympy.matrices import Matrix
+
+def test_type_B():
+    c = CartanType("B3")
+    m = Matrix(3, 3, [2, -1, 0, -1, 2, -2, 0, -1, 2])
+    assert m == c.cartan_matrix()
+    assert c.dimension() == 3
+    assert c.roots() == 18
+    assert c.simple_root(3) == [0, 0, 1]
+    assert c.basis() == 3
+    assert c.lie_algebra() == "so(6)"
+    diag = "0---0=>=0\n1   2   3"
+    assert c.dynkin_diagram() == diag
+    assert c.positive_roots() ==  {1: [1, -1, 0], 2: [1, 1, 0], 3: [1, 0, -1],
+            4: [1, 0, 1], 5: [0, 1, -1], 6: [0, 1, 1], 7: [1, 0, 0],
+            8: [0, 1, 0], 9: [0, 0, 1]}

venv/lib/python3.12/site-packages/sympy/liealgebras/tests/test_type_C.py

@@ -0,0 +1,22 @@
+from sympy.liealgebras.cartan_type import CartanType
+from sympy.matrices import Matrix
+
+def test_type_C():
+    c = CartanType("C4")
+    m = Matrix(4, 4, [2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -2, 2])
+    assert c.cartan_matrix() == m
+    assert c.dimension() == 4
+    assert c.simple_root(4) == [0, 0, 0, 2]
+    assert c.roots() == 32
+    assert c.basis() == 36
+    assert c.lie_algebra() == "sp(8)"
+    t = CartanType(['C', 3])
+    assert t.dimension() == 3
+    diag = "0---0---0=<=0\n1   2   3   4"
+    assert c.dynkin_diagram() == diag
+    assert c.positive_roots() == {1: [1, -1, 0, 0], 2: [1, 1, 0, 0],
+            3: [1, 0, -1, 0], 4: [1, 0, 1, 0], 5: [1, 0, 0, -1],
+            6: [1, 0, 0, 1], 7: [0, 1, -1, 0], 8: [0, 1, 1, 0],
+            9: [0, 1, 0, -1], 10: [0, 1, 0, 1], 11: [0, 0, 1, -1],
+            12: [0, 0, 1, 1], 13: [2, 0, 0, 0], 14: [0, 2, 0, 0], 15: [0, 0, 2, 0],
+            16: [0, 0, 0, 2]}

venv/lib/python3.12/site-packages/sympy/liealgebras/tests/test_type_D.py

@@ -0,0 +1,19 @@
+from sympy.liealgebras.cartan_type import CartanType
+from sympy.matrices import Matrix
+
+
+
+def test_type_D():
+    c = CartanType("D4")
+    m = Matrix(4, 4, [2, -1, 0, 0, -1, 2, -1, -1, 0, -1, 2, 0, 0, -1, 0, 2])
+    assert c.cartan_matrix() == m
+    assert c.basis() == 6
+    assert c.lie_algebra() == "so(8)"
+    assert c.roots() == 24
+    assert c.simple_root(3) == [0, 0, 1, -1]
+    diag = "    3\n    0\n    |\n    |\n0---0---0\n1   2   4"
+    assert diag == c.dynkin_diagram()
+    assert c.positive_roots() == {1: [1, -1, 0, 0], 2: [1, 1, 0, 0],
+            3: [1, 0, -1, 0], 4: [1, 0, 1, 0], 5: [1, 0, 0, -1], 6: [1, 0, 0, 1],
+            7: [0, 1, -1, 0], 8: [0, 1, 1, 0], 9: [0, 1, 0, -1], 10: [0, 1, 0, 1],
+            11: [0, 0, 1, -1], 12: [0, 0, 1, 1]}

venv/lib/python3.12/site-packages/sympy/liealgebras/tests/test_type_E.py

@@ -0,0 +1,22 @@
+from sympy.liealgebras.cartan_type import CartanType
+from sympy.matrices import Matrix
+from sympy.core.backend import Rational
+
+def test_type_E():
+    c = CartanType("E6")
+    m = Matrix(6, 6, [2, 0, -1, 0, 0, 0, 0, 2, 0, -1, 0, 0,
+        -1, 0, 2, -1, 0, 0, 0, -1, -1, 2, -1, 0, 0, 0, 0,
+        -1, 2, -1, 0, 0, 0, 0, -1, 2])
+    assert c.cartan_matrix() == m
+    assert c.dimension() == 8
+    assert c.simple_root(6) == [0, 0, 0, -1, 1, 0, 0, 0]
+    assert c.roots() == 72
+    assert c.basis() == 78
+    diag = " "*8 + "2\n" + " "*8 + "0\n" + " "*8 + "|\n" + " "*8 + "|\n"
+    diag += "---".join("0" for i in range(1, 6))+"\n"
+    diag += "1   " + "   ".join(str(i) for i in range(3, 7))
+    assert c.dynkin_diagram() == diag
+    posroots = c.positive_roots()
+    assert posroots[8] == [1, 0, 0, 0, 1, 0, 0, 0]
+    assert posroots[21] == [Rational(1,2),Rational(1,2),Rational(1,2),Rational(1,2),
+                            Rational(1,2),Rational(-1,2),Rational(-1,2),Rational(1,2)]

venv/lib/python3.12/site-packages/sympy/liealgebras/tests/test_type_F.py

@@ -0,0 +1,24 @@
+from sympy.liealgebras.cartan_type import CartanType
+from sympy.matrices import Matrix
+from sympy.core.backend import S
+
+def test_type_F():
+    c = CartanType("F4")
+    m = Matrix(4, 4, [2, -1, 0, 0, -1, 2, -2, 0, 0, -1, 2, -1, 0, 0, -1, 2])
+    assert c.cartan_matrix() == m
+    assert c.dimension() == 4
+    assert c.simple_root(1) == [1, -1, 0, 0]
+    assert c.simple_root(2) == [0, 1, -1, 0]
+    assert c.simple_root(3) == [0, 0, 0, 1]
+    assert c.simple_root(4) == [-S.Half, -S.Half, -S.Half, -S.Half]
+    assert c.roots() == 48
+    assert c.basis() == 52
+    diag = "0---0=>=0---0\n" + "   ".join(str(i) for i in range(1, 5))
+    assert c.dynkin_diagram() == diag
+    assert c.positive_roots() == {1: [1, -1, 0, 0], 2: [1, 1, 0, 0], 3: [1, 0, -1, 0],
+            4: [1, 0, 1, 0], 5: [1, 0, 0, -1], 6: [1, 0, 0, 1], 7: [0, 1, -1, 0],
+            8: [0, 1, 1, 0], 9: [0, 1, 0, -1], 10: [0, 1, 0, 1], 11: [0, 0, 1, -1],
+            12: [0, 0, 1, 1], 13: [1, 0, 0, 0], 14: [0, 1, 0, 0], 15: [0, 0, 1, 0],
+            16: [0, 0, 0, 1], 17: [S.Half, S.Half, S.Half, S.Half], 18: [S.Half, -S.Half, S.Half, S.Half],
+            19: [S.Half, S.Half, -S.Half, S.Half], 20: [S.Half, S.Half, S.Half, -S.Half], 21: [S.Half, S.Half, -S.Half, -S.Half],
+            22: [S.Half, -S.Half, S.Half, -S.Half], 23: [S.Half, -S.Half, -S.Half, S.Half], 24: [S.Half, -S.Half, -S.Half, -S.Half]}

venv/lib/python3.12/site-packages/sympy/liealgebras/tests/test_type_G.py

@@ -0,0 +1,16 @@
+# coding=utf-8
+from sympy.liealgebras.cartan_type import CartanType
+from sympy.matrices import Matrix
+
+def test_type_G():
+    c = CartanType("G2")
+    m = Matrix(2, 2, [2, -1, -3, 2])
+    assert c.cartan_matrix() == m
+    assert c.simple_root(2) == [1, -2, 1]
+    assert c.basis() == 14
+    assert c.roots() == 12
+    assert c.dimension() == 3
+    diag = "0≡<≡0\n1   2"
+    assert diag == c.dynkin_diagram()
+    assert c.positive_roots() == {1: [0, 1, -1], 2: [1, -2, 1], 3: [1, -1, 0],
+            4: [1, 0, 1], 5: [1, 1, -2], 6: [2, -1, -1]}

venv/lib/python3.12/site-packages/sympy/liealgebras/tests/test_weyl_group.py

@@ -0,0 +1,35 @@
+from sympy.liealgebras.weyl_group import WeylGroup
+from sympy.matrices import Matrix
+
+def test_weyl_group():
+    c = WeylGroup("A3")
+    assert c.matrix_form('r1*r2') == Matrix([[0, 0, 1, 0], [1, 0, 0, 0],
+        [0, 1, 0, 0], [0, 0, 0, 1]])
+    assert c.generators() == ['r1', 'r2', 'r3']
+    assert c.group_order() == 24.0
+    assert c.group_name() == "S4: the symmetric group acting on 4 elements."
+    assert c.coxeter_diagram() == "0---0---0\n1   2   3"
+    assert c.element_order('r1*r2*r3') == 4
+    assert c.element_order('r1*r3*r2*r3') == 3
+    d = WeylGroup("B5")
+    assert d.group_order() == 3840
+    assert d.element_order('r1*r2*r4*r5') == 12
+    assert d.matrix_form('r2*r3') ==  Matrix([[0, 0, 1, 0, 0], [1, 0, 0, 0, 0],
+        [0, 1, 0, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]])
+    assert d.element_order('r1*r2*r1*r3*r5') == 6
+    e = WeylGroup("D5")
+    assert e.element_order('r2*r3*r5') == 4
+    assert e.matrix_form('r2*r3*r5') == Matrix([[1, 0, 0, 0, 0], [0, 0, 0, 0, -1],
+        [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, -1, 0]])
+    f = WeylGroup("G2")
+    assert f.element_order('r1*r2*r1*r2') == 3
+    assert f.element_order('r2*r1*r1*r2') == 1
+
+    assert f.matrix_form('r1*r2*r1*r2') == Matrix([[0, 1, 0], [0, 0, 1], [1, 0, 0]])
+    g = WeylGroup("F4")
+    assert g.matrix_form('r2*r3') == Matrix([[1, 0, 0, 0], [0, 1, 0, 0],
+        [0, 0, 0, -1], [0, 0, 1, 0]])
+
+    assert g.element_order('r2*r3') == 4
+    h = WeylGroup("E6")
+    assert h.group_order() == 51840