euclidean_vector.ipynb

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     "text": [
      "The algebra is\n",
      "[[[ 0.  0.  0.  0.  0.  0.]\n",
      "  [ 0.  0.  1.  0.  0.  0.]\n",
      "  [ 0. -1. -0.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  1.]\n",
      "  [ 0.  0.  0.  0. -1.  0.]]\n",
      "\n",
      " [[ 0.  0. -1.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]\n",
      "  [ 1.  0.  0.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0. -1.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  1.  0.  0.]]\n",
      "\n",
      " [[ 0.  1.  0.  0.  0.  0.]\n",
      "  [-1.  0.  0.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  1.  0.]\n",
      "  [ 0.  0.  0. -1.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]]\n",
      "\n",
      " [[ 0.  0.  0.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  1.]\n",
      "  [ 0.  0.  0.  0. -1.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]]\n",
      "\n",
      " [[ 0.  0.  0.  0.  0. -1.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  1.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]]\n",
      "\n",
      " [[ 0.  0.  0.  0.  1.  0.]\n",
      "  [ 0.  0.  0. -1.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]\n",
      "  [ 0.  0.  0.  0.  0.  0.]]]\n",
      "Found 1 solutions for vector x vector -> scalar\n",
      "[[[0.577 0.    0.    0.   ]\n",
      "  [0.    0.577 0.    0.   ]\n",
      "  [0.    0.    0.577 0.   ]\n",
      "  [0.    0.    0.    0.   ]]]\n",
      "Found 1 solutions for vector x vector -> vector\n",
      "[[[0.    0.    0.    0.   ]\n",
      "  [0.    0.    0.    0.   ]\n",
      "  [0.    0.    0.    0.   ]\n",
      "  [0.    0.    0.    0.   ]]\n",
      "\n",
      " [[0.    0.    0.    0.   ]\n",
      "  [0.    0.    0.    0.   ]\n",
      "  [0.    0.    0.    0.   ]\n",
      "  [0.    0.    0.    0.   ]]\n",
      "\n",
      " [[0.    0.    0.    0.   ]\n",
      "  [0.    0.    0.    0.   ]\n",
      "  [0.    0.    0.    0.   ]\n",
      "  [0.    0.    0.    0.   ]]\n",
      "\n",
      " [[1.155 0.    0.    0.   ]\n",
      "  [0.    1.155 0.    0.   ]\n",
      "  [0.    0.    1.155 0.   ]\n",
      "  [0.    0.    0.    0.   ]]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from lie_nn import GenericRep\n",
    "from lie_nn.irreps import SO3Rep\n",
    "from lie_nn.util import round_to_sqrt_rational\n",
    "np.set_printoptions(precision=3, suppress=True)\n",
    "\n",
    "# Generators of rotations\n",
    "r = SO3Rep(1).continuous_generators()\n",
    "r = np.pad(r, ((0, 0), (0, 1), (0, 1)))  # pad with zeros to make it 4 dimensional\n",
    "\n",
    "# Generators of translations\n",
    "t = np.array(\n",
    "    [\n",
    "        [\n",
    "            [0, 0, 0, 1],\n",
    "            [0, 0, 0, 0],\n",
    "            [0, 0, 0, 0],\n",
    "            [0, 0, 0, 0.0],\n",
    "        ],\n",
    "        [\n",
    "            [0, 0, 0, 0],\n",
    "            [0, 0, 0, 1],\n",
    "            [0, 0, 0, 0],\n",
    "            [0, 0, 0, 0.0],\n",
    "        ],\n",
    "        [\n",
    "            [0, 0, 0, 0],\n",
    "            [0, 0, 0, 0],\n",
    "            [0, 0, 0, 1],\n",
    "            [0, 0, 0, 0.0],\n",
    "        ],\n",
    "    ]\n",
    ")\n",
    "\n",
    "# Generators of rotations and translations\n",
    "X = np.concatenate([r, t], axis=0)\n",
    "\n",
    "\n",
    "# Create generic representations (this will automatically compute the algebra and test it)\n",
    "vector = GenericRep.from_generators(X, round_fn=round_to_sqrt_rational)\n",
    "if vector is not None:\n",
    "    print(\"The algebra is\")\n",
    "    print(vector.algebra())\n",
    "\n",
    "scalar = GenericRep(\n",
    "    A=vector.algebra(),\n",
    "    X=np.zeros((6, 1, 1)),\n",
    "    H=np.zeros((0, 1, 1)),\n",
    ")\n",
    "\n",
    "# Compute Clebsch-Gordan coefficients to go from vector x vector -> scalar\n",
    "cg = GenericRep.clebsch_gordan(vector, vector, scalar, round_fn=round_to_sqrt_rational)\n",
    "print(f\"Found {cg.shape[0]} solutions for vector x vector -> scalar\")\n",
    "for c in cg:\n",
    "    print(np.moveaxis(c, -1, 0))\n",
    "\n",
    "cg = GenericRep.clebsch_gordan(vector, vector, vector, round_fn=round_to_sqrt_rational)\n",
    "print(f\"Found {cg.shape[0]} solutions for vector x vector -> vector\")\n",
    "for c in cg:\n",
    "    print(np.moveaxis(c, -1, 0))\n"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is no cross-product $x \\wedge y$ in the Euclidean group !! Weird, isn't it ??"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": []
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