Compute in an exterior algebra and induce a linear map on two-forms¶
Continuously checked
This is the exact source run by make test-examples and the ordinary Python test gate.
Domains: commalg · homalg · linalg · root
Source: examples/40_exterior_algebra.py
# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Compute in an exterior algebra and induce a linear map on two-forms."""
from noether import Q
from noether.commalg import PolynomialRing
from noether.homalg import FreeModule, ModuleElement, ModuleHomomorphism
from noether.linalg import Matrix, Vector
def main() -> None:
ring = PolynomialRing(Q, ["t"])
module = FreeModule(ring, 3, basis_labels=["e0", "e1", "e2"])
exterior = module.exterior_algebra()
e0, e1, e2 = [exterior.generator(position) for position in range(3)]
volume = e0 * e1 * e2
assert volume == exterior.alternating_product(
[
ModuleElement(module, Vector.from_values(ring, [1, 0, 0])),
ModuleElement(module, Vector.from_values(ring, [0, 1, 0])),
ModuleElement(module, Vector.from_values(ring, [0, 0, 1])),
]
)
assert e1 * e0 == -(e0 * e1)
assert e0 * e0 == exterior.zero()
e0_dual = ModuleElement(module.dual(), Vector.from_values(ring, [1, 0, 0]))
assert volume.contract(e0_dual) == e1 * e2
linear_map = ModuleHomomorphism(
module,
module,
Matrix.from_rows(ring, [[1, 1, 0], [0, 1, 0], [0, 0, 2]]),
)
on_two_forms = linear_map.exterior_power(2)
on_two_forms.verify()
assert on_two_forms.apply_exterior(e1 * e2) == 2 * (e0 * e2 + e1 * e2)
print(
"Λ² has basis",
exterior.basis(2),
"and the induced map sends e1∧e2 to",
on_two_forms.apply_exterior(e1 * e2).terms,
)
if __name__ == "__main__":
main()