liesym

Lie Algebras using Sympy and backend powered by Rust's pyO3 and ndarray

Overview

In an effort to supply python with the same computer algebra software (CAS) capabilities, SymPy was written. This python library is well written and allows an open source alternative to proprietary choices like Mathematica/WolframLanguage and Maple. Due to the nature of how SymPy was written, certain symbolic calculation can be extremely unoptimized in python. Even using numpy could offer little speed ups as it is not geared towards rational numbers (fractions). Sympy does currently offer a liealgebras module, but due to the performance limitations, certain tradeoffs had to be made such as locking the basis for the classic lie algebras in favor of speed. This is a fair trade off, but would require anyone using a different basis to hand calculate the representations of the algebra all over again. An alternative to solve this problem would be to use a compiled backend that supports generics (and isn't a pain to build with python).

Rust has good python binding support through py03 and allows easy communication through numpy using rust-numpy as well as numpy like api inside rust using ndarray.

Install

pip install liesym

Examples

See also example notebook

import liesym as ls
from sympy import Matrix
from IPython.display import display, Markdown
from sympy.printing.str import StrPrinter

Cartan Matrix

A3 = ls.A(3)
A3.cartan_matrix

$$\displaystyle \left[\begin{matrix}2 & -1 & 0\\ -1 & 2 & -1\\0 & -1 & 2\end{matrix}\right]$$

Positive Roots

for i in A3.positive_roots():
    display(i)

$$\displaystyle \left[\begin{matrix}1 & 0 & 1\end{matrix}\right]$$

$$\displaystyle \left[\begin{matrix}-1 & 1 & 1\end{matrix}\right]$$

$$\displaystyle \left[\begin{matrix}1 & 1 & -1\end{matrix}\right]$$

$$\displaystyle \left[\begin{matrix}-1 & 2 & -1\end{matrix}\right]$$

$$\displaystyle \left[\begin{matrix}0 & -1 & 2\end{matrix}\right]$$

$$\displaystyle \left[\begin{matrix}2 & -1 & 0\end{matrix}\right]$$

Simple Roots

for i in A3.simple_roots():
    display(i)

$$\displaystyle \left[\begin{matrix}1 & -1 & 0 & 0\end{matrix}\right]$$

$$\displaystyle \left[\begin{matrix}0 & 1 & -1 & 0\end{matrix}\right]$$

$$\displaystyle \left[\begin{matrix}0 & 0 & 1 & -1\end{matrix}\right]$$

Fundamental Weights

for i in A3.fundamental_weights(): # defaulted to orthogonal basis
    display(i)

$$\displaystyle \left[\begin{matrix}\frac{3}{4} & - \frac{1}{4} & - \frac{1}{4} & - \frac{1}{4}\end{matrix}\right]$$

$$\displaystyle \left[\begin{matrix}\frac{1}{2} & \frac{1}{2} & - \frac{1}{2} & - \frac{1}{2}\end{matrix}\right]$$

$$\displaystyle \left[\begin{matrix}\frac{1}{4} & \frac{1}{4} & \frac{1}{4} & - \frac{3}{4}\end{matrix}\right]$$

Dimension of representation

table = """\
| Dim | Irrep Name | Rep (Omega) |
| :-: | :-: | :-: |
"""
for i in A3.fundamental_weights(basis="omega"):
    table += f"""\
| {A3.dim(i)} | $${A3.dim_name(i)}$$  | $${i.table(StrPrinter())}$$|
"""
Markdown(table)

Dim	Irrep Name	Rep (Omega)
4	$$4$$	$$[1, 0, 0]$$
6	$$6$$	$$[0, 1, 0]$$
4	$$\bar{4}$$	$$[0, 0, 1]$$

Name of rep

Commonly in literature (especially physics), names of the reps are the dimension rather than the matrix rep.

A3.dim_name(Matrix([[0, 0, 1]]))

$$\displaystyle \bar{4}$$

A3.irrep_lookup(r"\bar{4}")

$$\displaystyle \left[\begin{matrix}0 & 0 & 1\end{matrix}\right]$$

Tensor product decomps

The decomp of irreps from a product of irreps

results = A3.tensor_product_decomposition([
    Matrix([[1,0,0]]),
    Matrix([[1,0,0]]),
])

table = """\
| Rep | Dim name |
| :-: | :-: |
"""
for i in results:
    table += f"""\
| $${i.table(StrPrinter())}$$ | $${A3.dim_name(i)}$$ |  
"""
Markdown(table)

Rep	Dim name
$$[0, 1, 0]$$	$$6$$
$$[2, 0, 0]$$	$$\bar{10}$$

Lie Groups

Currently supports SU(N), SO(N), Sp(N)

su2 = ls.SU(2)
su2.generators()

[Matrix([
 [  0, 1/2],
 [1/2,   0]]),
 Matrix([
 [  0, -I/2],
 [I/2,    0]]),
 Matrix([
 [1/2,    0],
 [  0, -1/2]])]

# cartan generators
su2.generators(cartan_only=True)

[Matrix([
 [1/2,    0],
 [  0, -1/2]])]

Structure constants. SU(2) structure constants are $e_{ijk}$

su2.structure_constants()

$$\displaystyle \left[\begin{matrix}\left[\begin{matrix}0 & 0 & 0\\0 & 0 & 1\\0 & -1 & 0\end{matrix}\right] & \left[\begin{matrix}0 & 0 & -1\\0 & 0 & 0\\1 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}0 & 1 & 0\\ -1 & 0 & 0\\0 & 0 & 0\end{matrix}\right]\end{matrix}\right]$$

A1 = ls.A(1)
for x in A1.simple_roots(basis="omega"):
    display(x)

$$\displaystyle \left[\begin{matrix}2\end{matrix}\right]$$

Quadratic Casimir

s = ls.Sp(6)
r = s.algebra.fundamental_weights()[0]
s.quadratic_casimir(r)

$$\displaystyle \frac{7}{2}$$