Usl3_module.py

This is a small Computer Algebra System written in Python. It provides functionality for simplifying expressions, parsing paths, and determining nilpotent actions and checking if elements are cycling for singular and non-singular Whittaker modules $Y_{\chi,\eta}$ defined by Kostant in his 1978 paper.

Features

Simplifying Expressions

The simplify(string) function takes a string representation of an element in the module written as a polynomial in the PBW basis and simplifies it to a unique expression. The simplification process involves repeatedly moving the positive roots ei's to the right using the commutator and evaluating them with the Whittaker function when the are at right end i.e. moving across the tensor product.

Nilpotent Actions

The find_nilpotent_action(sum, x) function finds the nilpotent action of a given sum on the generator 'x'. It repeatedly applies the action of a module element on 'x' until it reduces to a number. The function returns the nilpotent index and the result of the action.

Cyclic Properties

The is_cyclic(sum, path = []) function checks if a given module element is a cyclic element for the module.

Pretty Printing

This is a nice feature that allows the user to print the results of the above functions in a more readable format.

Usage

This module can be used in three different ways:

1. Simplify a Single Expression: To simplify a single expression, pass the expression as a command line argument. For example:

python Usl3_module.py "e1 + e2 + e3*e2"

Please note that brackets are not implemented yet. The carat (^) applies at the end of each multiplication (*) term and not individual summation (+) terms or generators. For clarity, use carats only at the end of each multiplication term. For example, "e1^2e2 * e1^3e3+1" is equivalent to "(e1e2)^2 (e1e3+1)^3". Instead, writing "e1e2 ^2 * e1e3+1 ^3", which is equivalent, is more clear.

2. Interactive Mode: To use the module in interactive mode, use the run command. In this mode, you can enter expressions to be simplified one at a time.

python Usl3_module.py run

3. Cyclic Check Mode: To use the module in cyclic check mode, use the cyclic command. In this mode, you can enter expressions to check if they are cyclic.

python Usl3_module.py cyclic

In both interactive and cyclic check modes, simply type the expression and press enter to get the result.

Note

This module uses a matrix representation of the commutative relations between the generators e1, e2, e3, h1, h2, f1, f2, f3, represented by 0, 1, 2, 3, 4, 5, 6, 7 respectively. The matrix C is used to define these relations.