Interface to LiE#

LiE is a software package under development at CWI since January 1988. Its purpose is to enable mathematicians and physicists to obtain on-line information as well as to interactively perform computations of a Lie group theoretic nature. It focuses on the representation theory of complex semisimple (reductive) Lie groups and algebras, and on the structure of their Weyl groups and root systems.

Type lie.[tab] for a list of all the functions available from your LiE install. Type lie.[tab]? for LiE’s help about a given function. Type lie(...) to create a new LiE object, and lie.eval(...) to run a string using LiE (and get the result back as a string).

To access the LiE interpreter directly, run lie_console().

EXAMPLES:

sage: a4 = lie('A4')             # optional - lie
sage: lie.diagram('A4')          # optional - lie
O---O---O---O
1   2   3   4
A4

sage: lie.diagram(a4)            # optional - lie
O---O---O---O
1   2   3   4
A4

sage: a4.diagram()               # optional - lie
O---O---O---O
1   2   3   4
A4

sage: a4.Cartan()                # optional - lie
     [[ 2,-1, 0, 0]
     ,[-1, 2,-1, 0]
     ,[ 0,-1, 2,-1]
     ,[ 0, 0,-1, 2]
     ]
sage: lie.LR_tensor([3,1],[2,2]) # optional - lie
1X[5,3]

>>> from sage.all import *
>>> a4 = lie('A4')             # optional - lie
>>> lie.diagram('A4')          # optional - lie
O---O---O---O
1   2   3   4
A4

>>> lie.diagram(a4)            # optional - lie
O---O---O---O
1   2   3   4
A4

>>> a4.diagram()               # optional - lie
O---O---O---O
1   2   3   4
A4

>>> a4.Cartan()                # optional - lie
     [[ 2,-1, 0, 0]
     ,[-1, 2,-1, 0]
     ,[ 0,-1, 2,-1]
     ,[ 0, 0,-1, 2]
     ]
>>> lie.LR_tensor([Integer(3),Integer(1)],[Integer(2),Integer(2)]) # optional - lie
1X[5,3]

Tutorial#

The following examples are taken from Section 2.1 of the LiE manual.

You can perform basic arithmetic operations in LiE.

sage: # optional - lie
sage: lie.eval('19+68')
'87'
sage: a = lie('1111111111*1111111111')
sage: a
1234567900987654321
sage: a/1111111111
1111111111
sage: a = lie('345')
sage: a^2+3*a-5
120055
sage: _ / 7*a
5916750

>>> from sage.all import *
>>> # optional - lie
>>> lie.eval('19+68')
'87'
>>> a = lie('1111111111*1111111111')
>>> a
1234567900987654321
>>> a/Integer(1111111111)
1111111111
>>> a = lie('345')
>>> a**Integer(2)+Integer(3)*a-Integer(5)
120055
>>> _ / Integer(7)*a
5916750

Vectors in LiE are created using square brackets. Notice that the indexing in LiE is 1-based, unlike Python/Sage which is 0-based.

sage: # optional - lie
sage: v = lie('[3,2,6873,-38]')
sage: v
[3,2,6873,-38]
sage: v[3]
6873
sage: v+v
[6,4,13746,-76]
sage: v*v
47239586
sage: v+234786
[3,2,6873,-38,234786]
sage: v-3
[3,2,-38]
sage: v^v
[3,2,6873,-38,3,2,6873,-38]

>>> from sage.all import *
>>> # optional - lie
>>> v = lie('[3,2,6873,-38]')
>>> v
[3,2,6873,-38]
>>> v[Integer(3)]
6873
>>> v+v
[6,4,13746,-76]
>>> v*v
47239586
>>> v+Integer(234786)
[3,2,6873,-38,234786]
>>> v-Integer(3)
[3,2,-38]
>>> v**v
[3,2,6873,-38,3,2,6873,-38]

You can also work with matrices in LiE.

sage: m = lie('[[1,0,3,3],[12,4,-4,7],[-1,9,8,0],[3,-5,-2,9]]') # optional - lie
sage: m # optional - lie
     [[ 1, 0, 3,3]
     ,[12, 4,-4,7]
     ,[-1, 9, 8,0]
     ,[ 3,-5,-2,9]
     ]
sage: print(lie.eval('*'+m._name))  # optional - lie
     [[1,12,-1, 3]
     ,[0, 4, 9,-5]
     ,[3,-4, 8,-2]
     ,[3, 7, 0, 9]
     ]

sage: # optional - lie
sage: m^3
     [[ 220,   87, 81, 375]
     ,[-168,-1089, 13,1013]
     ,[1550,  357,-55,1593]
     ,[-854, -652, 98,-170]
     ]
sage: v*m
[-6960,62055,55061,-319]
sage: m*v
[20508,-27714,54999,-14089]
sage: v*m*v
378549605
sage: m+v
     [[ 1, 0,   3,  3]
     ,[12, 4,  -4,  7]
     ,[-1, 9,   8,  0]
     ,[ 3,-5,  -2,  9]
     ,[ 3, 2,6873,-38]
     ]

sage: m-2 # optional - lie
     [[ 1, 0, 3,3]
     ,[-1, 9, 8,0]
     ,[ 3,-5,-2,9]
     ]

>>> from sage.all import *
>>> m = lie('[[1,0,3,3],[12,4,-4,7],[-1,9,8,0],[3,-5,-2,9]]') # optional - lie
>>> m # optional - lie
     [[ 1, 0, 3,3]
     ,[12, 4,-4,7]
     ,[-1, 9, 8,0]
     ,[ 3,-5,-2,9]
     ]
>>> print(lie.eval('*'+m._name))  # optional - lie
     [[1,12,-1, 3]
     ,[0, 4, 9,-5]
     ,[3,-4, 8,-2]
     ,[3, 7, 0, 9]
     ]

>>> # optional - lie
>>> m**Integer(3)
     [[ 220,   87, 81, 375]
     ,[-168,-1089, 13,1013]
     ,[1550,  357,-55,1593]
     ,[-854, -652, 98,-170]
     ]
>>> v*m
[-6960,62055,55061,-319]
>>> m*v
[20508,-27714,54999,-14089]
>>> v*m*v
378549605
>>> m+v
     [[ 1, 0,   3,  3]
     ,[12, 4,  -4,  7]
     ,[-1, 9,   8,  0]
     ,[ 3,-5,  -2,  9]
     ,[ 3, 2,6873,-38]
     ]

>>> m-Integer(2) # optional - lie
     [[ 1, 0, 3,3]
     ,[-1, 9, 8,0]
     ,[ 3,-5,-2,9]
     ]

LiE handles multivariate (Laurent) polynomials.

sage: # optional - lie
sage: lie('X[1,2]')
1X[1,2]
sage: -3*_
-3X[1,2]
sage: _ + lie('4X[-1,4]')
4X[-1,4] - 3X[ 1,2]
sage: _^2
16X[-2,8] - 24X[ 0,6] +  9X[ 2,4]
sage: lie('(4X[-1,4]-3X[1,2])*(X[2,0]-X[0,-4])')
-4X[-1, 0] + 3X[ 1,-2] + 4X[ 1, 4] - 3X[ 3, 2]
sage: _ - _
0X[0,0]

>>> from sage.all import *
>>> # optional - lie
>>> lie('X[1,2]')
1X[1,2]
>>> -Integer(3)*_
-3X[1,2]
>>> _ + lie('4X[-1,4]')
4X[-1,4] - 3X[ 1,2]
>>> _**Integer(2)
16X[-2,8] - 24X[ 0,6] +  9X[ 2,4]
>>> lie('(4X[-1,4]-3X[1,2])*(X[2,0]-X[0,-4])')
-4X[-1, 0] + 3X[ 1,-2] + 4X[ 1, 4] - 3X[ 3, 2]
>>> _ - _
0X[0,0]

You can call LiE’s built-in functions using lie.functionname.

sage: lie.partitions(6) # optional - lie
     [[6,0,0,0,0,0]
     ,[5,1,0,0,0,0]
     ,[4,2,0,0,0,0]
     ,[4,1,1,0,0,0]
     ,[3,3,0,0,0,0]
     ,[3,2,1,0,0,0]
     ,[3,1,1,1,0,0]
     ,[2,2,2,0,0,0]
     ,[2,2,1,1,0,0]
     ,[2,1,1,1,1,0]
     ,[1,1,1,1,1,1]
     ]
sage: lie.diagram('E8') # optional - lie
        O 2
        |
        |
O---O---O---O---O---O---O
1   3   4   5   6   7   8
E8

>>> from sage.all import *
>>> lie.partitions(Integer(6)) # optional - lie
     [[6,0,0,0,0,0]
     ,[5,1,0,0,0,0]
     ,[4,2,0,0,0,0]
     ,[4,1,1,0,0,0]
     ,[3,3,0,0,0,0]
     ,[3,2,1,0,0,0]
     ,[3,1,1,1,0,0]
     ,[2,2,2,0,0,0]
     ,[2,2,1,1,0,0]
     ,[2,1,1,1,1,0]
     ,[1,1,1,1,1,1]
     ]
>>> lie.diagram('E8') # optional - lie
        O 2
        |
        |
O---O---O---O---O---O---O
1   3   4   5   6   7   8
E8

You can define your own functions in LiE using lie.eval . Once you’ve defined a function (say f), you can call it using lie.f ; however, user-defined functions do not show up when using tab-completion.

sage: # optional - lie
sage: lie.eval('f(int x) = 2*x')
''
sage: lie.f(984)
1968
sage: lie.eval('f(int n) = a=3*n-7; if a < 0 then a = -a fi; 7^a+a^3-4*a-57')
''
sage: lie.f(2)
-53
sage: lie.f(5)
5765224

>>> from sage.all import *
>>> # optional - lie
>>> lie.eval('f(int x) = 2*x')
''
>>> lie.f(Integer(984))
1968
>>> lie.eval('f(int n) = a=3*n-7; if a < 0 then a = -a fi; 7^a+a^3-4*a-57')
''
>>> lie.f(Integer(2))
-53
>>> lie.f(Integer(5))
5765224

LiE’s help can be accessed through lie.help(‘functionname’) where functionname is the function you want to receive help for.

sage: print(lie.help('diagram'))  # optional - lie
diagram(g).   Prints the Dynkin diagram of g, also indicating
   the type of each simple component printed, and labeling the nodes as
   done by Bourbaki (for the second and further simple components the
   labels are given an offset so as to make them disjoint from earlier
   labels). The labeling of the vertices of the Dynkin diagram prescribes
   the order of the coordinates of root- and weight vectors used in LiE.

>>> from sage.all import *
>>> print(lie.help('diagram'))  # optional - lie
diagram(g).   Prints the Dynkin diagram of g, also indicating
   the type of each simple component printed, and labeling the nodes as
   done by Bourbaki (for the second and further simple components the
   labels are given an offset so as to make them disjoint from earlier
   labels). The labeling of the vertices of the Dynkin diagram prescribes
   the order of the coordinates of root- and weight vectors used in LiE.

This can also be accessed with lie.functionname? .

With the exception of groups, all LiE data types can be converted into native Sage data types by calling the .sage() method.

Integers:

sage: a = lie('1234') # optional - lie
sage: b = a.sage(); b # optional - lie
1234
sage: type(b) # optional - lie
<class 'sage.rings.integer.Integer'>

>>> from sage.all import *
>>> a = lie('1234') # optional - lie
>>> b = a.sage(); b # optional - lie
1234
>>> type(b) # optional - lie
<class 'sage.rings.integer.Integer'>

Vectors:

sage: a = lie('[1,2,3]') # optional - lie
sage: b = a.sage(); b # optional - lie
[1, 2, 3]
sage: type(b) # optional - lie
<... 'list'>

>>> from sage.all import *
>>> a = lie('[1,2,3]') # optional - lie
>>> b = a.sage(); b # optional - lie
[1, 2, 3]
>>> type(b) # optional - lie
<... 'list'>

Matrices:

sage: a = lie('[[1,2],[3,4]]') # optional - lie
sage: b = a.sage(); b # optional - lie
[1 2]
[3 4]
sage: type(b) # optional - lie
<class 'sage.matrix.matrix_integer_dense.Matrix_integer_dense'>

>>> from sage.all import *
>>> a = lie('[[1,2],[3,4]]') # optional - lie
>>> b = a.sage(); b # optional - lie
[1 2]
[3 4]
>>> type(b) # optional - lie
<class 'sage.matrix.matrix_integer_dense.Matrix_integer_dense'>

Polynomials:

sage: a = lie('X[1,2] - 2*X[2,1]') # optional - lie
sage: b = a.sage(); b              # optional - lie
-2*x0^2*x1 + x0*x1^2
sage: type(b)                      # optional - lie
<class 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>

>>> from sage.all import *
>>> a = lie('X[1,2] - 2*X[2,1]') # optional - lie
>>> b = a.sage(); b              # optional - lie
-2*x0^2*x1 + x0*x1^2
>>> type(b)                      # optional - lie
<class 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>

Text:

sage: a = lie('"text"') # optional - lie
sage: b = a.sage(); b # optional - lie
'text'
sage: type(b) # optional - lie
<... 'str'>

>>> from sage.all import *
>>> a = lie('"text"') # optional - lie
>>> b = a.sage(); b # optional - lie
'text'
>>> type(b) # optional - lie
<... 'str'>

LiE can be programmed using the Sage interface as well. Section 5.1.5 of the manual gives an example of a function written in LiE’s language which evaluates a polynomial at a point. Below is a (roughly) direct translation of that program into Python / Sage.

sage: # optional - lie
sage: def eval_pol(p, pt):
....:     s = 0
....:     for i in range(1,p.length().sage()+1):
....:         m = 1
....:         for j in range(1,pt.size().sage()+1):
....:             m *= pt[j]^p.expon(i)[j]
....:         s += p.coef(i)*m
....:     return s
sage: a = lie('X[1,2]')
sage: b1 = lie('[1,2]')
sage: b2 = lie('[2,3]')
sage: eval_pol(a, b1)
4
sage: eval_pol(a, b2)
18

>>> from sage.all import *
>>> # optional - lie
>>> def eval_pol(p, pt):
...     s = Integer(0)
...     for i in range(Integer(1),p.length().sage()+Integer(1)):
...         m = Integer(1)
...         for j in range(Integer(1),pt.size().sage()+Integer(1)):
...             m *= pt[j]**p.expon(i)[j]
...         s += p.coef(i)*m
...     return s
>>> a = lie('X[1,2]')
>>> b1 = lie('[1,2]')
>>> b2 = lie('[2,3]')
>>> eval_pol(a, b1)
4
>>> eval_pol(a, b2)
18

AUTHORS:

  • Mike Hansen 2007-08-27

  • William Stein (template)

class sage.interfaces.lie.LiE(maxread=None, script_subdirectory=None, logfile=None, server=None)[source]#

Bases: ExtraTabCompletion, Expect

Interface to the LiE interpreter.

Type lie.[tab] for a list of all the functions available from your LiE install. Type lie.[tab]? for LiE’s help about a given function. Type lie(...) to create a new LiE object, and lie.eval(...) to run a string using LiE (and get the result back as a string).

console()[source]#

Spawn a new LiE command-line session.

EXAMPLES:

sage: lie.console()                    # not tested
LiE version 2.2.2 created on Sep 26 2007 at 18:13:19
Authors: Arjeh M. Cohen, Marc van Leeuwen, Bert Lisser.
Free source code distribution
...

>>> from sage.all import *
>>> lie.console()                    # not tested
LiE version 2.2.2 created on Sep 26 2007 at 18:13:19
Authors: Arjeh M. Cohen, Marc van Leeuwen, Bert Lisser.
Free source code distribution
...
eval(code, strip=True, **kwds)[source]#

EXAMPLES:

sage: lie.eval('2+2')  # optional - lie
'4'

>>> from sage.all import *
>>> lie.eval('2+2')  # optional - lie
'4'
function_call(function, args=None, kwds=None)[source]#

EXAMPLES:

sage: lie.function_call("diagram", args=['A4']) # optional - lie
O---O---O---O
1   2   3   4
A4

>>> from sage.all import *
>>> lie.function_call("diagram", args=['A4']) # optional - lie
O---O---O---O
1   2   3   4
A4
get(var)[source]#

Get the value of the variable var.

EXAMPLES:

sage: lie.set('x', '2')  # optional - lie
sage: lie.get('x')       # optional - lie
'2'

>>> from sage.all import *
>>> lie.set('x', '2')  # optional - lie
>>> lie.get('x')       # optional - lie
'2'
get_using_file(var)[source]#

EXAMPLES:

sage: lie.get_using_file('x')
Traceback (most recent call last):
...
NotImplementedError

>>> from sage.all import *
>>> lie.get_using_file('x')
Traceback (most recent call last):
...
NotImplementedError
help(command)[source]#

Return a string of the LiE help for command.

EXAMPLES:

sage: lie.help('diagram') # optional - lie
'diagram(g)...'

>>> from sage.all import *
>>> lie.help('diagram') # optional - lie
'diagram(g)...'
read(filename)[source]#

EXAMPLES:

sage: filename = tmp_filename()
sage: with open(filename, 'w') as f:
....:     _ = f.write('x = 2\n')
sage: lie.read(filename)  # optional - lie
sage: lie.get('x')        # optional - lie
'2'
sage: import os
sage: os.unlink(filename)

>>> from sage.all import *
>>> filename = tmp_filename()
>>> with open(filename, 'w') as f:
...     _ = f.write('x = 2\n')
>>> lie.read(filename)  # optional - lie
>>> lie.get('x')        # optional - lie
'2'
>>> import os
>>> os.unlink(filename)
set(var, value)[source]#

Set the variable var to the given value.

EXAMPLES:

sage: lie.set('x', '2')  # optional - lie
sage: lie.get('x')       # optional - lie
'2'

>>> from sage.all import *
>>> lie.set('x', '2')  # optional - lie
>>> lie.get('x')       # optional - lie
'2'
version()[source]#

EXAMPLES:

sage: lie.version() # optional - lie
'2.2'

>>> from sage.all import *
>>> lie.version() # optional - lie
'2.2'
class sage.interfaces.lie.LiEElement(parent, value, is_name=False, name=None)[source]#

Bases: ExtraTabCompletion, ExpectElement

type()[source]#

EXAMPLES:

sage: m = lie('[[1,0,3,3],[12,4,-4,7],[-1,9,8,0],[3,-5,-2,9]]') # optional - lie
sage: m.type() # optional - lie
'mat'

>>> from sage.all import *
>>> m = lie('[[1,0,3,3],[12,4,-4,7],[-1,9,8,0],[3,-5,-2,9]]') # optional - lie
>>> m.type() # optional - lie
'mat'
class sage.interfaces.lie.LiEFunction(parent, name)[source]#

Bases: ExpectFunction

class sage.interfaces.lie.LiEFunctionElement(obj, name)[source]#

Bases: FunctionElement

sage.interfaces.lie.is_LiEElement(x)[source]#

EXAMPLES:

sage: from sage.interfaces.lie import is_LiEElement
sage: is_LiEElement(2)
doctest:...: DeprecationWarning: the function is_LiEElement is deprecated; use isinstance(x, sage.interfaces.abc.LiEElement) instead
See https://github.com/sagemath/sage/issues/34804 for details.
False
sage: l = lie(2) # optional - lie
sage: is_LiEElement(l) # optional - lie
True

>>> from sage.all import *
>>> from sage.interfaces.lie import is_LiEElement
>>> is_LiEElement(Integer(2))
doctest:...: DeprecationWarning: the function is_LiEElement is deprecated; use isinstance(x, sage.interfaces.abc.LiEElement) instead
See https://github.com/sagemath/sage/issues/34804 for details.
False
>>> l = lie(Integer(2)) # optional - lie
>>> is_LiEElement(l) # optional - lie
True
sage.interfaces.lie.lie_console()[source]#

Spawn a new LiE command-line session.

EXAMPLES:

sage: from sage.interfaces.lie import lie_console
sage: lie_console()                    # not tested
LiE version 2.2.2 created on Sep 26 2007 at 18:13:19
Authors: Arjeh M. Cohen, Marc van Leeuwen, Bert Lisser.
Free source code distribution
...

>>> from sage.all import *
>>> from sage.interfaces.lie import lie_console
>>> lie_console()                    # not tested
LiE version 2.2.2 created on Sep 26 2007 at 18:13:19
Authors: Arjeh M. Cohen, Marc van Leeuwen, Bert Lisser.
Free source code distribution
...
sage.interfaces.lie.lie_version()[source]#

EXAMPLES:

sage: from sage.interfaces.lie import lie_version
sage: lie_version() # optional - lie
'2.2'

>>> from sage.all import *
>>> from sage.interfaces.lie import lie_version
>>> lie_version() # optional - lie
'2.2'
sage.interfaces.lie.reduce_load_lie()[source]#

EXAMPLES:

sage: from sage.interfaces.lie import reduce_load_lie
sage: reduce_load_lie()
LiE Interpreter

>>> from sage.all import *
>>> from sage.interfaces.lie import reduce_load_lie
>>> reduce_load_lie()
LiE Interpreter