universalEnvelopingAlgebra -- computes the universal enveloping algebra of a Lie algebra

Usage:

universalEnvelopingAlgebra(g)
Inputs:
- LAB, an instance of the type LieAlgebraBasis,
Outputs:
- S, a sequence,

Description

Let $\mathfrak{g}$ be a Lie algebra. Let $T(\mathfrak{g})$ be the tensor algebra on $\mathfrak{g}$. The universal enveloping algebra $U(\mathfrak{g})$ is the quotient of $T(\mathfrak{g})$ by the two-sided ideal generated by all relations of the form $X_1 X_2 - X_2 X_1 - [X_1,X_2]$.

We construct $T(\mathfrak{g})$ and $U(\mathfrak{g})$ using the AssociativeAlgebras package. The generators of $T(\mathfrak{g})$ are the elements of a Lie algebra basis of $\mathfrak{g}$. Let $h_1,\ldots,H_n$ be the Cartan subalgebra basis elements, $x_1,\ldots,x_l$ the positive root vectors, and $y_1,\ldots,y_n$ the negative root vectors. Then the term order on $T(\mathfrak{g})$ we use is degree lex with the variables ordered $x_l,\ldots,x_1,h_n,\ldots,h_1,y_l,\ldots,y_1$, so that the basis used for the quotient $U(\mathfrak{g})$ consists of monomials of the form $y_1^{a_1}\cdots y_l^{a_l} h_1^{b_1}\cdots h_n^{b_n} x_1^{c_1} \cdots x_n^{c_n}$.

The function universalEnvelopingAlgebra returns a sequence consisting of the universal enveloping algebra, the permutation $\sigma$ matching elements of the Lie algebra basis to the generators of the tensor algebra $T(\mathfrak{g})$, and the inverse permutation $\sigma^{-1}$.

In the following example, we express the monomial $x_1 y_2 y_3$ in $U(sl_3)$. We have $x_1 = E_{(1,2)}$, $y_2 = E_{(3,2)}$ and $y_3 = E_{(3,1)}$. Thus $x_1y_2 = y_2x_1$ and $x_1 y_3 = y_3 x_1 - y_2$, so that $x_1 y_2 y_3 = y_2 y_3 x_1-y_2^2$.

i1 : sl3 = simpleLieAlgebra("A",2)

o1 = sl3

o1 : simple LieAlgebra

i2 : S = universalEnvelopingAlgebra(sl3)
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

                                                                             
                                                                             
o2 = (-----------------------------------------------------------------------
      (0, - h h  + h h , - x h  + h x  - 2x , - x h  + h x  + x , - x h  + h 
             2 1    1 2     1 1    1 1     1     2 1    1 2    2     3 1    1
     ------------------------------------------------------------------------
                                                                             
                                                                             
     ------------------------------------------------------------------------
     x  - x , h y  - y h  + 2y , h y  - y h  - y , h y  - y h  + y , h h  - h
      3    3   1 1    1 1     1   1 2    2 1    2   1 3    3 1    3   2 1    
     ------------------------------------------------------------------------
                                                                             
                                                                             
     ------------------------------------------------------------------------
      h , 0, - x h  + h x  + x , - x h  + h x  - 2x , - x h  + h x  - x , h y
     1 2        1 2    2 1    1     2 2    2 2     2     3 2    2 3    3   2 
     ------------------------------------------------------------------------
                                                                             
                                                                             
     ------------------------------------------------------------------------
       - y h  - y , h y  - y h  + 2y , h y  - y h  + y , x h  - h x  + 2x , x
     1    1 2    1   2 2    2 2     2   2 3    3 2    3   1 1    1 1     1   
     ------------------------------------------------------------------------
                                                                             
                                                                             
     ------------------------------------------------------------------------
      h  - h x  - x , 0, - x x  + x x  - x , - x x  + x x , x y  - y x  - h ,
     1 2    2 1    1        2 1    1 2    3     3 1    1 3   1 1    1 1    1 
     ------------------------------------------------------------------------
                                                                             
                                                                             
     ------------------------------------------------------------------------
      x y  - y x , x y  - y x  + y , x h  - h x  - x , x h  - h x  + 2x , x x
       1 2    2 1   1 3    3 1    2   2 1    1 2    2   2 2    2 2     2   2 
     ------------------------------------------------------------------------
                                                                          QQ 
                                                                             
     ------------------------------------------------------------------------
       - x x  + x , 0, - x x  + x x , x y  - y x , x y  - y x  - h , x y  - y
     1    1 2    3        3 2    2 3   2 1    1 2   2 2    2 2    2   2 3    
     ------------------------------------------------------------------------
     <|x , x , x , h , h , y , y , y |>
        3   2   1   2   1   3   2   1
     ------------------------------------------------------------------------
      x  - y , x h  - h x  + x , x h  - h x  + x , x x  - x x , x x  - x x , 
     3 2    1   3 1    1 3    3   3 2    2 3    3   3 1    1 3   3 2    2 3  
     ------------------------------------------------------------------------
     ------------------------------------------------------------------------
     0, x y  - y x  + x , x y  - y x  - x , x y  - y x  - h  - h , - h y  + y
         3 1    1 3    2   3 2    2 3    1   3 3    3 3    2    1     1 1    
     ------------------------------------------------------------------------
     ------------------------------------------------------------------------
      h  - 2y , - h y  + y h  + y , - x y  + y x  + h , - x y  + y x , - x y 
     1 1     1     2 1    1 2    1     1 1    1 1    1     2 1    1 2     3 1
     ------------------------------------------------------------------------
     ------------------------------------------------------------------------
      + y x  - x , 0, - y y  + y y  + y , - y y  + y y , - h y  + y h  + y , 
         1 3    2        2 1    1 2    3     3 1    1 3     1 2    2 1    2  
     ------------------------------------------------------------------------
     ------------------------------------------------------------------------
     - h y  + y h  - 2y , - x y  + y x , - x y  + y x  + h , - x y  + y x  + 
        2 2    2 2     2     1 2    2 1     2 2    2 2    2     3 2    2 3   
     ------------------------------------------------------------------------
     ------------------------------------------------------------------------
     x , y y  - y y  - y , 0, - y y  + y y , - h y  + y h  - y , - h y  + y h
      1   2 1    1 2    3        3 2    2 3     1 3    3 1    3     2 3    3 
     ------------------------------------------------------------------------
     ------------------------------------------------------------------------
       - y , - x y  + y x  - y , - x y  + y x  + y , - x y  + y x  + h  + h ,
     2    3     1 3    3 1    2     2 3    3 2    1     3 3    3 3    2    1 
     ------------------------------------------------------------------------
     -----------------------------, {4, 3, 2, 1, 0, 7, 6, 5}, {4, 3, 2, 1, 0,
      y y  - y y , y y  - y y , 0)
       3 1    1 3   3 2    2 3
     ------------------------------------------------------------------------
     7, 6, 5})

o2 : Sequence

i3 : U = first S

o3 = U

o3 : FreeAlgebraQuotient

i4 : x_1*y_2*y_3

               2
o4 = y y x  - y
      2 3 1    2

o4 : U

Ways to use universalEnvelopingAlgebra:

universalEnvelopingAlgebra(LieAlgebra)
universalEnvelopingAlgebra(LieAlgebraBasis)

For the programmer

The object universalEnvelopingAlgebra is a method function.

The source of this document is in LieAlgebraRepresentations/documentation.m2:1479:0.