Universal Enveloping Algebras

This section describes the functionality for universal enveloping algebras of Lie algebras. If a Lie algebra is semisimple and defined over a field of characteristic \(0\), then it is possible to write down an integral basis of the universal enveloping algebra that has nice properties. To accommodate this possibility, two constructions of a universal enveloping algebra are provided: a general construction, and one in which this integral basis is used. First we briefly describe the theoretical background behind universal enveloping algebras.

In Magma, universal enveloping algebras have type AlgUE and their elements have type AlgUEElt. Integral universal enveloping algebras have type AlgIUE and their elements have type AlgIUEElt. General algebras having a PBW basis (see below) have type AlgPBW (elements type AlgPBWElt) which inherit from types Alg and Rng. Consequently, the type AlgIUE inherits from AlgPBW.

Background

Universal Enveloping Algebras

Let \(L\) be a Lie algebra over the field \(F\) having basis \(x_1,\ldots, x_n\). The universal enveloping algebra \(U(L)\) of \(L\) is the associative algebra with identity, generated by \(n\) symbols which are also denoted by \(x_1,\ldots, x_n\). These generators satisfy the relations

\[x_jx_i -x_ix_j = [ x_j,x_i], 1\leq i,j\leq n>.\]

Here \([x_j,x_i]\) is the product in the Lie algebra \(L\), so it is a certain linear combination of the \(x_k\).

The theorem of Poincaré-Birkhoff-Witt states that a basis of \(U(L)\) is formed by the set of all elements

\[x_1^{k_1}\cdots x_n^{k_n},\]

where the \(k_i\) are non-negative integers. Furthermore, the product of two such basis elements may be rewritten as a linear combination of basis elements using the defining relations \(x_jx_i -x_ix_j = [ x_j,x_i]\) for \(j>i\).

The Integral Form of a Universal Enveloping Algebra

If the Lie algebra \(L\) happens to be (split) semisimple and of characteristic \(0\), then the universal enveloping algebra has a nice basis described by [Kos66]. The first step in constructing this basis involves taking a Chevalley basis of \(L\), consisting of the elements \(y_1,\ldots, y_s\), \(h_1,\ldots, h_r\) and \(x_1,\ldots, x_s\). Here the \(y_i\) and \(x_i\) are root vectors belonging to negative roots and positive roots, respectively. The \(h_i\) are basis elements of a Cartan subalgebra. In the universal enveloping algebra we use the divided powers

\[y_i^{(n)} = {y_i^n\over n!}, ~~ x_i^{(n)} = {x_i^n\over n!},\]

and the binomials

\[{h_i \choose k } = { h_i(h_i-1)\cdots (h_i-k+1)\over k!}.\]

A basis of \(U(L)\) is formed by the elements

\[y_1^{(m_1)}\cdots y_s^{(m_s)} {h_1\choose k_1} \cdots {h_r\choose k_r} x_1^{(n_1)}\cdots x_s^{(n_s)}.\]

This basis has the useful property that if we multiply two basis elements, the structure constants will be integers (usually of quite moderate size). So this is a basis of an integral form of the universal enveloping algebra.

Construction of Universal Enveloping Algebras

UniversalEnvelopingAlgebra(L): AlgLie -> AlgUE: This creates the universal enveloping algebra \(U\) of the Lie algebra \(L\). Here the \(i\)-th basis element of \(L\) (i.e., L.i) corresponds to the \(i\)-th generator of \(U\) (i.e., U.i). Every product of generators is rewritten as a linear combination of Poincaré-Birkhoff-Witt monomials (cf. Section Universal Enveloping Algebras).

IntegralUEA(L): AlgLie -> AlgIUE

IntegralUEAlgebra(L): AlgLie -> AlgIUE

IntegralUniversalEnvelopingAlgebra(L): AlgLie -> AlgIUE

Given a semisimple Lie algebra \(L\) of characteristic \(0\), create the integral universal enveloping algebra \(U\) of \(L\). The basis described in Section The Integral Form of a Universal Enveloping Algebra is used.

Let \(x\), \(y\), and \(h\) denote the output of ChevalleyBasis(L). Let \(s\) be the length of \(x\), and \(r\) the length of \(h\). Then every generator of \(U\) corresponds to an element of \(x\), \(y\), or \(h\). If \(1\leq i\leq s\) then the \(i\)-th generator of \(U\) (i.e., U.i) corresponds to the \(i\)-th element of \(y\). It is printed as y_i. If \(s+1\leq i\leq s+r\), then the \(i\)-th generator of \(U\) corresponds to the \(k\)-th element of \(h\), where \(k=i-s\). It is printed as [ h_k ; 1 ] (i.e., h_k choose 1). Finally, if \(s+r+1\leq i\leq 2s+r\), then the \(i\)-th generator corresponds to the \(k\)-th element of \(x\), where \(k=i-s-r\). It is printed as x_k.

Using this form of the universal enveloping algebra has two advantages. Firstly, the structure constants are integers which usually remain relatively small. Secondly, multiplication of elements is, in general, much faster than is the case with universal enveloping algebras that employ PBW bases.

Example: Constructing a Universal Enveloping Algebra (ex-c78d02)

> T:= [ <4,1,1,1>, <1,4,1,-1>, <4,1,3,1>, <1,4,3,-1>, <4,2,2,1>, <2,4,2,-1>,
> <4,3,1,1>, <3,4,1,-1>, <3,1,2,1>, <1,3,2,-1> ];
> L:= LieAlgebra< Rationals(), 4 | T >;
> U:= UniversalEnvelopingAlgebra(L);
> U.4*U.1;
x_1*x_4 + x_1 + x_3
> L:= LieAlgebra("F4", Rationals());
> U:= IntegralUEA(L);
> U.29*U.1;
y_1*x_1 + [ h_1 ; 1 ]
> (1/4)*U.29^2*U.1^2;
y_1^(2)*x_1^(2) + y_1*[ h_1 ; 1 ]*x_1 - 2*y_1*x_1 + [ h_1 ; 2 ]

In the last example we divided by \(4\) because U.29^2 = 2 U.29^(2), and likewise for U.1^2.

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AssignNames(~U, Q): AlgPBW, [ MonStgElt ]: Assign the names in the sequence \(Q\) to the generators of the algebra \(U\).

ChangeRing(U, S): AlgUE, Rng -> AlgUE: Given a universal enveloping algebra \(U\) with base ring \(R\), together with a ring \(S\), construct the algebra \(U'\) with base ring \(S\) obtained by coercing the coefficients of elements of \(U\) into \(S\).

Elements of Universal Enveloping Algebras

Most functions in this section are applicable both to universal enveloping algebras and to integral universal enveloping algebras. Therefore, they are only documented once. An exception is the function HBinomial, which is only applicable to integral universal enveloping algebras.

Creation of Elements

U ! 0: AlgPBW, RngIntElt -> AlgPBWElt
Zero(U): AlgPBW -> AlgPBWElt: The zero element of the universal enveloping algebra \(U\).

U ! 1: AlgPBW, RngIntElt -> AlgPBWElt
One(U): AlgPBW -> AlgPBWElt: The identity element of the universal enveloping algebra \(U\).

U . i: AlgPBW, RngIntElt -> AlgPBWElt: The \(i\)-th generator of the universal enveloping algebra \(U\).

U ! r: AlgPBW, Any -> AlgPBWElt: Returns \(r\) as an element of the enveloping algebra \(U\) where \(r\) may be anything coercible into the coefficient ring of \(U\) or an element of another enveloping algebra of the same type as \(U\) whose coefficients can be coerced into the coefficient ring of \(U\).

HBinomial(U, i, n): AlgIUE, RngIntElt, RngIntElt -> AlgIUEElt
HBinomial(h, n): AlgIUEElt, RngIntElt -> AlgIUEElt: This function is applicable only in the case of integral universal enveloping algebras. It is used for constructing the “binomial” elements \(h_i\) choose \(n\). In the first form \(U\) is an integral universal enveloping algebra, and \(i\) is an index between \(1\) and the rank of the root datum. In the second form, the element \(h\) is simply U.(s+i), where \(s\) is the number of positive roots.

Example: Binomial Elements of an Integral Enveloping Algebra (ex-4c79c2)

> L:= LieAlgebra("E6",Rationals());
> U:= IntegralUEA(L);
> HBinomial(U, 4, 10);
[ h_4 ; 10 ]

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Operations on Elements

x + y: AlgPBWElt, AlgPBWElt -> AlgPBWElt
x - y: AlgPBWElt, AlgPBWElt -> AlgPBWElt
x * y: AlgPBWElt, AlgPBWElt -> AlgPBWElt
c * x: RngElt, AlgPBWElt -> AlgPBWElt
x * c: AlgPBWElt, RngElt -> AlgPBWElt
x ^ n: AlgPBWElt, RngIntElt -> AlgPBWElt
Monomials(u): AlgPBWElt -> SeqEnum: The sequence of the monomials that occur in the element \(u\) of a universal enveloping algebra.

Coefficients(u): AlgPBWElt -> SeqEnum: The sequence of coefficients of the monomials in the element \(u\) of a universal enveloping algebra. The \(k\)-th element of this sequence corresponds exactly to the \(k\)-th monomial in the sequence returned by Monomials(u).

Degree(u, i): AlgPBWElt, RngIntElt -> RngIntElt: Given an element \(u\) of a universal enveloping algebra \(U\) and an integer \(i\), this function returns the degree of \(u\) in the \(i\)-th generator of \(U\).

Example: Operations on Enveloping Algebra Elements (ex-81e4e4)

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> L:= LieAlgebra("G2",Rationals());
> U:= IntegralUEA(L);

> c:= U.7*U.2;c;
y_2*[ h_1 ; 1 ] + 3*y_2
> Monomials(c);
[
    y_2*[ h_1 ; 1 ],
    y_2
]
> Coefficients(c);
[ 1, 3 ]
> c:= U.10*U.7*U.2; c;
y_2*[ h_1 ; 1 ]*x_2 + 6*y_2*x_2 + [ h_1 ; 1 ]*[ h_2 ; 1 ] + 3*[ h_2 ; 1 ]
> Degree(c, 2);
1
> Degree(c, 7);
1
> Degree(c, 8);
1

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Universal Enveloping Algebras#

Background#