Free Lie algebra

In the mathematical theory of Lie algebras , a free Lie algebra or freely generated Lie algebra is a Lie algebra that is free in the category of Lie algebras and Lie homomorphisms. This allows Lie algebras to be constructed with given generators and relations .

definition

We consider a solid body as a coefficient domain . Let the freely generated associative K -algebra be over for a given set , be the inclusion map. Through the lie clamp ${\displaystyle K}$ ${\displaystyle X\not =\emptyset }$ ${\displaystyle A(X)}$ ${\displaystyle X}$ ${\displaystyle i\colon X\rightarrow A(X)}$

${\displaystyle [a,b]:=a\cdot b-b\cdot a,\quad a,b\in A(X)}$

becomes a Lie algebra. Be in it ${\displaystyle A(X)}$

${\displaystyle FL(X):=\bigcap \{L|L\subset A(X){\text{ ist Lie-Unteralgebra mit }}i(X)\subset L\}}$

is the average of all contained Lie subalgebras of . This is not the average over an empty set, because it is a Lie sub-algebra that contains. ${\displaystyle i(X)}$ ${\displaystyle A(X)}$ ${\displaystyle A(X)}$ ${\displaystyle i(X)}$

${\displaystyle FL(X)}$is called free Lie algebra over . ${\displaystyle X}$

According to construction , that means we can also understand it as an inclusion mapping . ${\displaystyle i(X)\subset FL(X)}$ ${\displaystyle i}$ ${\displaystyle X\rightarrow FL(X)}$

Universal property

The free Lie algebra over satisfies the following universal property : ${\displaystyle FL(X)}$ ${\displaystyle X}$

Let be a mapping from into a Lie algebra . Then there is exactly one Lie algebra homomorphism with . ${\displaystyle \theta \colon X\rightarrow L}$ ${\displaystyle X}$ ${\displaystyle L}$ ${\displaystyle \varphi \colon FL(X)\rightarrow L}$ ${\displaystyle \varphi \circ i=\theta }$

This justifies the term free Lie algebra .

Alternative construction

At Bourbaki there is an alternative construction of the free Lie algebra. For a non-empty set, let the free magma over and the free over vector space generated with the linear multiplication from. In it consider the ideal , that of all expressions of form ${\displaystyle X}$ ${\displaystyle M(X)}$ ${\displaystyle X}$ ${\displaystyle \mathrm {Lib} (X)}$ ${\displaystyle M(X)}$ ${\displaystyle K}$ ${\displaystyle M(X)}$ ${\displaystyle I\subset \mathrm {Lib} (X)}$

${\displaystyle a\cdot a,\quad a\in \mathrm {Lib} (X)}$

${\displaystyle a\cdot (b\cdot c)+b\cdot (c\cdot a)+c\cdot (a\cdot b),\quad a,b,c\in \mathrm {Lib} (X)}$

is produced. Then the freely generated over is called Lie algebra. Through the transition to quotient algebra, the listed elements become the zero element, because by definition they are ideal . Therefore anti-commutativity and Jacobi identity apply , that is, a Lie algebra is obtained. This is shown to fulfill the above universal quality. Since every two Lie algebras that fulfill the same universal property with respect to must be isomorphic, this construction can be regarded as an alternative to the one given above. ${\displaystyle \mathrm {Lib} (X)/I}$ ${\displaystyle X}$ ${\displaystyle I}$ ${\displaystyle \mathrm {Lib} (X)/I}$ ${\displaystyle X}$

Examples

If one element is then isomorphic to the commutative polynomial algebra of all polynomials in the indefinite . The Lie algebra is therefore Abelian, that is, every subspace is a Lie subalgebra. Thus, by definition, is the smallest subspace that contains, and that is . So is ${\displaystyle X=\{x\}}$ ${\displaystyle A(\{x\})}$ ${\displaystyle x}$ ${\displaystyle A(\{x\})}$ ${\displaystyle FL(\{x\})}$ ${\displaystyle i(x)}$ ${\displaystyle K\cdot i(x)}$

${\displaystyle FL(\{x\})\cong K}$ the trivial one-dimensional Lie algebra.

The universal enveloping algebra of the free Lie algebra is isomorphic to the free associative algebra over , in formulas . ${\displaystyle FL(X)}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {U}}(FL(X))\cong A(X)}$

Generators and Relations

construction

Again be a non-empty set. A Lie word über is a finite linear combination of finite Lie monomials, that is, finite Lie products of elements . An example of a Lie monomial is ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$

${\displaystyle [[[x_{1},x_{2}],x_{1}],[x_{1},x_{3}]],\quad x_{1},x_{2},x_{3}\in X}$,

is an example of a lie word

${\displaystyle 7\cdot [[[x_{1},x_{2}],x_{1}],[x_{1},x_{3}]]-[[[[x_{1},x_{1}],x_{1}],x_{1}],x_{2}],\quad x_{1},x_{2},x_{3}\in X}$.

For a lot of Lie words about was that of generated Lie Ideal . Then the quotient algebra is called ${\displaystyle R}$ ${\displaystyle X}$ ${\displaystyle \langle R\rangle \subset FL(X)}$ ${\displaystyle R\subset FL(X)}$

${\displaystyle L(X;R):=FL(X)/\langle R\rangle }$

the Lie algebra generated by the set and the relations . ${\displaystyle X}$ ${\displaystyle R}$

As with the presentation of a group considered in group theory , one can also construct Lie algebras here with given properties; more precisely, each Lie word becomes an equation in . ${\displaystyle w}$ ${\displaystyle w=0}$ ${\displaystyle L(X;R)}$

Examples

• ${\displaystyle L(X;\emptyset )=FL(X)}$because is the ideal generated by the empty set . ${\displaystyle \{0\}}$

• Let be the set of all Lie words . Then is the abelian Lie algebra generated by , that is, the freely generated K- vector space with zero multiplication as a Lie bracket. ${\displaystyle R}$ ${\displaystyle [x_{1},x_{2}],\,x_{1},x_{2}\in X}$ ${\displaystyle L(X;R)}$ ${\displaystyle X}$ ${\displaystyle X}$

• Let and be certain real constants that are less than or equal to 0 for different indices. ${\displaystyle X=\{e_{1},\ldots ,e_{n},h_{1},\ldots ,h_{n},f_{1},\ldots ,f_{n}\}}$ ${\displaystyle A_{i,j}}$

Then it is the set of relations ${\displaystyle R}$

${\displaystyle [h_{i},h_{j}]}$

${\displaystyle [h_{i},e_{j}]-A_{i,j}e_{j}}$

${\displaystyle [h_{i},f_{j}]+A_{i,j}f_{j}}$

${\displaystyle [e_{i},f_{i}]-h_{i}}$

${\displaystyle [e_{i},f_{j}]}$   For   ${\displaystyle i\not =j}$

${\displaystyle [e_{i},[e_{i}[\ldots [e_{i},e_{j}]]]}$   with and occurrences of ${\displaystyle i\not =j}$ ${\displaystyle 1-A_{i,j}}$ ${\displaystyle e_{i}}$

${\displaystyle [f_{i},[f_{i}[\ldots [f_{i},f_{j}]]]}$   with and occurrences of ${\displaystyle i\not =j}$ ${\displaystyle 1-A_{i,j}}$ ${\displaystyle f_{i}}$

Then the Lie algebra plays an important role in the proof of the existence theorem for semisimple Lie algebras . If these are the coefficients of a Cartan matrix , then there is a finite-dimensional Lie algebra with this same Cartan matrix. This is Serre's proof of the existence theorem. These same techniques are also used to define Kac-Moody algebras . ${\displaystyle L(X;R)}$ ${\displaystyle A_{i,j}}$ ${\displaystyle L(X;R)}$

Individual evidence

1. ^ Roger Carter : Lie Algebras of Finite and Affine Type , Cambridge studies in advanced mathematics 96 (2005), ISBN 978-0-521-85138-1 , Chapter 9.3: Free Lie algebras
2. ^ Roger Carter: Lie Algebras of Finite and Affine Type , Cambridge studies in advanced mathematics 96 (2005), ISBN 978-0-521-85138-1 , sentence 9.9
3. ^ N. Bourbaki: Lie Groups and Lie Algebras: Chapters 1-3 , Springer Verlag (1989), ISBN 3-540-64242-0 , Chapter II, §2.2: Construction of the free Lie algebra
4. ^ N. Bourbaki: Lie Groups and Lie Algebras: Chapters 1-3 , Springer Verlag (1989), ISBN 3-540-64242-0 , Chapter II, §2.2, sentence 1
5. ^ N. Bourbaki: Lie Groups and Lie Algebras: Chapters 1-3 , Springer Verlag (1989), ISBN 3-540-64242-0 , Chapter II, §2.2, note on page 123
6. ^ Roger Carter: Lie Algebras of Finite and Affine Type , Cambridge studies in advanced mathematics 96 (2005), ISBN 978-0-521-85138-1 , sentence 9.10
7. ^ Roger Carter: Lie Algebras of Finite and Affine Type , Cambridge studies in advanced mathematics 96 (2005), ISBN 978-0-521-85138-1 , example 9.12
8. ↑ James E. Humphreys: Introduction to Lie Algebras and Representation Theory , Springer-Verlag (1972), ISBN 0-387-90052-7 , Chapter 18.3: Serre's Theorem
9. Jump up ↑ Roger Carter : Lie Algebras of Finite and Affine Type , Cambridge studies in advanced mathematics 96 (2005), ISBN 978-0-521-85138-1 , Chapter 14: Generalized Cartan matrices and Kac-Moody algebras