Free Lie algebras and Witt's formula

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Problem 50 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Section I.3 "Categories and Functors"

Prove that there exists a universal object for the category $LA_n(K)$ of Lie algebras with $n$ distinguished generators over the field $K$ . The corresponding object is called the free Lie algebra with $n$ generators.

Remark.

An object $A$ of a category is called a universal repelling object (or, briefly, a universal object) if for every object $B$ there is exactly one morphism from $A$ to $B$ .

Problem 51 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Section I.3 "Categories and Functors"

Let $\mathfrak G$ be a Lie algebra over a field $K$ of characteristic zero. We consider the category $K(\mathfrak G)$ whose objects are those linear mappings $\varphi$ from $\mathfrak G$ into associative algebras (depending on the object) having the property that

\varphi ([x,y])=\varphi (x)\varphi (y)-\varphi (y)\varphi (x).

A morphism of an object $\varphi \colon \mathfrak G\to A$ to an object $\psi \colon \mathfrak G\to B$ is defined to be a homomorphism $\chi \colon A\to B$ which makes the following diagram commute:

Prove that $K(\mathfrak G)$ has a universal object $\varphi_0\colon \mathfrak G\to U(\mathfrak G)$ . The algebra $U(\mathfrak G)$ is called the associative hull or enveloping algebra of $\mathfrak G$ .

Remark.

An object $A$ of a category is called a universal repelling object (or, briefly, a universal object) if for every object $B$ there is exactly one morphism from $A$ to $B$ .

Problem 52 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Section I.3 "Categories and Functors"

Prove that the associative hull (see Problem 51) of the free Lie algebra with $n$ generators is isomorphic to the tensor algebra over an $n$ -dimensional space.

Remark.

An object $A$ of a category is called a universal repelling object (or, briefly, a universal object) if for every object $B$ there is exactly one morphism from $A$ to $B$ .

Problem III.5.2.9 from Кострикин А.И. "Основные структуры алгебры" [Кос04]More problems in Section III.5.2 "Finite fields"

Let $A(q)=\operatorname {Ass}_{F}\left(X_{1}, \ldots , X_{q}\right)$ be the free associative algebra over a field $F$ , generated by $q$ free generators — noncommuting variables $X_{1}, \ldots , X_{q}$ . Setting

A_{m}(q)= \left\langle X_{i_{1}} X_{i_{2}} \ldots X_{i_{m}} \mid 1 \leqslant i_{j} \leqslant q\right\rangle _{F}, \quad \operatorname {dim} A_{m}(q)=q^{m}

we note that $A(q)$ is a graded algebra:

A(q)=F \cdot 1 \oplus A_{1}(q) \oplus A_{2}(q) \oplus A_{3}(q) \oplus \ldots

$A(q)$ contains the free Lie algebra $L(q)=\operatorname {Lie}\left(X_{1}, \ldots , X_{q}\right)$ with the same free generators and the commutator operation $[U V]=U V-V U$ . The algebra $L(q)$ is also graded:

L(q)=L_{1}(q) \oplus L_{2}(q) \oplus L_{3}(q) \oplus \ldots

where

L_{1}(q)= \left\langle X_{1}, \ldots , X_{q}\right\rangle _{F}, \quad L_{2}(q)= \left\langle \left[X_{i}, X_{j}\right] \mid i<j\right\rangle _{F}, \quad \ldots

Using the Jacobi identity, verify that

L_{3}(q)= \left\langle \left[\left[X_{i}, X_{j}\right], X_{k}\right]\right\rangle , \quad i<j, k \leqslant j ; \quad \operatorname {dim} L_{3}(q)=\frac{1}{3}\left(q^{3}-q\right)

In fact, the general Witt formula holds

\operatorname {dim} L_{m}(q)=\Psi _{m}(q)=\frac{1}{m} \sum _{d \mid m} \mu \left(\frac{m}{d}\right) q^{d}

coinciding exactly with formula (12). The essential difference is only that in (12') $q$ is an arbitrary natural number, while in (12) it is a power of a prime number.

The number of necklaces from Problem 2 in the preamble to Ch. 3 is expressed by the same formula.

References

[KiGv88] Kirillov A.A., Gvishiani A.D. Теоремы и задачи функционального анализа. 2-е изд., Наука, 1988.

[Кос04] Кострикин А.И. Основные структуры алгебры. ФИЗМАТЛИТ, 2004.