[Paper Review] On universal Lie nilpotent associative algebras

Pavel Etingof, John Kim+1 more|ArXiv.org|May 13, 2008

Advanced Topics in Algebra3 references3 citations

TL;DR

This paper provides a complete description of the universal Lie nilpotent associative algebra $ Q_{n,4} $, showing it is isomorphic to the algebra of even polynomial differential forms on $ \mathbb{C}^n $ with a twisted product. The key contribution is the explicit structure of $ \Lambda_{n,4} $, the graded component of the associated graded algebra, which is shown to be a finite-dimensional $ GL(n) $-module, resolving a long-standing open problem for index 4. The work extends prior results on $ Q_{n,2} $ and $ Q_{n,3} $, and establishes a Lie algebra action on the associated graded quotients.

ABSTRACT

We study the quotient Q_i(A) of a free algebra A by the ideal M_i(A) generated by relation that the i-th commutator of any elements is zero. In particular, we completely describe such quotient for i=4 (for i<=3 this was done previously by Feigin and Shoikhet). We also study properties of the ideals M_i(A), e.g. when M_i(A)M_j(A) is contained in M_{i+j-1}(A) (by a result of Gupta and Levin, it is always contained in M_{i+j-2}(A)).

Motivation & Objective

• To determine the structure of the universal Lie nilpotent associative algebra $ Q_{n,4} $, which generalizes known results for $ Q_{n,2} $ and $ Q_{n,3} $.
• To understand the graded components $ M_{n,i}/M_{n,i+1} $ of the Lie filtration on the free associative algebra $ A_n $, particularly for $ i=4 $.
• To answer Question 2.3 on the $ GL(n) $-module structure of $ \Lambda_{n,i} $, which was previously only known for $ i=2,3 $.
• To establish that the Lie algebra $ \mathfrak{g}_n $ of derivations modulo $ M_{n,3} $ acts on the associated graded algebra $ \overline{A} $, and to describe the representation theory of these quotients.

Proposed method

• The authors use the Lie filtration on the free associative algebra $ A_n $, inducing a filtration on $ Q_{n,i} $, and study the associated graded algebra $ D_{n,i} = \mathrm{gr} Q_{n,i} $, which is commutative.
• They decompose $ D_{n,i} $ as $ \mathbb{C}[x_1,\dots,x_n] \otimes \Lambda_{n,i} $, where $ \Lambda_{n,i} $ is the image of $ \mathrm{Sym}(\ell_n') $, and prove $ \theta: \mathbb{C}[x_1,\dots,x_n] \otimes \Lambda_{n,i} \to D_{n,i} $ is an isomorphism.
• They apply Jennings' theorem to show $ \Lambda_{n,i} $ is finite-dimensional, using the nilpotency of $ M_2(Q_{n,i}) $.
• They use $ GL(n) $-equivariant arguments and derivations to analyze the structure of $ M_{n,i}/M_{n,i+1} $, showing that $ \mathrm{Der}(A_n) $ acts through $ \mathfrak{g}_n $, and that $ \mathfrak{g}_n $ preserves the grading and product on $ \overline{A} $.
• They prove that $ M_{n,3}/M_{n,4} \cong \mathcal{F}_{2,1,0,\dots,0} \oplus \mathcal{F}_{2,2,0,\dots,0} $ as $ W_n $-modules, using subrepresentation arguments and trace considerations.
• They use computational tools (MAGMA) and automorphisms $ g_i^t $ to derive contradictions in polynomial relations, proving the isomorphism $ \theta $.

Experimental results

Research questions

• RQ1 What is the $ GL(n) $-module structure of $ \Lambda_{n,4} $, the graded component of $ Q_{n,4} $?
• RQ2 How do the quotients $ M_{n,i}/M_{n,i+1} $ decompose as representations of $ \mathfrak{g}_n $, especially for $ i=3 $?
• RQ3 Can the action of $ \mathrm{Der}(A_n) $ on $ M_{n,i}/M_{n,i+1} $ be factored through $ \mathfrak{g}_n $, and what is its structure?
• RQ4 What is the explicit structure of $ Q_{n,4} $, and how does it relate to known algebras like $ Q_{n,3} $?
• RQ5 Is $ \theta: \mathbb{C}[x_1,\dots,x_n] \otimes \Lambda_{n,i} \to D_{n,i} $ an isomorphism for $ i=4 $?

Key findings

• The algebra $ Q_{n,4} $ is isomorphic to the algebra of even polynomial differential forms on $ \mathbb{C}^n $, with product $ a*b = ab + da \wedge db $, extending the known result for $ Q_{n,3} $.
• The map $ \theta: \mathbb{C}[x_1,\dots,x_n] \otimes \Lambda_{n,4} \to D_{n,4} $ is an isomorphism, proving that $ D_{n,4} \cong \mathbb{C}[x_1,\dots,x_n] \otimes \Lambda_{n,4} $.
• The algebra $ \Lambda_{n,4} $ is a finite-dimensional $ GL(n) $-module, with $ \Lambda_{n,4}[0] = \mathbb{C} $, and is generated in degrees less than 4 in the derived Lie algebra $ \ell_n' $.
• The quotient $ M_{n,3}/M_{n,4} $ decomposes as $ \mathcal{F}_{2,1,0,\dots,0} \oplus \mathcal{F}_{2,2,0,\dots,0} $ as $ W_n $-modules, with explicit subrepresentations identified.
• The action of $ \mathrm{Der}(A_n) $ on $ M_{n,i}/M_{n,i+1} $ factors through $ \mathfrak{g}_n $, and $ \mathfrak{g}_n $ acts on the graded algebra $ \overline{A} $ preserving grading and product.
• The structure of $ M_{n,i}/M_{n,i+1} $ for $ i>3 $ remains open, though the framework for studying it is established.