[Paper Review] Representations of n-Lie algebras

Askar Dzhumadil’daev|ArXiv.org|Feb 5, 2002

Advanced Topics in Algebra3 citations

TL;DR

This paper classifies finite-dimensional representations of the n-Lie algebra $V_n$, the vector product algebra on $n+1$ generators over $ b{C} $. It proves that all such representations are completely reducible and that irreducible modules correspond precisely to highest weight $t ilde{ ho}$ for $t eq 0$, where $ ilde{ ho}$ is the fundamental weight of $so_{n+1}$, generalizing the $sl_2$-like structure of $V_2$. The key result is a complete characterization of irreducible $n$-Lie $V_n$-modules via highest weight $t ilde{ ho}$, with dimension formula $\frac{n+2t-1}{n+t-1}\binom{n+t-1}{t}$.

ABSTRACT

Let $V_n=$ be a vector products n-Lie algebra with n-Lie commutator $[e_1,...,\hat{e_i},...,e_{n+1}]=(-1)^ie_i$ over the field of complex numbers. Any finite-dimensional n-Lie $V_n$-module is completely reducible. Any finite-dimensional irreducible n-Lie $V_n$-module is isomorphic to a n-Lie extension of $so_{n+1}$-module with highest weight $tπ_1$ for some nonnegative integer t.

Motivation & Objective

• To classify all finite-dimensional representations of the $n$-Lie algebra $V_n$, the vector product algebra on $n+1$ generators over $\bb{C}$.
• To determine which irreducible $so_{n+1}$-modules can be lifted to $n$-Lie modules over $V_n$.
• To establish a complete structural analogy between $V_n$ and $sl_2$ in representation theory, with highest weight $t\pi_1$ as the key invariant.
• To provide a dimension formula for irreducible $n$-Lie $V_n$-modules and characterize their existence via the $q(M)$-invariant.

Proposed method

• Construct the basic Lie algebra $L(V_n) \cong so_{n+1}$ from the $n$-Lie algebra $V_n$ via the $\wedge^{n-1}$-construction.
• Define the universal enveloping algebra $U(L(V_n))$ and quotient by the ideal $Q(V_n)$ generated by $n$-Lie relations to obtain $U(V_n)$.
• Define $n$-Lie modules as $U(L(V_n))$-modules on which $Q(V_n)$ acts trivially, ensuring closure under semi-direct sum.
• Use the isomorphism $L(V_n) \cong so_{n+1}$ to reduce the classification of $n$-Lie $V_n$-modules to $so_{n+1}$-module theory.
• Apply the $q(M)$-invariant to classify which $so_{n+1}$-modules lift to $n$-Lie modules: only those with $q(M) = 1$ (i.e., $M \cong M(t\pi_1)$) do.
• Verify triviality of $Q(V_n)$-action on $M(t\pi_1)$ via explicit computation of the operator $R_{ijsk}$ on monomials in the polynomial realization of $M(t\pi_1)$.

Experimental results

Research questions

• RQ1 Which finite-dimensional $so_{n+1}$-modules can be lifted to $n$-Lie modules over the vector product $n$-Lie algebra $V_n$?
• RQ2 What is the complete classification of finite-dimensional irreducible $n$-Lie $V_n$-modules over $\bb{C}$?
• RQ3 How does the representation theory of $V_n$ compare to that of $sl_2$ for $n=2$, and what generalizes to $n>2$?
• RQ4 What is the dimension of an irreducible $n$-Lie $V_n$-module with highest weight $t\pi_1$?
• RQ5 What role does the $q(M)$-invariant play in determining whether an $so_{n+1}$-module lifts to a $V_n$-module?

Key findings

• Any finite-dimensional $n$-Lie $V_n$-module is completely reducible, as $L(V_n) \cong so_{n+1}$ is semisimple and Weyl's theorem applies.
• For $n=3$, an irreducible $so_4$-module $M_{t,r}$ lifts to a $V_3$-module if and only if $t = r$, i.e., it is symmetric in the two $sl_2$-factors.
• For $n > 3$, an irreducible $so_{n+1}$-module $M$ lifts to a $V_n$-module if and only if its highest weight is $t\pi_1$ for some nonnegative integer $t$.
• The dimension of an irreducible $n$-Lie $V_n$-module with highest weight $t\pi_1$ is $\frac{n+2t-1}{n+t-1}\binom{n+t-1}{t}$.
• For $n=3$, the dimension of an irreducible $V_3$-module with highest weight $t$ is $(t+1)^2$, matching the dimension of $M_{t,t}$.
• The $q(M)$-invariant detects non-liftable modules: if $q(M) > 1$, then $M$ cannot be a $V_n$-module, as shown by nontrivial action of $R_{ijsk}$ on monomials.