[Paper Review] Noncanonical Polynomial Representations of Classical Lie Algebras

Cuiling Luo|ArXiv.org|Apr 2, 2008

Algebraic structures and combinatorial models6 references3 citations

TL;DR

This paper constructs noncanonical polynomial representations of classical Lie algebras—specifically $\mathfrak{sl}(n,\mathbb{C})$, $\mathfrak{so}(n,\mathbb{C})$, and $\mathfrak{sp}(n,\mathbb{C})$—by exploiting skew-symmetry between differential and multiplication operators in canonical representations. It decomposes polynomial spaces into infinite-dimensional irreducible submodules and constructs explicit bases, yielding new infinite-dimensional irreducible modules for symplectic algebras that are not of highest weight type.

ABSTRACT

Using the skew-symmetry of the differential operators and multiplication operators in the canonical representations of finite-dimensional classical Lie algebras, we obtain some noncanonical polynomial representations of the classical Lie algebras. The representation spaces of all polynomials are decomposed into irreducible submodules, which are infinite-dimensional. Bases for the irreducible submodules are constructed. In particular, we obtain some new infinite-dimensional irreducible modules of symplectic Lie algebras that are not of highest weight type.

Motivation & Objective

• To extend polynomial representations of classical Lie algebras beyond canonical forms, particularly to infinite-dimensional irreducible modules.
• To address the lack of explicit bases and representation formulas for infinite-dimensional irreducible modules, especially non-highest weight types.
• To decompose polynomial representation spaces into irreducible submodules using skew-symmetry of differential and multiplication operators.
• To construct explicit bases for these irreducible submodules, especially for $\mathfrak{sp}(n,\mathbb{C})$ and $\mathfrak{so}(n,\mathbb{C})$.
• To provide new examples of infinite-dimensional irreducible modules for symplectic Lie algebras that are not of highest weight type.

Proposed method

• Utilizes skew-symmetry relations $[\partial_{x_i}, x_j] = \delta_{ij} = [-x_j, \partial_{x_i}]$ to define noncanonical representations of $\mathfrak{gl}(n,\mathbb{C})$.
• Restricts the noncanonical representation to subalgebras $\mathfrak{sl}(n,\mathbb{C})$, $\mathfrak{so}(n,\mathbb{C})$, and $\mathfrak{sp}(n,\mathbb{C})$ via permutation actions on indices.
• Introduces weighted polynomial spaces $\mathcal{A}_{\langle r\rangle}$ and $\mathcal{B}_{l_1,l_2}$ with integer grading to decompose representation spaces.
• Applies Xu’s work on flag partial differential equations to analyze harmonic submodules in the $\mathfrak{so}(n,\mathbb{C})$ case.
• Uses differential operators and integration techniques to construct singular vectors and bases for irreducible submodules.
• Employs operator identities involving $\eta = y_s\partial_{x_s} + v$ and $\Delta$ to derive explicit basis formulas for $\mathcal{H}_{l_1,l_2}$.

Experimental results

Research questions

• RQ1 Can noncanonical polynomial representations of classical Lie algebras be constructed such that both representation formulas and bases are explicitly given?
• RQ2 Are there infinite-dimensional irreducible modules for symplectic Lie algebras that are not of highest weight type?
• RQ3 How can the space of all polynomials be decomposed into irreducible submodules under noncanonical representations of $\mathfrak{so}(n,\mathbb{C})$ and $\mathfrak{sp}(n,\mathbb{C})$?
• RQ4 What are the explicit bases for irreducible submodules in noncanonical representations of $\mathfrak{sl}(n,\mathbb{C})$, $\mathfrak{so}(n,\mathbb{C})$, and $\mathfrak{sp}(n,\mathbb{C})$?
• RQ5 Can the structure of harmonic submodules $\mathcal{H}_{l_1,l_2}$ in $\mathcal{B} = \mathbb{C}[x_1,\dots,x_n,y_1,\dots,y_n]$ be fully characterized via explicit basis formulas?

Key findings

• The polynomial space $\mathcal{A}_{\langle r\rangle}$ is an infinite-dimensional irreducible highest weight module for $\mathfrak{sl}(n,\mathbb{C})$, with $x_m^r$ or $x_{m+1}^{-r}$ as a highest weight vector depending on the sign of $r$.
• For $\mathfrak{so}(n,\mathbb{C})$, the space $\mathcal{B}_{l_1,l_2}$ decomposes as $\mathcal{H}_{l_1,l_2} \oplus \eta \mathcal{B}_{l_1-1,l_2-1}$, where $\mathcal{H}_{l_1,l_2}$ is an irreducible highest weight module with a given basis.
• A basis for $\mathcal{H}_{l_1,l_2}$ is explicitly constructed using multinomial coefficients and differential operators, valid under conditions $l_1 + l_2 \leq -(n-s-1)$ or $l_1 > -(n-s-1)$.
• The highest weight of $\mathcal{H}_{l_1,l_2}$ is $-l_1\lambda_{s-1} + (l_1 - 1)\lambda_s + l_2\lambda_{n-1}$ when $l_1 + l_2 \leq n - s - 1$, with highest weight vector $x_s^{-l_1}y_n^{l_2}$.
• For $\mathfrak{sp}(n,\mathbb{C})$, the paper constructs new infinite-dimensional irreducible modules that are not of highest weight type, extending beyond known classification.
• The basis formula for $\mathcal{H}_{l_1,l_2}$ involves a sum over multi-indices with combinatorial coefficients and constraints $\alpha_n\beta_n = 0$, $\sum_{j=s+1}^n \alpha_j - \sum_{i=1}^s \alpha_i = l_1$, and $\sum_{i=1}^n \beta_i = l_2$.