[Paper Review] Quasi-particle bases of principal subspaces for the affine Lie algebras of types $B_{l}^{(1)}$ and $C_{l}^{(1)}$

Marijana Butorac|arXiv (Cornell University)|May 3, 2015

Algebraic structures and combinatorial models16 references3 citations

TL;DR

This paper constructs quasi-particle bases for principal subspaces of standard and generalized Verma modules at level $k \geq 1$ for affine Lie algebras of types $B_l^{(1)}$ and $C_l^{(1)}$, generalizing earlier results for $B_2^{(1)}$. Using vertex operator relations and projection techniques, it derives explicit character formulas that yield new Rogers-Ramanujan-type identities for $l \geq 3$. The key contribution is a combinatorial description of the graded dimensions of these subspaces via energy conditions on quasi-particles of different colors and charges.

ABSTRACT

Generalizing our earlier work, we construct quasi-particle bases of principal subspaces of standard module $L_{X_l^{(1)}}(kΛ_0)$ and generalized Verma module $N_{X_l^{(1)}}(kΛ_0)$ at level $k\geq 1$ in the case of affine Lie algebras of types $B_l^{(1)}$ and $C_l^{(1)}$. As a consequence, from quasi-particle bases, we obtain the graded dimensions of these subspaces.

Motivation & Objective

• To extend the construction of quasi-particle bases from $B_2^{(1)}$ to higher-rank affine Lie algebras of types $B_l^{(1)}$ and $C_l^{(1)}$ for $l \geq 3$ at level $k \geq 1$.
• To establish character formulas for principal subspaces of generalized Verma modules $N_{X_l^{(1)}}(k\Lambda_0)$ and irreducible modules $L_{X_l^{(1)}}(k\Lambda_0)$.
• To derive new Rogers-Ramanujan-type identities by analyzing the graded dimensions obtained from the quasi-particle bases.
• To generalize the method of projection onto tensor products of $\mathfrak{h}$-weight subspaces used in prior work to prove linear independence of the spanning sets.

Proposed method

• Construct spanning sets of principal subspaces using quasi-particles $x_{r\alpha_i}(m)$ of color $i$, charge $r \geq 1$, and energy $-m$, defined via residue extraction from vertex operators.
• For $B_l^{(1)}$, the monomials are ordered as $b(\alpha_l)b(\alpha_{l-1})\cdots b(\alpha_1)$, with energy difference conditions matching those in $A_{l-1}^{(1)}$ for $i \leq l-2$ and $B_2^{(1)}$ for $i = l-1,l$.
• For $C_l^{(1)}$, the monomials are ordered as $b(\alpha_1)\cdots b(\alpha_{l-1})b(\alpha_l)$, with energy conditions for $l-1,l$ matching $B_2^{(1)}$ and for $i \leq l-2$ matching $A_{l-1}^{(1)}$ at level $2k$.
• Use a projection map onto tensor products of $\mathfrak{h}$-weight subspaces of standard modules to reduce linear independence to known results from $B_2^{(1)}$ and $A_{l-1}^{(1)}$ cases.
• Apply induction on the order of quasi-particle monomials, leveraging coefficients of intertwining operators, simple current operators, and Weyl group translation on level-one modules.
• Derive character formulas via Poincaré-Birkhoff-Witt theorem applied to a basis of $U(\mathcal{L}(\mathfrak{n}_+))_{<0}$, leading to infinite product expressions in terms of $q$-Pochhammer symbols.

Experimental results

Research questions

• RQ1 How can quasi-particle bases be systematically constructed for principal subspaces of $B_l^{(1)}$ and $C_l^{(1)}$ affine Lie algebras at level $k \geq 1$ for $l \geq 3$?
• RQ2 What are the energy difference conditions that govern the linear independence of quasi-particle monomials in these bases for $B_l^{(1)}$ and $C_l^{(1)}$?
• RQ3 Can the graded dimensions of principal subspaces for generalized Verma modules $N_{X_l^{(1)}}(k\Lambda_0)$ be expressed as infinite product identities?
• RQ4 Do these character formulas yield new Rogers-Ramanujan-type identities beyond those known for $A_1^{(1)}$ and $B_2^{(1)}$?
• RQ5 How can the linear independence of the spanning sets be proven using projections and known results from lower-rank cases?

Key findings

• The paper establishes a quasi-particle basis for the principal subspace of the generalized Verma module $N_{B_l^{(1)}}(k\Lambda_0)$, with monomials ordered as $b(\alpha_l)b(\alpha_{l-1})\cdots b(\alpha_1)$, where energy conditions for colors $1$ to $l-2$ match those in $A_{l-1}^{(1)}$ at level $k$, and for colors $l-1$ and $l$ match those in $B_2^{(1)}$ at level $k$.
• For $C_l^{(1)}$, the basis is ordered as $b(\alpha_1)\cdots b(\alpha_{l-1})b(\alpha_l)$, with energy conditions for colors $l-1$ and $l$ matching $B_2^{(1)}$ at level $k$, and for colors $1$ to $l-2$ matching $A_{l-1}^{(1)}$ at level $2k$.
• The character formula for the principal subspace of $N_{B_l^{(1)}}(k\Lambda_0)$ is given by an infinite product: $\prod_{m>0} \frac{1}{(1-q^m y_1)(1-q^m y_1 y_2)\cdots(1-q^m y_1\cdots y_{l-1})(1-q^m y_1 y_2^2 \cdots y_l^2)} \cdots \frac{1}{(1-q^m y_l)}$, with a corresponding sum over partitions with specific quadratic and bilinear terms in the exponent.
• Similarly, the character formula for $N_{C_l^{(1)}}(k\Lambda_0)$ is expressed as a product over $m>0$ of terms $\frac{1}{(1-q^m y_1)}\cdots\frac{1}{(1-q^m y_1 y_2^2 \cdots y_l^2)}\cdots\frac{1}{(1-q^m y_l)}$, matching the structure of the $B_l^{(1)}$ case but with different ordering and weight dependencies.
• The character formulas are proven to be equivalent to sums over partitions $r_i^{(j)}$ with specific exponents: $q^{\sum (r_i^{(j)})^2 - \sum r_{i-1}^{(j)} r_i^{(j)}}$ divided by $q$-Pochhammer symbols, yielding new identities of Rogers-Ramanujan type.
• The linear independence of the spanning sets is established via a projection technique that reduces the problem to known linear independence results in $B_2^{(1)}$ and $A_{l-1}^{(1)}$ cases, using coefficients of intertwining operators and Weyl group translation.