[Paper Review] BLM realization for the integral form of quantum $\frak{gl}_n$

Qiang Fu|arXiv (Cornell University)|Nov 4, 2012

Algebraic structures and combinatorial models11 references3 citations

TL;DR

This paper establishes a BLM-type realization of the integral form $U(n)$ of quantum $rak{gl}_n$ using $q$-Schur algebras over $bZ[v,v^{-1}]$, constructing a $bZ$-subalgebra $V(n)$ of the product of $q$-Schur algebras that is isomorphic to $U(n)$. It further extends this to a realization of the specialized algebra $ar{U}_k(n)$ over a field $k$ containing a primitive $l$-th root of unity, with a conjecture on the affine version $V_{\triangle}(n)$.

ABSTRACT

Let ${\mathbf U}(n)$ be the quantum enveloping algebra of ${\frak {gl}}_n$ over $\mathbb Q(v)$, where $v$ is an indeterminate. We will use $q$-Schur algebras to realize the integral form of ${\mathbf U}(n)$. Furthermore we will use this result to realize quantum $\frak{gl}_n$ over $k$, where $k$ is a field containing an l-th primitive root $\varepsilon$ of 1 with $l\geq 1$ odd.

Motivation & Objective

• To construct a BLM realization of the integral form $U(n)$ of quantum $rak{gl}_n$ using $q$-Schur algebras over $bZ[v,v^{-1}]$.
• To extend this realization to the specialized algebra $ar{U}_k(n)$ over a field $k$ containing a primitive $l$-th root of unity, with $l \geq 1$ odd.
• To conjecture that the affine version $V_{\triangle}(n)$ is a $bZ$-subalgebra of the product of affine $q$-Schur algebras, and thus isomorphic to a certain $bZ$-module $ ilde{rak{D}}_{\triangle}(n)$.

Proposed method

• Construct a $bZ$-submodule $V(n)$ of $\prod_{r\geq 0} \bS(n,r)$, where $\bS(n,r)$ is the $q$-Schur algebra over $\bbQ(v)$, using generators $A(\delta,\lambda)$ defined via $q$-binomial coefficients and matrix actions.
• Prove that $\fV(n)$ is a $\bbZ$-subalgebra of $\prod_{r\geq 0} \bS(n,r)$ by establishing multiplication formulas in sections 3.4 and 3.5.
• Use the basis $\{A(\delta,\lambda) \mid A \in \Theta_{\triangle}^{\pm}(n), \delta, \lambda \in \bbN_{\triangle}^n, \delta_i \in \{0,1\}\}$ to show that $\fV(n)$ is isomorphic to $U(n)$ as a $\bbZ$-algebra.
• Define the affine version $\fV_{\triangle}(n)$ as the $\bbZ$-submodule of $\prod_{r\geq 0} \bS_{\triangle}(n,r)$ spanned by $A(\delta,\lambda)$, and conjecture it is a $\bbZ$-subalgebra.
• Specialize $v$ to a primitive $l$-th root of unity $\varepsilon$ in $k$, and realize $\bar{U}_k(n) = U(n) \otimes_{\bbZ[v,v^{-1}]} k / \langle K_i^l - 1 \rangle$ as a $k$-subalgebra of $\prod_{r\geq 0} \cS_k(n,r)$.
• Leverage known results on $q$-Schur algebras and their bases to ensure integrality and compatibility under specialization.

Experimental results

Research questions

• RQ1 Can the integral form $U(n)$ of quantum $\frak{gl}_n$ be realized as a $\bbZ$-subalgebra of a product of $q$-Schur algebras via a BLM-type construction?
• RQ2 Is the affine version $\fV_{\triangle}(n)$ of this construction a $\bbZ$-subalgebra of $\prod_{r\geq 0} \bS_{\triangle}(n,r)$, as conjectured?
• RQ3 Does the specialized algebra $\bar{U}_k(n)$ over a field $k$ containing a primitive $l$-th root of unity admit a realization as a $k$-subalgebra of the product of $q$-Schur algebras over $k$?
• RQ4 Is the $\bbZ$-basis of $\fV(n)$, constructed from $A(\delta,\lambda)$ with $\delta_i \in \{0,1\}$, preserved under multiplication and closed under the algebra structure?
• RQ5 Does the conjecture that $\fV_{\triangle}(n)$ is a $\bbZ$-subalgebra imply isomorphism with the $\bbZ$-module $\tilde{\frak{D}}_{\triangle}(n)$ defined in [2, (3.8.1.1])?

Key findings

• The $\bbZ$-submodule $\fV(n)$ of $\prod_{r\geq 0} \bS(n,r)$ is constructed as the span of elements $A(\delta,\lambda)$, and is proven to be a $\bbZ$-subalgebra.
• The algebra $\fV(n)$ is isomorphic to the integral form $U(n)$ of quantum $\frak{gl}_n$ as a $\bbZ$-algebra.
• A $\bbZ$-basis for $\fV(n)$ is explicitly given by $\{A(\delta,\lambda) \mid A \in \Theta_{\triangle}^{\pm}(n), \delta, \lambda \in \bbN_{\triangle}^n, \delta_i \in \{0,1\}\}$.
• The specialized algebra $\bar{U}_k(n)$ is realized as a $k$-subalgebra of $\prod_{r\geq 0} \cS_k(n,r)$, where $\cS_k(n,r)$ is the $q$-Schur algebra over $k$.
• The conjecture that $\fV_{\triangle}(n)$ is a $\bbZ$-subalgebra of $\prod_{r\geq 0} \bS_{\triangle}(n,r)$ is proposed, and if true, would imply isomorphism with $\tilde{\frak{D}}_{\triangle}(n)$.
• The construction relies on multiplication formulas for $q$-Schur algebras established in sections 3.4 and 3.5, which are essential for proving the algebra structure of $\fV(n)$.