Presenting Lusztig Algebras

Abstract

Abstract Let ${\textbf {U}}_{\!\vartriangle }(n)$ be the quantized enveloping algebra of $\widehat {\mathfrak {g}\mathfrak {l}}_n^{\prime}$ over ${\mathbb {Q}}(v)$, where $\widehat {\mathfrak {g}\mathfrak {l}}_n^{\prime}=\mathfrak {g}\mathfrak {l}_n\oplus \bigoplus _{l\in {\mathbb {Z}},\,l\not =0}\texttt {t}^l\mathfrak {s}\mathfrak {l}_n$. There is a natural algebra homomorphism $\zeta _r$ from ${\textbf {U}}_{\!\vartriangle }(n)$ to the affine quantum Schur algebra ${\boldsymbol {{\mathcal {S}}}}_{\!\vartriangle }(n,r)$. Let $ U_{\!\vartriangle }(n)_{{{\mathcal {Z}}}}$ be the Lusztig ${{\mathcal {Z}}}$-form of ${\textbf {U}}_{\!\vartriangle }(n)$, where ${{\mathcal {Z}}}={\mathbb {Z}}[v,v^{-1}]$. The algebras ${\textbf {U}}_{\!\vartriangle }(n,r):=\zeta _r({\textbf {U}}_{\!\vartriangle }(n))$ and $U_{\!\vartriangle }(n,r)_{{\mathcal {Z}}}:=\zeta _r( U_{\!\vartriangle }(n)_{{{\mathcal {Z}}}})$ are called Lusztig algebras. Let $\mathpzc {K}$ be a commutative ring containing an invertible element $\epsilon $. Let $U_{\!\vartriangle }(n,r)_{\mathpzc {K}}=U_{\!\vartriangle }(n,r)_{{\mathcal {Z}}}\otimes _{{\mathcal {Z}}}\mathpzc {K}$, where $\mathpzc {K}$ is regarded as a ${{\mathcal {Z}}}$-module by specializing $v$ to $\epsilon $. In this paper, we give a presentation for the Lusztig algebra $U_{\!\vartriangle }(n,r)_{\mathpzc {K}}$ for any commutative ring $\mathpzc {K}$.