6,937 research outputs found

### BLM realization for Frobenius--Lusztig Kernels of type A

The infinitesimal quantum $\frak{gl}_n$ was realized in \cite[\S 6]{BLM}. We
will realize Frobenius--Lusztig Kernels of type $A$ in this paper

### BLM realization for the integral form of quantum $\frak{gl}_n$

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

### BLM realization for ${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$

In 1990, Beilinson-Lusztig-MacPherson (BLM) discovered a realization
\cite[5.7]{BLM} for quantum $\frak{gl}_n$ via a geometric setting of quantum
Schur algebras. We will generailze their result to the classical affine case.
More precisely, we first use Ringel-Hall algebras to construct an integral form
${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$ of ${\mathcal
U}(\hat{\frak{gl}}_n)$, where ${\mathcal U}(\hat{\frak{gl}}_n)$ is the
universal enveloping algebra of the loop algebra
$\hat{\frak{gl}}_n:=\frak{gl}_n(\mathbb Q)\otimes\mathbb Q[t,t^{-1}]$. We then
establish the stabilization property of multiplication for the classical affine
Schur algebras. This stabilization property leads to the BLM realization of
${\mathcal U}(\hat{\frak{gl}}_n)$ and ${\mathcal U}_{\mathbb
Z}(\hat{\frak{gl}}_n)$. In particular, we conclude that ${\mathcal U}_{\mathbb
Z}(\hat{\frak{gl}}_n)$ is a $\mathbb Z$-Hopf subalgebra of ${\mathcal
U}(\hat{\frak{gl}}_n)$. As a bonus, this method leads to an explicit $\mathbb
Z$-basis for ${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$, and it yields
explicit multiplication formulas between generators and basis elements for
${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$. As an application, we will prove
that the natural algebra homomorphism from ${\mathcal U}_{\mathbb
Z}(\hat{\frak{gl}}_n)$ to the affine Schur algebra over $\mathbb Z$ is
surjective.Comment: 33 page

### Integral affine Schur-Weyl reciprocity

Let ${\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ be the double Ringel--Hall
algebra of the cyclic quiver $\triangle(n)$ and let $\dot{\boldsymbol{\mathfrak
D}_{\vartriangle}}(n)$ be the modified quantum affine algebra of
${\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$. We will construct an integral
form $\dot{{\mathfrak D}_{\vartriangle}}(n)$ for $\dot{\boldsymbol{\mathfrak
D}_{\vartriangle}}(n)$ such that the natural algebra homomorphism from
$\dot{{\mathfrak D}_{\vartriangle}}(n)$ to the integral affine quantum Schur
algebra is surjective. Furthermore, we will use Hall algebras to construct the
integral form ${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$ of the universal
enveloping algebra ${\mathcal U}(\hat{\frak{gl}}_n)$ of the loop algebra
$\hat{\frak{gl}}_n=\frak{gl}_n({\mathbb Q})\otimes\mathbb Q[t,t^{-1}]$, and
prove that the natural algebra homomorphism from ${\mathcal U}_\mathbb
Z(\hat{\frak{gl}}_n)$ to the affine Schur algebra over $\mathbb Z$ is
surjective.Comment: 20 page

### On the hyperalgebra of the loop algebra ${\widehat{\frak{gl}}_n}$

Let $\widetilde{\mathcal U}_{\mathbb Z}({\widehat{\frak{gl}}_n})$ be the
Garland integral form of ${\mathcal U}(\widehat{{\frak{gl}}}_n)$ introduced by
Garland \cite{Ga}, where ${\mathcal U}(\widehat{{\frak{gl}}}_n)$ is the
universal enveloping algebra of ${\widehat{{\frak{gl}}}_n}$. Using Ringel--Hall
algebras, a certain integral form ${\mathcal U}_{\mathbb
Z}(\widehat{{\frak{gl}}}_n)$ of ${\mathcal U}(\widehat{{\frak{gl}}}_n)$ was
constructed in \cite{Fu13}. We prove that the Garland integral form
$\widetilde{\mathcal U}_{\mathbb Z}({\widehat{{\frak{gl}}}_n})$ coincides with
${\mathcal U}_{\mathbb Z}(\widehat{{\frak{gl}}}_n)$. Let {\mathpzc k} be a
commutative ring with unity and let {\mathcal U}_{\mathpzc
k}(\widehat{{\frak{gl}}}_n)={\mathcal U}_{\mathbb
Z}(\widehat{{\frak{gl}}}_n)\otimes{\mathpzc k}. For $h\geq 1$, we use
Ringel--Hall algebras to construct a certain subalgebra, denoted by
${{\mathtt{u}}}_{\!\vartriangle\!}(n)_h$, of {\mathcal U}_{\mathpzc
k}(\widehat{{\frak{gl}}}_n). The algebra
${{\mathtt{u}}}_{\!\vartriangle\!}(n)_h$ is the affine analogue of
${\mathtt{u}}({{\frak{gl}}}_n)_h$, where ${\mathtt{u}}({{\frak{gl}}}_n)_h$ is a
certain subalgebra of the hyperalgebra associated with ${\frak{gl}}_n$
introduced by Humhpreys \cite{Hum}. The algebra
${\mathtt{u}}({{\frak{gl}}}_n)_h$ plays an important role in the modular
representation theory of ${\frak{gl}}_n$. In this paper we give a realization
of ${{\mathtt{u}}}_{\!\vartriangle\!}(n)_h$ using affine Schur algebras.Comment: 30 Page

### Affine quantum Schur algebras at roots of unity

We will classify finite dimensional irreducible modules for affine quantum
Schur algebras at roots of unity and generalize \cite[(6.5f) and (6.5g)]{Gr80}
to the affine case in this paper.Comment: 16 pages, Corollary 3.7 and subsection 4.5 were adde

### The comultiplication of modified quantum affine $\frak{sl}_n$

Let $\dot{\mathbf{U}}(\widehat{\frak{sl}}_n)$ be the modified quantum affine
$\frak{sl}_n$ and let ${\bf U}(\widehat{\frak{sl}}_N)^+$ be the positive part
of quantum affine $\frak{sl}_N$. Let $\dot{\mathbf{B}}(n)$ be the canonical
basis of $\dot{\mathbf{U}}(\widehat{\frak{sl}}_n)$ and let
$\mathbf{B}(N)^{\mathrm{ap}}$ be the canonical basis of ${\bf
U}(\widehat{\frak{sl}}_N)^+$. It is proved in \cite{FS} that each structure
constant for the multiplication with respect to $\dot{\mathbf{B}}(n)$ coincide
with a certain structure constant for the multiplication with respect to
$\mathbf{B}(N)^{\mathrm{ap}}$ for $n<N$. In this paper we use the theory of
affine quantum Schur algebras to prove that the structure constants for the
comultiplication with respect to $\dot{\mathbf{B}}(n)$ are determined by the
structure constants for the comultiplication with respect to
$\mathbf{B}(N)^{\mathrm{ap}}$ for $n<N$. In particular, the positivity property
for the comultiplication of $\dot{\mathbf{U}}(\widehat{\frak{sl}}_n)$ follows
from the positivity property for the comultiplication of ${\bf
U}(\widehat{\frak{sl}}_N)^+$.Comment: 10 page

### Dividing Line between Decidable PDA's and Undecidable Ones

Senizergues has proved that language equivalence is decidable for disjoint
epsilon-deterministic PDA. Stirling has showed that strong bisimilarity is
decidable for PDA. On the negative side Srba demonstrated that the weak
bisimilarity is undecidable for normed PDA. Later Jancar and Srba established
the undecidability of the weak bisimilarity for disjoint epsilon-pushing PDA
and disjoint epsilon-popping PDA. These decidability and undecidability results
are extended in the present paper. The extension is accomplished by looking at
the equivalence checking issue for the branching bisimilarity of several
variants of PDA.Comment: 26 pages, 9 figure

### Small Representations for Affine q-Schur Algebras

When the parameter $q\in\mathbb C^*$ is not a root of unity, simple modules
of affine $q$-Schur algebras have been classified in terms of Frenkel--Mukhin's
dominant Drinfeld polynomials (\cite[4.6.8]{DDF}). We compute these Drinfeld
polynomials associated with the simple modules of an affine $q$-Schur algebra
which come from the simple modules of the corresponding $q$-Schur algebra via
the evaluation maps.Comment: 21 Page

### Presenting affine Schur Algebras

The universal enveloping algebra ${\mathcal U}({\widehat{\frak{gl}}_n})$ of
${\widehat{\frak{gl}}_n}$ was realized in \cite[Ch. 6]{DDF} using affine Schur
algebras. In particular some explicit multiplication formulas in affine Schur
algebras were derived. We use these formulas to study the structure of affine
Schur algebras. In particular, we give a presentation of the affine Schur
algebra ${\mathcal S}_{{\!\vartriangle}}(n,r)_{\mathbb Q}$.Comment: 17 page

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