6,937 research outputs found

    BLM realization for Frobenius--Lusztig Kernels of type A

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

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

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

    BLM realization for UZ(gl^n){\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)

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    In 1990, Beilinson-Lusztig-MacPherson (BLM) discovered a realization \cite[5.7]{BLM} for quantum gln\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 UZ(gl^n){\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n) of U(gl^n){\mathcal U}(\hat{\frak{gl}}_n), where U(gl^n){\mathcal U}(\hat{\frak{gl}}_n) is the universal enveloping algebra of the loop algebra gl^n:=gln(Q)βŠ—Q[t,tβˆ’1]\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 U(gl^n){\mathcal U}(\hat{\frak{gl}}_n) and UZ(gl^n){\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n). In particular, we conclude that UZ(gl^n){\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n) is a Z\mathbb Z-Hopf subalgebra of U(gl^n){\mathcal U}(\hat{\frak{gl}}_n). As a bonus, this method leads to an explicit Z\mathbb Z-basis for UZ(gl^n){\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n), and it yields explicit multiplication formulas between generators and basis elements for UZ(gl^n){\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n). As an application, we will prove that the natural algebra homomorphism from UZ(gl^n){\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n) to the affine Schur algebra over Z\mathbb Z is surjective.Comment: 33 page

    Integral affine Schur-Weyl reciprocity

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    Let Dβ–³(n){\boldsymbol{\mathfrak D}_{\vartriangle}}(n) be the double Ringel--Hall algebra of the cyclic quiver β–³(n)\triangle(n) and let Dβ–³Λ™(n)\dot{\boldsymbol{\mathfrak D}_{\vartriangle}}(n) be the modified quantum affine algebra of Dβ–³(n){\boldsymbol{\mathfrak D}_{\vartriangle}}(n). We will construct an integral form Dβ–³Λ™(n)\dot{{\mathfrak D}_{\vartriangle}}(n) for Dβ–³Λ™(n)\dot{\boldsymbol{\mathfrak D}_{\vartriangle}}(n) such that the natural algebra homomorphism from Dβ–³Λ™(n)\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 UZ(gl^n){\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n) of the universal enveloping algebra U(gl^n){\mathcal U}(\hat{\frak{gl}}_n) of the loop algebra gl^n=gln(Q)βŠ—Q[t,tβˆ’1]\hat{\frak{gl}}_n=\frak{gl}_n({\mathbb Q})\otimes\mathbb Q[t,t^{-1}], and prove that the natural algebra homomorphism from UZ(gl^n){\mathcal U}_\mathbb Z(\hat{\frak{gl}}_n) to the affine Schur algebra over Z\mathbb Z is surjective.Comment: 20 page

    On the hyperalgebra of the loop algebra gl^n{\widehat{\frak{gl}}_n}

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

    Affine quantum Schur algebras at roots of unity

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    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 sln\frak{sl}_n

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

    Dividing Line between Decidable PDA's and Undecidable Ones

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    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

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    When the parameter q∈Cβˆ—q\in\mathbb C^* is not a root of unity, simple modules of affine qq-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 qq-Schur algebra which come from the simple modules of the corresponding qq-Schur algebra via the evaluation maps.Comment: 21 Page

    Presenting affine Schur Algebras

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    The universal enveloping algebra U(gl^n){\mathcal U}({\widehat{\frak{gl}}_n}) of gl^n{\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 S ⁣△(n,r)Q{\mathcal S}_{{\!\vartriangle}}(n,r)_{\mathbb Q}.Comment: 17 page
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