Integral bases for twisted multiloop algebras with diagram automorphism actions
We construct integral forms for the universal enveloping algebras of certain twisted multiloop algebras and explicit integral bases for these integral forms.
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arXiv subjects
Angelo Bianchi
Publications and source records attributed to Angelo Bianchi.
On the tame isotropy group of a derivation
We introduce the tame isotropy group of a derivation of a polynomial ring. We study this group for certain triangular derivations up to three variables, for simple derivations in two variables, and for simple Shamsuddin derivations in any polynomial ring.
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An Integral Basis for the Universal Enveloping Algebra of the Onsager Algebra
We construct an integral form for the universal enveloping algebra of the Onsager algebra and an explicit integral basis for this integral form. We also formulate straightening identities among some products of basis elements.
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Weyl modules and Weyl functors for hyper-map algebras
We investigate the representations of the hyperalgebras associated to the map algebras $\mathfrak g\otimes \mathcal A$, where $\mathfrak g$ is any finite-dimensional complex simple Lie algebra and $\mathcal A$ is any associative commutative unitary algebra with a multiplicatively closed basis. We consider the natural definition of the local and global Weyl modules, and the Weyl functor for these algebras. Under certain conditions, we prove that these modules satisfy certain universal properties, and we also give conditions for the local or global Weyl modules to be finite-dimensional or finitely generated, respectively.
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Demazure and local Weyl modules for twisted hyper current algebras
In this paper, we study local graded Weyl modules and Demazure modules for twisted hypercurrent algebras. We prove that local graded Weyl modules for a twisted hypercurrent algebra are isomorphic to the corresponding level 1 Demazure modules, and moreover, that they are restrictions of corresponding local graded Weyl modules for the untwisted hypercurrent algebra.
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Finite-dimensional representations of hyper multicurrent and multiloop algebras
We investigate the categories of finite-dimensional representations of multicurrent and multiloop hyperalgebras in positive characteristic, i.e., the hyperalgebras associated to the multicurrent algebras $\mathfrak g\otimes\mathbb{C}[t_1,\ldots,t_n]$ and to the multiloop algebras $\mathfrak g\otimes\mathbb{C}[t_1^{\pm1},\ldots,t_n^{\pm 1}]$, where $\mathfrak g$ is any finite-dimensional complex simple Lie algebra. The main results are the construction of the universal finite-dimensional highest-weight modules and a classification of irreducible modules in each category. In the characteristic zero setting we also provide a relationship between them.
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Identities and isomorphisms of finite-dimensional graded simple algebras
Let $\mathbb F$ be an algebraically closed field, $G$ be an abelian group, and let $A$ and $B$ be arbitrary finite-dimensional $G$-graded simple algebras over $\mathbb F$. We prove that $A$ and $B$ are isomorphic if, and only if, they satisfy the same graded polynomial identities.
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Bases for local Weyl modules for the hyper and truncated current $\mathfrak{sl}_2$-algebras
We use the theory of Gr\"obner-Shirshov bases for ideals to construct linear bases for graded local Weyl modules for the (hyper) current and the truncated current algebras associated to the finite-dimensional complex simple Lie algebra $\mathfrak{sl}_2$. The main result is a characteristic-free construction of bases for this important family of modules for the hyper current $\mathfrak{sl}_2$-algebra. In the positive characteristic setting this work represents the first construction in the literature. In the characteristic zero setting, the method provides a different construction of the Chari-Pressley (also Kus-Littelmann) bases and the Chari-Venkatesh bases for local Weyl modules for the current $\mathfrak{sl}_2$-algebra. Our construction allows us to obtain new bases for the local Weyl modules for truncated current $\mathfrak{sl}_2$-algebras with very particular properties.
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On Demazure and local Weyl modules for affine hyperalgebras
We establish the existence of Demazure flags for graded local Weyl modules for hyper current algebras in positive characteristic. If the underlying simple Lie algebra is simply laced, the flag has length one, i.e., the graded local Weyl modules are isomorphic to Demazure modules. This extends to the positive characteristic setting results of Fourier-Littelmann and Naoi for current algebras in characteristic zero. Using this result, we prove that the character of local Weyl modules for hyper loop algebras depend only on the highest weight, but not on the (algebraically closed) ground field, and deduce a tensor product factorization for them.
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Tilting Modules in Truncated Categories
We begin the study of a tilting theory in certain truncated categories of modules $\mathcal G(Γ)$ for the current Lie algebra associated to a finite-dimensional complex simple Lie algebra, where $Γ= P^+ \times J$, $J$ is an interval in $\mathbb Z$, and $P^+$ is the set of dominant integral weights of the simple Lie algebra. We use this to put a tilting theory on the category $\mathcal G(Γ')$ where $Γ' = P' \times J$, where $P'\subseteq P^+$ is saturated. Under certain natural conditions on $Γ'$, we note that $\mathcal G(Γ')$ admits full tilting modules.
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On multigraded generalizations of Kirillov-Reshetikhin modules
We study the category of Z^l-graded modules with finite-dimensional graded pieces for certain Z+^l-graded Lie algebras. We also consider certain Serre subcategories with finitely many isomorphism classes of simple objects. We construct projective resolutions for the simple modules in these categories and compute the Ext groups between simple modules. We show that the projective covers of the simple modules in these Serre subcategories can be regarded as multigraded generalizations of Kirillov-Reshetikhin modules and give a recursive formula for computing their graded characters.
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Finite-dimensional representations of twisted hyper loop algebras
We investigate the category of finite-dimensional representations of twisted hyper loop algebras, i.e., the hyperalgebras associated to twisted loop algebras over finite-dimensional simple Lie algebras. The main results are the classification of the irreducible modules, the definition of the universal highest-weight modules, called the Weyl modules, and, under a certain mild restriction on the characteristic of the ground field, a proof that the simple modules and the Weyl modules for the twisted hyper loop algebras are isomorphic to appropriate simple and Weyl modules for the non-twisted hyper loop algebras, respectively, via restriction of the action.
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