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Register a new userSimple bounded weight modules of $displaystyle{mathfrak{sl}(infty)}$, $displaystyle{mathfrak{o}(infty)}$, $displaystyle{mathfrak{sp}(infty)}$
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We classify the simple bounded weight modules of ${mathfrak{sl}(infty})$, ${mathfrak{o}(infty)}$ and ${mathfrak{sp}(infty)}$, and compute their annihilators in $U({mathfrak{sl}(infty}))$, $U({mathfrak{o}(infty))}$, $U({mathfrak{sp}(infty))}$, respectively.
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We use analogues of Enrights and Arkhipovs functors to determine the quiver and relations for a category of $mathfrak{sl}_2 ltimes L(4)$-modules which are locally finite (and with finite multiplicities) over $mathfrak{sl}_2$. We also outline serious obstacles to extend our result to $mathfrak{sl}_2 ltimes L(k)$, for $k>4$.
Let $n>1$ be an integer, $alphain{mathbb C}^n$, $bin{mathbb C}$, and $V$ a $mathfrak{gl}_n$-module. We define a class of weight modules $F^alpha_{b}(V)$ over $sl_{n+1}$ using the restriction of modules of tensor fields over the Lie algebra of vector fields on $n$-dimensional torus. In this paper we consider the case $n=2$ and prove the irreducibility of such 5-parameter $mathfrak{sl}_{3}$-modules $F^alpha_{b}(V)$ generically. All such modules have infinite dimensional weight spaces and lie outside of the category of Gelfand-Tsetlin modules. Hence, this construction yields new families of irreducible $mathfrak{sl}_{3}$-modules.
With the aid of the exponentiation functor and Fourier transform we introduce a class of modules $T(g,V,S)$ of $mathfrak{sl} (n+1)$ of mixed tensor type. By varying the polynomial $g$, the $mathfrak{gl}(n)$-module $V$, and the set $S$, we obtain important classes of weight modules over the Cartan subalgebra $mathfrak h$ of $mathfrak{sl} (n+1)$, and modules that are free over $mathfrak h$. Furthermore, these modules are obtained through explicit presentation of the elements of $mathfrak{sl} (n+1)$ in terms of differential operators and lead to new tensor coherent families of $mathfrak{sl} (n+1)$. An isomorphism theorem and simplicity criterion for $T(g,V,S)$ is provided.
Kostants weight $q$-multiplicity formula is an alternating sum over a finite group known as the Weyl group, whose terms involve the $q$-analog of Kostants partition function. The $q$-analog of the partition function is a polynomial-valued function defined by $wp_q(xi)=sum_{i=0}^k c_i q^i$, where $c_i$ is the number of ways the weight $xi$ can be written as a sum of exactly $i$ positive roots of a Lie algebra $mathfrak{g}$. The evaluation of the $q$-multiplicity formula at $q = 1$ recovers the multiplicity of a weight in an irreducible highest weight representation of $mathfrak{g}$. In this paper, we specialize to the Lie algebra $mathfrak{sp}_6(mathbb{C})$ and we provide a closed formula for the $q$-analog of Kostants partition function, which extends recent results of Shahi, Refaghat, and Marefat. We also describe the supporting sets of the multiplicity formula (known as the Weyl alternation sets of $mathfrak{sp}_6(mathbb{C})$), and use these results to provide a closed formula for the $q$-multiplicity for any pair of dominant integral weights of $mathfrak{sp}_6(mathbb{C})$. Throughout this work, we provide code to facilitate these computations.
Let $mathfrak{a},mathfrak{b}$ be two ideals of a commutative noetherian ring $R$ and $M$ a finitely generated $R$-module.~We continue to study $textrm{f}textrm{-}mathrm{grad}_R(mathfrak{a},mathfrak{b},M)$ which was introduced in [Bull. Malays. Math. Sci. Soc. 38 (2015) 467--482], some computations and bounds of $textrm{f}textrm{-}mathrm{grad}_R(mathfrak{a},mathfrak{b},M)$ are provided.~We also give the structure of $(mathfrak{a},mathfrak{b})$-$mathrm{f}$-modules,~various properties which are analogous to those of Cohen Macaulay modules are discovered.
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