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Register a new userTensor product weight representations of the Neveu-Schwarz algebra
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Authors
Xiufu Zhang
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In this paper, the tensor product of highest weight modules with intermediate series modules over the Neveu-Schwarz algebra is studied. The weight spaces of such tensor products are all infinitely dimensional if the highest weight module is nontrivial. We find that all such tensor products are indecomposable. We give the necessary and sufficient conditions for these tensor product modules to be irreducible by using shifting technique established for the Virasoro case in [13]. The necessary and sufficient conditions for any two such tensor products to be isomorphic are also determined.
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We show that the support of a simple weight module over the Neveu-Schwarz algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all non-trivial weight spaces of such module are infinite-dimensional. As a corollary we obtain that every simple weight module over the Neveu-Schwarz algebra, having a non-trivial finite-dimensional weight space, is a Harish-Chandra module (and hence is either a highest or lowest weight module, or else a module of the intermediate series). This result generalizes a theorem which was originally given on the Virasoro algebra.
For the Drinfeld double $D_n$ of the Taft algebra $A_n$ defined over an algebraically closed field $mathbb k$ of characteristic zero using a primitive $n$th root of unity $q in mathbb k$ for $n$ odd, $nge3$, we determine the ribbon element of $D_n$ explicitly. We use the R-matrix and ribbon element of $D_n$ to construct an action of the Temperley-Lieb algebra $mathsf{TL}_k(xi)$ with $xi = -(q^{frac{1}{2}}+q^{-frac{1}{2}})$ on the $k$-fold tensor product $V^{otimes k}$ of any two-dimensional simple $D_n$-module $V$. When $V$ is the unique self-dual two-dimensional simple module, we develop a diagrammatic algorithm for computing the $mathsf{TL}_k(xi)$-action. We show that this action on $V^{otimes k}$ is faithful for arbitrary $k ge 1$ and that $mathsf{TL}_k(xi)$ is isomorphic to the centralizer algebra $text{End}_{D_n}(V^{otimes k})$ for $1 le kle 2n-2$.
We classify all Rota-Baxter operators of nonzero weight on the matrix algebra of order three over an algebraically closed field of characteristic zero which are not arisen from the decompositions of the entire algebra into a direct vector space sum of two subalgebras.
In this paper, we establish Composition-Diamond lemma for tensor product $k< X> otimes k< Y>$ of two free algebras over a field. As an application, we construct a Groebner-Shirshov basis in $k< X> otimes k< Y>$ by lifting a Groebner-Shirshov basis in $k[X] otimes k< Y>$, where $k[X]$ is a commutative algebra.
In this paper, conjugate-linear anti-involutions and unitary Harish-Chandra modules over the Schr{o}dinger-Virasoro algebra are studied. It is proved that there are only two classes conjugate-linear anti-involutions over the Schr{o}dinger-Virasoro algebra. The main result of this paper is that a unitary Harish-Chandra module over the Schr{o}dinger-Virasoro algebra is simply a unitary Harish-Chandra module over the Virasoro algebra.
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