Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFactorizable Module Algebras
162
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The aim of this paper is to introduce and study a large class of $mathfrak{g}$-module algebras which we call factorizable by generalizing the Gauss factorization of (square or rectangular) matrices. This class includes coordinate algebras of corresponding reductive groups $G$, their parabolic subgroups, basic affine spaces and many others. It turns out that tensor products of factorizable algebras are also factorizable and it is easy to create a factorizable algebra out of virtually any $mathfrak{g}$-module algebra. We also have quant
rate research
Read More
For a symmetrizable GCM $C$ and its symmetrizer $D$, Geiss-Leclerc-Schroer [Invent. Math. 209 (2017)] has introduced a generalized preprojective algebra $Pi$ associated to $C$ and $D$, that contains a class of modules, called locally free modules. We show that any basic support $tau$-tilting $Pi$-module is locally free and gives a classification theorem of torsion-free classes in $operatorname{mathbf{rep}}{Pi}$ as the generalization of the work of Mizuno [Math. Z. 277 (2014)].
Let $U_q(mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let $mathcal{C}^0_{mathfrak{g}}$ be Hernandez-Leclercs category. For a duality datum $mathcal{D}$ in $mathcal{C}^0_{mathfrak{g}}$, we denote by $mathcal{F}_{mathcal{D}}$ the quantum affine Weyl-Schur duality functor. We give sufficient conditions for a duality datum $mathcal{D}$ to provide the functor $mathcal{F}_{mathcal{D}}$ sending simple modules to simple modules. Then we introduce the notion of cuspidal modules in $mathcal{C}^0_{mathfrak{g}}$, and show that all simple modules in $mathcal{C}^0_{mathfrak{g}}$ can be constructed as the heads of ordered tensor products of cuspidal modules.
Let $mathfrak{g}_0$ be a simple Lie algebra of type ADE and let $U_q(mathfrak{g})$ be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group $B(mathfrak{g}_0)$ on the quantum Grothendieck ring $K_t(mathfrak{g})$ of Hernandez-Leclercs category $C_{mathfrak{g}}^0$. Focused on the case of type $A_{N-1}$, we construct a family of monoidal autofunctors ${mathscr{S}_i}_{iin mathbb{Z}}$ on a localization $T_N$ of the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{infty}$. Under an isomorphism between the Grothendieck ring $K(T_N)$ of $T_N$ and the quantum Grothendieck ring $K_t({A^{(1)}_{N-1}})$, the functors ${mathscr{S}_i}_{1le ile N-1}$ recover the action of the braid group $B(A_{N-1})$. We investigate further properties of these functors.
Let $R := R_{2}(p)=mathbb{C}[t^{pm 1}, u : u^2 = t(t-alpha_1)cdots (t-alpha_{2n})] $ be the coordinate ring of a nonsingular hyperelliptic curve and let $mathfrak{g}otimes R$ be the corresponding current Lie algebra. color{black} Here $mathfrak g$ is a finite dimensional simple Lie algebra defined over $mathbb C$ and begin{equation*} p(t)= t(t-alpha_1)cdots (t-alpha_{2n})=sum_{k=1}^{2n+1}a_kt^k. end{equation*} In earlier work, Cox and Im gave a generator and relations description of the universal central extension of $mathfrak{g}otimes R$ in terms of certain families of polynomials $P_{k,i}$ and $Q_{k,i}$ and they described how the center $Omega_R/dR$ of this universal central extension decomposes into a direct sum of irreducible representations when the automorphism group was the cyclic group $C_{2k}$ or the dihedral group $D_{2k}$. We give examples of $2n$-tuples $(alpha_1,dots,alpha_{2n})$, which are the automorphism groups $mathbb G_n=text{Dic}_{n}$, $mathbb U_ncong D_n$ ($n$ odd), or $mathbb U_n$ ($n$ even) of the hyperelliptic curves begin{equation} S=mathbb{C}[t, u: u^2 = t(t-alpha_1)cdots (t-alpha_{2n})] end{equation} given in [CGLZ17]. In the work below, we describe this decomposition when the automorphism group is $mathbb U_n=D_n$, where $n$ is odd.
Motivated by the relation between Schur algebra and the group algebra of a symmetric group, along with other similar examples in algebraic Lie theory, Min Fang and Steffen Koenig addressed some behaviour of the endomorphism algebra of a generator over a symmetric algebra, which they called gendo-symmetric algebra. Continuing this line of works, we classify in this article the representation-finite gendo-symmetric algebras that have at most one isomorphism class of indecomposable non-injective projective module. We also determine their almost { u}-stable derived equivalence classes in the sense of Wei Hu and Changchang Xi. It turns out that a representative can be chosen as the quotient of a representation-finite symmetric algebra by the socle of a certain indecomposable projective module.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
