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Register a new userMorita equivalences on Brauer algebras and BMW algebras of simply-laced types
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Authors
Shoumin Liu
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The Morita equivalences of classical Brauer algebras and classical Birman-Murakami-Wenzl algebras have been well studied. Here we study the Morita equivalence problems on these two kinds of algebras of simply-laced type, especially for them with the generic parameters. We show that Brauer algebras and Birman-Murakami-Wenzl algebras of simply-laced type are Morita equivalent to the direct sums of some group algebras of Coxeter groups and some Hecke algebras of some Coxeter groups, respectively.
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It is known that the recently discovered representations of the Artin groups of type A_n, the braid groups, can be constructed via BMW algebras. We introduce similar algebras of type D_n and E_n which also lead to the newly found faithful representations of the Artin groups of the corresponding types. We establish finite dimensionality of these algebras. Moreover, they have ideals I_1 and I_2 with I_2 contained in I_1 such that the quotient with respect to I_1 is the Hecke algebra and I_1/I_2 is a module for the corresponding Artin group generalizing the Lawrence-Krammer representation. Finally we give conjectures on the structure, the dimension and parabolic subalgebras of the BMW algebra, as well as on a generalization of deformations to Brauer algebras for simply laced spherical type other than A_n.
In this paper, we will define the Brauer algebras of Weyl types, and describe some propositions of these algebras. Especially, we prove the result of type $G_2$ to accomplish our project of Brauer algebras of non-simply laced types.
Let $U_q(mathfrak{g})$ be a quantum affine algebra with an indeterminate $q$ and let $mathscr{C}_{mathfrak{g}}$ be the category of finite-dimensional integrable $U_q(mathfrak{g})$-modules. We write $mathscr{C}_{mathfrak{g}}^0$ for the monoidal subcategory of $mathscr{C}_{mathfrak{g}}$ introduced by Hernandez-Leclerc. In this paper, we associate a simply-laced finite type root system to each quantum affine algebra $U_q(mathfrak{g})$ in a natural way, and show that the block decompositions of $mathscr{C}_{mathfrak{g}}$ and $mathscr{C}_{mathfrak{g}}^0$ are parameterized by the lattices associated with the root system. We first define a certain abelian group $mathcal{W}$ (resp. $mathcal{W}_0$) arising from simple modules of $ mathscr{C}_{mathfrak{g}}$ (resp. $mathscr{C}_{mathfrak{g}}^0$) by using the invariant $Lambda^infty$ introduced in the previous work by the authors. The groups $mathcal{W}$ and $mathcal{W}_0$ have the subsets $Delta$ and $Delta_0$ determined by the fundamental representations in $ mathscr{C}_{mathfrak{g}}$ and $mathscr{C}_{mathfrak{g}}^0$ respectively. We prove that the pair $( mathbb{R} otimes_mathbb{Z} mathcal{W}_0, Delta_0)$ is an irreducible simply-laced root system of finite type and the pair $( mathbb{R} otimes_mathbb{Z} mathcal{W}, Delta) $ is isomorphic to the direct sum of infinite copies of $( mathbb{R} otimes_mathbb{Z} mathcal{W}_0, Delta_0)$ as a root system.
Hecke-Hopf algebras were defined by A. Berenstein and D. Kazhdan. We give an explicit presentation of an Hecke-Hopf algebra when the parameter $m_{ij},$ associated to any two distinct vertices $i$ and $j$ in the presentation of a Coxeter group, equals $4,$ $5$ or $6$. As an application, we give a proof of a conjecture of Berenstein and Kazhdan when the Coxeter group is crystallographic and non-simply-laced. As another application, we show that another conjecture of Berenstein and Kazhdan holds when $m_{ij},$ associated to any two distinct vertices $i$ and $j,$ equals $4$ and that the conjecture does not hold when some $m_{ij}$ equals $6$ by giving a counterexample to it.
Let $Lambda$ be a finite-dimensional algebra over a fixed algebraically closed field $mathbf{k}$ of arbitrary characteristic, and let $V$ be a finitely generated $Lambda$-module. It follows from results previously obtained by F.M. Bleher and the third author that $V$ has a well-defined versal deformation ring $R(Lambda, V)$, which is a complete local commutative Noetherian $mathbf{k}$-algebra with residue field $mathbf{k}$. The third author also proved that if $Lambda$ is a Gorenstein $mathbf{k}$-algebra and $V$ is a Cohen-Macaulay $Lambda$-module whose stable endomorphism ring is isomorphic to $mathbf{k}$, then $R(Lambda, V)$ is universal. In this article we prove that the isomorphism class of a versal deformation ring is preserved under singular equivalence of Morita type between Gorenstein $mathbf{k}$-algebras.
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