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Register a new userNichols algebras over classical Weyl groups (II)
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Weicai Wu
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It is shown that except in three cases conjugacy classes of classical Weyl groups $W(B_{n})$ and $W(D_{n})$ are of type ${rm D}$. This proves that Nichols algebras of irreducible Yetter-Drinfeld modules over the classical Weyl groups $mathbb W_{n}$ (i.e. $H_{n}rtimes mathbb{S}_{n}$) are infinite dimensional, except the class of type $(2, 3),(1^{2}, 3)$ in $mathbb S_{5}$, and $(1^{n-2}, 2)$ in $mathbb S_{n}$ for $n >5$.
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We show that except in several cases conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type {rm D}. We prove that except in three cases Nichols algebras of irreducible Yetter-Drinfeld ({rm YD} in short )modules over the classical Weyl groups are infinite dimensional.
We show that except in several cases conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type {rm D}. We prove that except in three cases Nichols algebras of irreducible Yetter-Drinfeld ({rm YD} in short )modules over the classical Weyl groups are infinite dimensional. We establish the relationship between Fomin-Kirillov algebra $mathcal E_n$ and Nichols algebra $mathfrak{B} ({mathcal O}_{{(1, 2)}} , epsilon otimes {rm sgn})$ of transposition over symmetry group by means of quiver Hopf algebras. We generalize {rm FK } algebra. The characteristic of finiteness of Nichols algebras in thirteen ways and of {rm FK } algebras ${mathcal E}_n$ in nine ways is given. All irreducible representations of finite dimensional Nichols algebras %({rm FK } algebras ${mathcal E}_n$) and a complete set of hard super- letters of Nichols algebras of finite Cartan types are found. The sufficient and necessary condition for Nichols algebra $mathfrak B(M)$ of reducible {rm YD} module $M$ over $A rtimes mathbb{S}_n$ with ${rm supp } (M) subseteq A$ to be finite dimensional is given. % Some conditions for a braided vector space to become a {rm YD} module over finite commutative group are obtained. It is shown that hard braided Lie Lyndon word, standard Lyndon word, Lyndon basis path, hard Lie Lyndon word and standard Lie Lyndon word are the same with respect to $ mathfrak B(V)$, Cartan matrix $A_c$ and $U(L^+)$, respectively, where $V$ and $L$ correspond to the same finite Cartan matrix $A_c$.
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We construct a family of homomorphisms between Weyl modules for affine Lie algebras in characteristic p, which supports our conjecture on the strong linkage principle in this context. We also exhibit a large class of reducible Weyl modules beyond level one, for p not necessarily small.
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