A characterization of Nichols algebras of diagonal type which are free algebras. (English) Zbl 1543.16029
A Nichols algebra can be defined as the quotient of the tensor algebra by the direct sum of the kernels of the so-called braided symmetrizers. In this paper the authors provide all the irreducible factors of the determinant of each braided symmetrizer. This improves a result in the paper [Algebr. Represent. Theory 4, No. 1, 55–76 (2001; Zbl 0983.17010)] by D. Flores de Chela and J. A. Green. The present approach is based on an inequality for the number of Lyndon words and on an identity for the shuffle map.
As an application, based on certain identities involving shuffle maps in [G. Duchamp et al., Discrete Math. Theor. Comput. Sci. 1, No. 2, 159–216 (1997; Zbl 0930.05097)], the authors characterize the freeness of Nichols algebras of diagonal type and determine the dimension of the kernel of the shuffle map. The freeness of Nichols algebras of diagonal type with braiding matrix \((q_{m_{ij}} )_{1\leq i,j\leq n}\), \(m_{ij} \in \mathbb Z\), is related to solutions of a Diophantine equation.
As an application, based on certain identities involving shuffle maps in [G. Duchamp et al., Discrete Math. Theor. Comput. Sci. 1, No. 2, 159–216 (1997; Zbl 0930.05097)], the authors characterize the freeness of Nichols algebras of diagonal type and determine the dimension of the kernel of the shuffle map. The freeness of Nichols algebras of diagonal type with braiding matrix \((q_{m_{ij}} )_{1\leq i,j\leq n}\), \(m_{ij} \in \mathbb Z\), is related to solutions of a Diophantine equation.
Reviewer: Sonia Natale (Córdoba)
MSC:
| 16T05 | Hopf algebras and their applications |
| 16T20 | Ring-theoretic aspects of quantum groups |
| 17B22 | Root systems |
References:
| [1] | If P m (q) = 0, then there is a non-trivial relation in B(V) of degree m. |
| [2] | If P l (q) = 0 for all l ≤ m with |l| ≥ 2, then there is no non-trivial relation in B(V) of degree m. |
| [3] | Proof. (1) Assume that P m (q) = 0. Then det(ρ m (S 1,m−1 )|V m ) = 0 by Lemma 4.2(1). Thus det(ρ m (S m )|V m ) = 0 by the definition of S m , and the claim follows from Equation (16). |
| [4] | Assume that the Nichols algebra B(V) has a non-trivial relation in degree m. Let l ≤ m be such that B(V) has a non-trivial relation in degree l and no non-trivial relation in any degree < l. Let l = |l|. Then l ≥ 2, ker(ρ l (S 1,l−2 )|V l ) = 0, and ker(ρ l (S 1,l−1 )|V l ) = 0 by (16) and by the definition of S l . Hence P l (q) = 0 by Lemma 4.2(2). This proves (2). |
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