nLab quantum nilpotent algebra

Idea

Quantum nilpotent algebras are a class of associative algebras which are defined as iterated Ore extensions of a special type.

Definition

Let $K$ be a ground ring. An iterated Ore extension

$$K[x_1][x_2;\sigma_2,\delta_2][x_3;\sigma_3,\delta_3]\cdots [x_n;\sigma_n,\delta_n]$$

where $\sigma_j$ is an automorphism of

$$A_{j-1} = K[x_1][x_2;\sigma_2,\delta_2][x_3;\sigma_3,\delta_3]\cdots [x_{j-1};\sigma_{j-1},\delta_{j-1}]$$

and $\delta_j$ is the $\sigma_j$-derivation of $A_{j-1}$ is a quantum nilpotent algebra if it is equipped with a rational action of the torus $(K^\times)^l$ such that

• $x_1,\ldots,x_n$ are eigenvectors of the action

• each $\delta_j$ is a locally nilpotent $\sigma_j$-derivation of $A_{j-1}$

• For every $j\in\{1,\ldots,n\}$, there exists $h_j\in (K^\times)^l$ such that $(h_j\cdot )|_{A_{j-1}} = \sigma_j$ and $h_j\cdot x_j = q_j x_j$ for some $q_j\in K^\times$ which is not a root of unity.

It follows that $\sigma_j\circ\delta_j = q_j \delta_j\sigma_j$ (this was stated as a requirement in the original definition).

Properties

Quantum nilpotent algebras are a framework where the so called derivative elimination algorithm can be applied.

References

• K. R. Goodearl, M. T. Yakimov, Quantum cluster algebras and quantum nilpotent algebras, Proc. Natl. Acad. Sci. USA 111(27):9696–9703 (2014) doi
• S. Launois, T. H. Lenagan, B. M. Nolan, Total positivity is a quantum phenomenon: the Grassmannian case, Memoirs of the Amer. Math. Soc. 1448 (2023) 123 p.

category: algebra