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Bar-invariant bases of the quantum cluster algebra of type A 2 ( 2 )

Xueqing Chen;  Ming Ding;  Jie Sheng — 2011

Czechoslovak Mathematical Journal

We construct bar-invariant ℤ [ q ± 1 / 2 ] -bases of the quantum cluster algebra of the valued quiver A 2 ( 2 ) , one of which coincides with the quantum analogue of the basis of the corresponding cluster algebra discussed in P. Sherman, A. Zelevinsky: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J., 4, 2004, 947–974.

Fermionic Novikov algebras admitting invariant non-degenerate symmetric bilinear forms

Zhiqi Chen;  Xueqing Chen;  Ming Ding — 2020

Czechoslovak Mathematical Journal

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. Fermionic Novikov algebras correspond to a certain Hamiltonian superoperator in a supervariable. In this paper, we show that fermionic Novikov algebras equipped with invariant non-degenerate symmetric bilinear forms are Novikov algebras.

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