The proposed Ext-algebra presentation for logarithmic model modules

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Fix ss with 1 ⩽ s ⩽ p−11\,{\leqslant}\, s\,{\leqslant}\, p-1. Let Exts∙\mathrm{Ext}^\bullet_s be the associative Yoneda algebra formed from the extension spaces between the specified irreducible modules, and let xr,i±x^{\pm}_{r,i}, with i∈{0,1}i\in\{0,1\} and r ⩾ 1r\,{\geqslant}\, 1, be the chosen generators. Ext-algebra presentation conjecture. The algebra Exts∙\mathrm{Ext}^\bullet_s is generated by xr,i±x^{\pm}_{r,i} with defining relations

xr,i+xr′,j+=xr,i−xr′,j−=x1,1−x1,0+=0,x^+_{r,i}x^+_{r',j}=x^-_{r,i}x^-_{r',j}=x^-_{1,1}x^+_{1,0}=0, xr,0+xr,1−+xr+1,1+xr,0−=0,xr,0−xr+1,1++xr+1,1−xr+1,0+=0.x^+_{r,0}x^-_{r,1}+x^+_{r+1,1}x^-_{r,0}=0,\qquad x^-_{r,0}x^+_{r+1,1}+x^-_{r+1,1}x^+_{r+1,0}=0.

Here i,j∈{0,1}i,j\in\{0,1\} and r,r′ ⩾ 1r,r'\,{\geqslant}\, 1. This is presented as a proposed algebraic structure for the extension algebra, and the supplied text gives no evidence of resolution.

References

Primary source

P. V. Bushlanov, A. M. Gainutdinov and I. Yu. Tipunin, “Kazhdan-Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models”, arXiv:1102.0271 (2011).

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