On degenerations of Lie superalgebras

The Variety of Nilpotent Tortkara Algebras

“…We therefore have all non-split 1-dimensional central extensions with 1-dimensional annihilator of T 4 02 : T 5 08 : (T 4 02 ) 5,1 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , T 5 09 : (T 4 02 ) 5,2 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 3 = e 5 , T 5 10 : (T 4 02 ) 5,3 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 2 e 4 = e 5 .…”

“…Then, A is isomorphic to exactly one of the following algebras: T 5 01 : e 1 e 2 = e 3 , T 5 02 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , T 5 03 : e 1 e 2 = e 4 , e 1 e 3 = e 5 . T 5 04 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 2 e 3 = e 5 , T 5 05 : e 1 e 2 = e 5 , e 3 e 4 = e 5 , T 5 06 : e 1 e 2 = e 3 , e 1 e 4 = e 5 , e 2 e 3 = e 5 , T 5 07 : e 1 e 2 = e 3 , e 3 e 4 = e 5 , T 5 08 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , T 5 09 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 3 = e 5 , T 5 10 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 2 e 4 = e 5 .…”

“…Theorem 2.1. The variety of 5-dimensional nilpotent Tortkara algebras has one irreducible component defined by the rigid algebra T 5 10 . The graph of primary degenerations for 5-dimensional nilpotent Tortkara algebras has the following form:…”

The geometric classification of nilpotent Tortkara algebras

“…There are fewer works in which the full information about degenerations has been found for some variety of algebras. This problem has been solved for 2-dimensional pre-Lie algebras in [7], for 2-dimensional terminal algebras in [11], for 3-dimensional Novikov algebras in [8], for 3-dimensional Jordan algebras in [20], for 3-dimensional Jordan superalgebras in [6], for 3-dimensional Leibniz algebras in [25], for 3-dimensional anticommutative algebras in [25], for 4-dimensional Lie algebras in [10], for 4-dimensional Lie superalgebras in [5], for 4-dimensional Zinbiel algebras in [28], for 3-dimensional nilpotent algebras [17], for 4-dimensional nilpotent Leibniz algebras in [28], for 4-dimensional nilpotent commutative algebras [17], for 5-dimensional nilpotent Tortkara algebras in [19], for 5-dimensional nilpotent anticommutative algebras in [17], for 6-dimensional nilpotent Lie algebras in [21,34], for 6-dimensional nilpotent Malcev algebras in [29], for 7-dimensional 2-step nilpotent Lie algebras in [4], and for all 2dimensional algebras in [30].…”

Degenerations of nilpotent associative commutative algebras

“…There are fewer works in which the full information about degenerations was given for some variety of algebras. This problem was solved for 2-dimensional pre-Lie algebras [6], for 2-dimensional terminal algebras [9], for 3-dimensional Novikov algebras [7], for 3dimensional Jordan algebras [15], for 3-dimensional Jordan superalgebras [5], for 3-dimensional Leibniz and 3-dimensional anticommutative algebras [20], for 4-dimensional Lie algebras [8], for 4-dimensional Lie superalgebras [4], for 4-dimensional Zinbiel and 4-dimensional nilpotent Leibniz algebras [23], for 5-dimensional nilpotent Tortkara algebras [14], for 6-dimensional nilpotent Lie algebras [16,30], for 6dimensional nilpotent Malcev algebras [24], for 7-dimensional 2-step nilpotent Lie algebras [3], and for all 2-dimensional algebras [25]. Here we construct the graphs of primary degenerations for the variety of complex 5-dimensional nilpotent associative commutative algebras.…”