Some properties of c-covers of a pair of Lie algebras

On the $c$-nilpotent multiplier of a pair of Lie algebras

“…We prove next some results concerning perfect pairs of Lie algebras. In [3], it is shown that any perfect pair of Lie algebras has at least one c-covering pair. In the following theorem, we prove the converse of this result.…”

“…In [3] and [14], the first two authors characterized the structure of the cnilpotent multiplier of a pair of Lie algebras in terms of its c-covering pairs, and proved that every nilpotent pair of Lie algebras of class at most k with non-trivial c-nilpotent multiplier does not admit any c-covering pair if c > k. Moreover, they proved under some conditions that a relative c-central extension of a pair of Lie algebras is a homomorphic image of a c-covering pair, and showed that a c-cover of a pair of finite dimensional Lie algebras, under some assumptions, has a unique domain up to isomorphism, and also that any perfect pair of Lie algebras has at least one c-covering pair. Finally, some conditions are discussed under which every two c-covering pairs are c-isoclinic.…”

On multipliers of pairs of Lie superalgebras

“…Clearly, (i) means that |[x, y]| = |x| + |y| (modulo 2). Moreover, since 1 2 , 1 3 ∈ F, (ii) implies that [x, x] = 0 for all x ∈ L 0 , and (iii) implies that [x, [x, x]] = 0 for all x ∈ L. Hence the even part L 0 of a Lie superalgebra L is actually a Lie algebra, which means that if L 1 = 0, then L becomes a usual Lie algebra. Also, the odd part L 1 is an L 0 -module.…”

“…In 1996, Batten et al [3] discussed and studied the concept of the Schur multiplier of a Lie algebra, which is analogous to the Schur multiplier of a group introduced by Schur [25] in 1904. Moneyhun [17] proved for an m-dimensional Lie algebra L that dim M(L) ≤ 1 2 m(m − 1), where M(L) denotes the Schur multiplier of L. Also in [19], Saeedi et al generalized the Moneyhun's result to a pair of Lie algebras and proved that if (N, L) is a pair of Lie algebras in which N admits a complement in L and dim N = m, then dim M(N, L) ≤ 1 2 m m + 2 dim(L/N) − 1 . Recently in [18,28], the notion of the Schur multiplier has been extended to Lie superalgebras.…”

“…Recently in [18,28], the notion of the Schur multiplier has been extended to Lie superalgebras. In [18], Nayak shows that if L is a Lie superalgebra of dimension (m|n), then dim M(L) ≤ 1 2 (m + n) 2 + (n − m) . In the present paper, following [18,19,28] we study the Schur multiplier of a pair of Lie superalgebras and provide an upper bound for its dimension, which simultaneously extends all above bounds.…”