Symmetric polynomials in Leibniz algebras and their inner automorphisms

FINDIK Ş., YAPTI ÖZKURT Z.

TURKISH JOURNAL OF MATHEMATICS, cilt.44, sa.6, ss.2306-2311, 2020 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 44 Sayı: 6
  • Basım Tarihi: 2020
  • Doi Numarası: 10.3906/mat-2006-44
  • Dergi Adı: TURKISH JOURNAL OF MATHEMATICS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH, TR DİZİN (ULAKBİM)
  • Sayfa Sayıları: ss.2306-2311
  • Anahtar Kelimeler: Leibniz algebras, metabelian identity, automorphisms, symmetric polynomials
  • Çukurova Üniversitesi Adresli: Evet

Özet

Let L-n be the free metabelian Leibniz algebra generated by the set X-n = {x(1),..., x(n)} over a field K of characteristic zero. This is the free algebra of rank n in the variety of solvable of class 2 Leibniz algebras. We call an element s(X-n) is an element of L-n symmetric if s(x(sigma(1)),..., x(sigma(n))) = s(x(1),..., x(n)) for each permutation sigma of {1,..., n}. The set L-n(Sn) of symmetric polynomials of L-n is the algebra of invariants of the symmetric group S-n. Let K[X-n] be the usual polynomial algebra with indeterminates from X-n. The description of the algebra K[X-n](Sn) is well known, and the algebra (L-n')(Sn) in the commutator ideal L-n' is a right K[X-n](Sn)-module. We give explicit forms of elements of the K[X-n](Sn)-module (L-n')Sn. Additionally, we determine the description of the group Inn(L-n(Sn)) of inner automorphisms of the algebra L-n(Sn). The findings can be considered as a generalization of the recent results obtained for the free metabelian Lie algebra of rank n.