Tag - Other non-associative rings

Shavkat Ayupov: Local and 2-local derivations and automorphisms of Octonian algebras

15th April, 2021

The talk is devoted to description of local and 2-local derivations (respectively, automorphisms) on octonian algebras over fields with zero characteristics. We shall give a general form of local derivations on the real octonion algebra O(ℝ). This description implies that the space of all local derivations on O(ℝ) when equipped with Lie bracket is isomorphic to the Lie algebra 𝔰𝔬7(ℝ) of all real skew-symmetric 7 × 7-matrices. We also consider 2-local derivations on the octonion algebra O(F) over an algebraically closed field F and prove that every 2-local derivation on O(F) is a derivation. Further, we apply these results to problems for the simple 7-dimensional Malcev algebra. As a corollary we obtain that the real octonion algebra O(ℝ) and Malcev algebra M7(ℝ) are simple non-associative algebras which admit pure local derivations, that is, local derivations which are not derivation. Further, we shall give a general form of local automorphisms on the octonion algebra O(F) over a field F. This description implies that the group of all local automorphisms on O(F) is isomorphic to the group O7(F) of all orthogonal 7 × 7-matrices over F. We also consider 2-local automorphisms on the octonion algebra O(F) over an algebraically closed field F and prove that every 2-local automorphism on O(F) is an automorphism. As a corollary we obtain descriptions of local and 2-local automorphisms of seven dimensional simple Malcev algebra.

Pavel Kolesnikov: Derived algebras and their identities

9th July, 2020

In this talk we will consider a "differential counterpart" of the dendriform splitting procedure for operads. This problem has a very natural interpretation in the language of non-associative algebras. It is well-known that a (non-associative, in general) algebra equipped with a Rota-Baxter operator (a formalization of integration) gives rise to a system in a class of splitting algebras. The latter include dendriform (pre-associative), pre-Lie (left-symmetric), pre-Poisson, Zinbiel (pre-commutative) algebras, etc. What happens if we replace a Rota-Baxter operator with a derivation? The answer is well known for associative commutative algebras: the resulting class of systems obtained in this way coincides with the variety Nov of Novikov algebras. We will show in general that for an arbitrary binary operad Var the variety of derived Var-algebras coincides with the Manin white product of operads Var and Nov. If we allow the initial multiplication(s) to leave in the language of a derived algebra then the same sort of description can be obtained just by replacement of Nov with GD!, the Koszul dual to the operad of Gelfand-Dorfman algebras. We will also discuss similar statements for the "integral" case of Rota-Baxter operators.

Ivan Shestakov: Coordination Theorems for certain non-associative algebras

16th April, 2020

Coordinatization Theorems are very useful for classification problems. The classical Wedderburn Coordinatization Theorem claims that if a unital associative algebra A contains a matrix subalgebra Mn(F) with the same unit then A=Mn(B) for a certain subalgebra B. The Jacobson Coordinatization Theorems in the structure theories of alternative and Jordan algebras state similar results for octonions and Albert algebras. Various coordinatization theorems were proved for noncommutative Jordan algebras, for commutative power associative algebras, for alternative and Jordan superalgebras, etc. In our talk, we consider three coordinatization theorems:

1) for 2x2 matrices in the class of alternative algebras (Jacobson's problem),

2) for Jordan algebra of symmetric 2x2 matrices in the class of Jordan algebras,

3) for octonions in the class of right alternative algebras.