This research is motivated by universal algebraic geometry. We consider in universal algebraic geometry the some variety of universal algebras Θ and algebras H ∈Θ from this variety. One of the central question of the theory is the following: When do two algebras have the same geometry? What does it mean that the two algebras have the same geometry? The notion of geometric equivalence of algebras gives a sort of answer to this question. Algebras Hi and H_2 are called geometrically equivalent if and only if the H_1-closed sets coincide with the H_2-closed sets. The notion of automorphic equivalence is a generalization of the first notion. Algebras H_1 and H_2 are called automorphically equivalent if and only if the Hi-closed sets coincide with the H_2-closed sets after some "changing of coordinates". We can detect the difference between geometric and automorphic equivalence of algebras of the variety by researching of the automorphisms of the category Θ~0 of the finitely generated free algebras of the variety. Θ By [5] the automorphic equivalence of algebras provided by inner automorphism coincide with the geometric equivalence. So the various differences between geometric and automorphic equivalence of algebras can be found in the variety Θ if the factor group η/η is big. Here 2t is the group of all automorphisms of the category Θ~0 η is a normal subgroup of all inner automorphisms of the category Θ~0. In [6] the variety of all Lie algebras and the variety of all associative algebras over the infinite field k were studied. If the field k has not nontrivial automorphisms then group η/η in the first case is trivial and in the second case has order 2. We consider in this paper the variety of all linear algebras over the infinite field k. We prove that group η/η is isomorphic to the group (U(kS_2)/U(k{e}))λAutk,where S_2 is the symmetric group of the set which has 2 elements, U(kS_2) is the group of all invertible elements of the group algebra kS_2, e ∈ S_2, U (k{e}) is a group of all invertible elements of the subalgebra k{e}, Autk is the group of all automorphisms of the field k. So even the field k has not nontrivial automorphisms the group η/η is infinite. This kind of result is obtained for the first time. The example of two linear algebras which are automorphically equivalent but not geometrically equivalent is presented in the last section of this paper. This kind of example is also obtained for the first time.
展开▼
