On Endomorphs of Right-Symmetric Algebras
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NDOMORPHS OF RIGHT-SYMMETRIC ALGEBRAS A. P. Pozhidaev
UDC 512.57
Abstract: We introduce the notion of endomorph E(A ) of a (super)algebra A and prove that E(A ) is a simple (super)algebra if A is not an algebra of scalar multiplication. If A is a right-symmetric (super)algebra then E(A ) is right-symmetric as well. Thus, we construct a wide class of simple (rightsymmetric) (super)algebras which contains a matrix subalgebra with a common unity. We calculate the derivation algebra of the endomorph of a unital algebra A and the automorphism group of the simple right-symmetric algebra E(Vn ) (the endomorph of a direct sum of fields). DOI: 10.1134/S0037446620050092 Keywords: endomorph, right-symmetric algebra, left-symmetric algebra, simple algebra, derivation, automorphism, pre-Lie algebra
Introduction An algebra A is right-symmetric provided that the associator on A is right-symmetric; i. e., the associator is symmetric in the last two arguments: (x, y, z) = (x, z, y)
(1)
for all x, y, z ∈ A , where (x, y, z) := (xy)z − x(yz) is the associator of x, y, and z (the symbol := stands for equality by definition). Note that (1) may be written in operator form as [Ry , Rz ] = R[y,z] ,
(2)
where [y, z] := yz − zy is the commutator of y and z, while Ry is the right multiplication by y ∈ A ; i. e., xRy = xy for all x ∈ A . The right-symmetric algebras present an important generalization of associative algebras. These algebras are Lie-admissible, while arising naturally in various contexts. Their anti-isomorphic analogs are left-symmetric algebras. Apparently, the left-symmetric algebras were first introduced by Cayley in 1857. In 1961, Koszul used them in the study of actions of affine transformations; see [1]. In 1963, Vinberg applied the left-symmetric algebras to classify the convex homogeneous cones in [2], and Gerstenhaber used them to study the deformations of algebras in [3]. At present, many articles are devoted to the left-symmetric algebras which are also known under other names, for instance, as pre-Lie algebras (see, e. g., [4]). One of the principal examples of the right-symmetric algebras is as follows: Example. Let (A ; ·) be a commutative associative algebra, and let D be a derivation of A . Then the new product a • b = D(a) · b for all a, b ∈ A turns A into a right-symmetric algebra. In what follows, F stands for a ground field; and A , for a nonzero algebraover F . Unless otherwise aij eij where eij are the stated, given A in the matrix algebra Mn := Mn (F ), we use notation A = usual matrix units. We denote by Vn an n-dimensional vector space over F with the standard basis of ¯ := {1, . . . , n}. Also, we put (x, y, z)rs := (x, y, z) − (x, z, y). The notation Υ := ΥF is rows ei , i ∈ n used for the linear span of a set Υ over F , where we omit F if the field is clear from the context. Given a vector space V over F , we denote by V ∗ the dual vector space for V ; and by L(V ), the algebra of all F -linear operators on V . The work was carried out in the framework of the State Task to the Sobolev Institute o
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