Simple Lie Color Algebras of Weyl Type
نویسنده
چکیده
Abstract. For an (ǫ,Γ)-color-commutative associative algebra A with an identity element over a field F of characteristic not 2, and for a color-commutative subalgebraD of color-derivations of A, denote by A[D] the associative subalgebra of End(A) generated by A (regarding as operators on A via left multiplication) and D. It is easily proved that, as an associative algebra, A[D] is Γgraded simple if and only if A is Γ-graded D-simple. Suppose A is Γ-graded D-simple. Then, (a) A[D] is a free left A-module; (b) as a Lie color algebra, the subquotient [A[D], A[D]]/Z(A[D]) ∩ [A[D], A[D]] is simple (except one minor case), where Z(A[D]) is the color center of A[D]. The structure of this subquotient is explicitly described.
منابع مشابه
Simple Algebras of Weyl Type
Over a field IF of any characteristic, for a commutative associative algebra A with an identity element and for the polynomial algebra IF [D] of a commutative derivation subalgebra D of A, the associative and the Lie algebras of Weyl type on the same vector space A[D] = A ⊗ IF [D] are defined. It is proved that A[D], as a Lie algebra (modular its center) or as an associative algebra, is simple ...
متن کاملReflection Groups in Hyperbolic Spaces and the Denominator Formula for Lorentzian Kac–moody Lie Algebras
This is a continuation of our ”Lecture on Kac–Moody Lie algebras of the arithmetic type” [25]. We consider hyperbolic (i.e. signature (n, 1)) integral symmetric bilinear form S : M × M → Z (i.e. hyperbolic lattice), reflection group W ⊂ W (S), fundamental polyhedron M of W and an acceptable (corresponding to twisting coefficients) set P (M) ⊂ M of vectors orthogonal to faces of M (simple roots)...
متن کاملRealization of locally extended affine Lie algebras of type $A_1$
Locally extended affine Lie algebras were introduced by Morita and Yoshii in [J. Algebra 301(1) (2006), 59-81] as a natural generalization of extended affine Lie algebras. After that, various generalizations of these Lie algebras have been investigated by others. It is known that a locally extended affine Lie algebra can be recovered from its centerless core, i.e., the ideal generated by weight...
متن کاملLie-type higher derivations on operator algebras
Motivated by the intensive and powerful works concerning additive mappings of operator algebras, we mainly study Lie-type higher derivations on operator algebras in the current work. It is shown that every Lie (triple-)higher derivation on some classical operator algebras is of standard form. The definition of Lie $n$-higher derivations on operator algebras and related pot...
متن کاملGraded Simple Lie Algebras and Graded Simple Representations
For any finitely generated abelian group Q, we reduce the problem of classification of Q-graded simple Lie algebras over an algebraically closed field of “good” characteristic to the problem of classification of gradings on simple Lie algebras. In particular, we obtain the full classification of finite-dimensional Q-graded simple Lie algebras over any algebraically closed field of characteristi...
متن کامل