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Extra info for Algebra, with Arithmetic and mensuration,: From the Sanscrit of Brahmegupta and Bháscara

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1 in a very simpl e way: 24 S. ), Non-Associative Algebra and Its Applications, 24-30. © 1994 Kluwer Academic Publishers . 2. (Characteristic 0) The nonsolvable Lie algebras whose ideals are a 3-element chain are a semi-direct sum of an abelian ideal N of dimension at least 2 and a simple subalgebra S such that N is an irreducible S-module. The principal task in this paper is giving some explicit constructions of nonsolvable Lie algebras in which the ideals are a n-element chain with n = 4. 1, we can use two methods in order to get the construction of an algebra L of this type: I) Suppose the radical of L is known, identify the simple Lie algebras that can occur as its semisimple Levi factor.

4) Let {e,Ul , .. , Ur , ... , u;Uj } be a basis of Band p(c(e) , . , c(u;u - j)) be a polynomial of Ass[c(e), .. ,c(Uj Uj )]. It is enough to prove that th ere exists n E N such that any polynomial with more than n elements from N belongs to I n ker Ill' Let us suppose that p E Ass[c(e), . ,f( UjUj)] has mor e than k elements from N . By universal property of U(B) , 1J' (p(c(e), . ,c(u ;Uj)) p(1J'c(e), ... l(e), . T he element Jl( e) disappears or it is at the end of the polynomial by identities (4).

Similarly for any unital, commutative, associative Falgebra 21, and any unital 21-module W with a symmetric 21-bilinear form ( ,) : W x W ---+ 21, the operation w 0 x (w ,x)1 defines the structure of a Jordan algebra on J(W) = 211 EEl W . = Recognition Theorem for Type Bn • Let L be a Bn-graded Lie algebra for n 2: 3. Then there exists a unital, commutative, associative F-algebra 21 and a unital 21-module W with a symmetric 21-bilinear form ( , ) : W x W ---+ 21 such that L is centrally isogenous with T(J(V)j F, J(W)j21) « V, V> 0 21) EEl (V 0 W)EEl < W, W >, (B n 0 21) EEl (V 0 W)EEl < W, W », where V is a (2n + I)-dimensional F-vector space with a nondegenerate symmetric bilinear form (the defining representation for B n ) , < V, V > is the set of skewsymmetric transformations on V relative to the form on V which is a simple Lie algebra of type B n , and < W, W > is the set of skew-symmetric transformations on W relative to the form on W.

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