On the nilpotency degree of the algebra with identity x^n=0

Denote by C_{n,d} the nilpotency degree of a relatively free algebra generated by d elements and satisfying the identity x^n=0. Under assumption that the characteristic p of the base field is greater than n/2, it is shown that C_{n,d}<n^{log_2(3d+2)+1} and C_{n,d}<4 2^{n/2} d. In particular, it is established that the nilpotency degree C_{n,d} has a polynomial growth in case the number of generators d is fixed and p > n/2. For p\neq2 the nilpotency degree C_{4,d} is described with deviation 4 for all d. As an application, a finite generating set for the algebra R^{GL(n)} of GL(n)-invariants of d matrices is established in terms of C_{n,d}. Several conjectures are formulated.Comment: 17 pages; v3. References are update

Associative nil-algebras over finite fields

The nilpotency degree of a relatively free finitely generated associative algebra with the identity $x^n=0$ is studied over finite fields.Comment: 12 page

Minimal generating and separating sets for O(3)-invariants of several matrices

Given an algebra $F[H]^G$ of polynomial invariants of an action of the group $G$ over the vector space $H$, a subset $S$ of $F[H]^G$ is called separating if $S$ separates all orbits that can be separated by $F[H]^G$. A minimal separating set is found for some algebras of matrix invariants of several matrices over an infinite field of arbitrary characteristic different from two in case of the orthogonal group. Namely, we consider the following cases: 1) $GL(3)$-invariants of two matrices; 2) $O(3)$-invariants of $d>0$ skew-symmetric matrices; 3) $O(4)$-invariants of two skew-symmetric matrices; 4) $O(3)$-invariants of two symmetric matrices. A minimal generating set is also given for the algebra of orthogonal invariants of three $3\times 3$ symmetric matrices.Comment: 11 page

Identities of sum of two PI-algebras in the case of positive characteristic

FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOWe consider the following question posted by Beidar and Mikhalev in 1995 for an associative ring R = R1 + R2: is it true that if the subrings R1 and R2 satisfy polynomial identities, then R also satisfies a polynomial identity? Over a field of positive characteristic we establish new conditions on R1 and R2 that guarantee a positive answer to the question. We find upper and low bounds on the degrees of identities of R. © 2015 World Scientific Publishing Company.We consider the following question posted by Beidar and Mikhalev in 1995 for an associative ring R = R1 + R2: is it true that if the subrings R1 and R2 satisfy polynomial identities, then R also satisfies a polynomial identity? Over a field of positive characteristic we establish new conditions on R1 and R2 that guarantee a positive answer to the question. We find upper and low bounds on the degrees of identities of R.25812651273FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO2013/15539-2Bahturin, Y., Giambruno, A., Identities of sums of commutative subalgebras (1994) Rend. Cir. Mat. Palermo, 43 (2), pp. 250-258Beidar, K.I., Mikhalev, A.V., Generalised polynomial identities and rings which are sums of two subrings (1995) Algebrai Logica, 34, pp. 3-11. , (in Russian)(1995) Algebra Logic, 34, pp. 1-5Bokut, L.A., Embeddings in simple associative algebras (1976) Algebra i Logica, 15, pp. 117-142. , (in Russian)(1976) Algebra Logic, 15, pp. 73-90Felzenszwalb, B., Giambruno, A., Leal, G., On rings which are sums of two PIsubrings: A combinatorial approach (2003) Pacific J. Math., 209 (1), pp. 17-31Goto, M., Note on a characterization of solvable Lie algebras (1962) J. Sci. Hiroshima Univ. Ser. A-I, 26, pp. 1-2Kegel, O.H., Zur hilpotenz gewisser assoziativer ringe (1962) Math. Ann., 149, pp. 258-260Kemer, A.R., The standard identity in charecteristic p: A conjecture of I.B. Volichenko (1993) Israel J. Math., 81, pp. 343-355Kepczyk, M., On algebras that are sums of two subalgebras satisfying certain polynomial identities (2008) Publ. Math. Debrecen, 72 (3), pp. 257-267Kepczyk, M., Note on algebras which are sums of two PI subalgebras (2015) J. Algebra Appl., 14 (10), p. 1550149Kepczyk, M., Puczylowski, E.R., On radicals of rings which are sums of two subrings (1996) Arch. Math., 66, pp. 8-12Kepczyk, M., Puczylowski, E.R., Rings which are sums of two subrings (1998) J. Pure Applied Algebra, 133, pp. 151-162Kepczyk, M., Puczylowski, E.R., Rings which are sums of two subrings satisfying a polynomial identity (2001) Commun. Algebra, 29, pp. 2059-2065Kolman, B., Semi-modular Lie algebras (1965) J. Sci. Hiroshima Univ. Ser. A-I, 29, pp. 149-163Panyukov, V.V., On the solvability of Lie algebras of positive characteristic (1990) Russ. Math. Surv., 45 (4), pp. 181-182Pchelintsev, S.V., Kegel theorem for alternative algebras (1985) Sibirsk. Mat. Zh., 26, pp. 195-196Petravchuk, A.P., Lie algebras which can be decomposed into the sum of an abelian subalgebra and a nilpotent subalgebra (1988) Ukrain. Math. J., 40, pp. 331-334Regev, A., Existence of polynomial identities in A F B (1971) Bull. Amer. Math. Soc., 77, pp. 1067-1069Regev, A., Existence of identities in A B (1972) Israel J. Math., 11, pp. 131-152Rowen, L.H., Generalized polynomial identities II (1976) J. Algebra, 38, pp. 380-392Salwa, A., Rings that are sums of two locally nilpotent subrings (1996) Commun. Algebra, 24 (12), pp. 3921-3931Zusmanovich, P., A Lie algebra that can be written as a sum of two nilpotent subalgebras is solvable (1991) Math. Notes, 50, pp. 909-91

Conservative algebras of 2-dimensional algebras

FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOIn 1990 Kantor introduced the conservative algebra W(n) of all algebras (i.e. bilinear maps) on the n-dimensional vector space. In case n > 1 the algebra W(n) does not belong to well-known classes of algebras (such as associative, Lie, Jordan, Leibniz algebras). We describe the algebra of all derivations of W(2) and subalgebras of W(2) of codimension one. We also study similar problems for the algebra W-2 of all commutative algebras on the two-dimensional vector space and the algebra S-2 of all commutative algebras with trace zero multiplication on the two-dimensional space. (C) 2015 Elsevier Inc. All rights reserved.In 1990 Kantor introduced the conservative algebra W(n) of all algebras (i.e. bilinear maps) on the n-dimensional vector space. In case n > 1 the algebra W(n) does not belong to well-known classes of algebras (such as associative, Lie, Jordan, Leibniz algebras). We describe the algebra of all derivations of W(2) and subalgebras of W(2) of codimension one. We also study similar problems for the algebra W-2 of all commutative algebras on the two-dimensional vector space and the algebra S-2 of all commutative algebras with trace zero multiplication on the two-dimensional space.486255274FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO2011/5113