The central polynomials for the finite dimensional Grassmann algebras

In this note we describe the central polynomials for the finite dimensional unitary Grassmann algebras Gk over an infinite field F of characteristic ≠2. We exhibit a set of generators of C(Gk), the T-space of the central polynomials of Gk in a free associative F-algebra

Ideals of identities of representations of nilpotent Lie algebras

Let L be a Lie algebra, nilpotent of class 2, over an infinite field It, and suppose that the centre C of L is one dimensional; such Lie algebras are called Heisenberg algebras. Let rho:L --> hom(K) V be a finite dimensional representation of the Heisenberg algebra L such that rho(C) contains non-singular linear transformations of V, and denote I(rho) the ideal of identities for the representation rho. We prove that the ideals of identities of representations containing I(rho) and generated by multilinear polynomials satisfy the ACC. Let sl(2)(K) be the Lie algebra of the traceless 2 x 2 matrices over K, and suppose the characteristic of K equals 2. As a corollary we obtain that the ideals of identities of representations of Lie algebras containing that of the regular representation of sl(2)(K) and generated by multilinear polynomials, are finitely based. In addition we show that one cannot simply dispense with the condition of multilinearity. Namely, we show that the ACC is violated for the ideals of representations of Lie algebras lover an infinite field of characteristic 2) that contain the identities of the regular representation of sl(2)(K).2873095311

Finitely based ideals of weak polynomial identities

Let K be a field, char K not equal 2, and let V-k be a k-dimensional vector space over K equipped with a nondegenerate symmetric bilinear form. Denote C-k the Clifford algebra of V-k. We study the polynomial identities for the pair (C-k, V-k). A basis of the identities for this pair is found. It is proved that they are consequences of the single identity [x(2), y] = 0 when k = infinity. It is shown that when k < infinity the identities for (C-k, V-k) follow from [x(2), y] = 0 and Wk+1 = 0 where Wk+1 is an analog of the standard polynomial St(k+1). Denote M-2(K) the matrix algebra of order two over K, and let sl(2)(K) be the Lie algebra of all traceless 2 x 2 matrices over K. As an application, new proof of the fact that the identity [x(2), y] = 0 is a basis of the weak Lie identities for the pair (M-2(K),sl(2)(K)) is given.26103335335

Basis Of The Identities Of The Matrix Algebra Of Order Two Over A Field Of Characteristic P ≠ 2

In this paper we prove that the polynomial identities of the matrix algebra of order 2 over an infinite field of characteristic p≠2 admit a finite basis. We exhibit a finite basis consisting of four identities, and in "almost" all cases for p we describe a minimal basis consisting of two identities. The only possibilities for p where we do not exhibit minimal bases of these identities are p=3 and p=5. We show that when p=3 one needs at least three identities, and we conjecture a minimal basis in this case. In the course of the proof we construct an explicit basis of the vector space of the central commutator polynomials modulo the ideal of the identities of the matrix algebra of order two. © 2001 Academic Press.2411410434Asparouhov, T., Drensky, V., Koev, P., Tsiganchev, D., Generic 2 × 2 matrices in positive characteristic (2000) J. Algebra, 225, pp. 451-486De Concini, C., Procesi, C., A characteristic free approach to invariant theory (1976) Adv. in Math., 21, pp. 330-354Doubilet, P., Rota, G.-C., Stein, J., On the foundations of combinatorial theory (1974) Stud. Appl. Math., 3, pp. 185-216Drensky, V., A minimal basis of identities for a second-order matrix algebra over a field of characteristic 0 (1980) Algebra i Logika, 20, pp. 282-290Drensky, V., Computational techniques for PI-algebras (1990) Topics in Algebra Banach Center Publications, 26. , Warsaw: PWN. p. 17-44Drensky, V., Identities of representations of nilpotent Lie algebras (1997) Comm. Algebra, 25, pp. 2115-2127Filippov V., T., Varieties of Mal'tsev algebras (1981) Algebra i Logika, 20, pp. 300-314Formanek, E., The Polynomial Identities and Invariants of n × n Matrices (1991) CBMS Regional Conference Series Mathematics, 78. , Providence: American Mathematical SocietyGiambruno, A., Koshlukov, P., On the identities of the Grassmann algebras in characteristic p > 0 (2001) Israel J. Math.Koshlukov, P., Weak polynomial identities for the matrix algebra of order two (1997) J. Algebra, 188, pp. 610-625Koshlukov, P., Finitely based ideals of weak polynomial identities (1998) Comm. Algebra, 26, pp. 3335-3359Koshlukov, P., Ideals of identities of representations of nilpotent Lie algebras (2000) Comm. Algebra, 28, pp. 3095-3113Mal'tsev Yu., N., Kuz'min E., N., A basis for the identities of the algebra of second-order matrices over a finite field (1978) Algebra i Logika, 17, pp. 28-32Procesi, C., Computing with 2 × 2 matrices (1984) J. Algebra, 87, pp. 342-359Razmyslov Yu., P., Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero (1973) Algebra i Logika, 12, pp. 83-113Razmyslov Yu., P., Identities of Algebras and Their Representations (1994) Translations of Mathematical Monographs, 138. , Providence: American Mathematical SocietyRegev, A., On the Codimensions of Matrix Algebras (1988) Lecture Notes in Mathematics, 1352. , New York/Berlin: Springer-Verlag. p. 162-172Specht, W., Gesetze in Ringen, I (1950) Math. Z., 52, pp. 557-589Tki B., T., On the basis of the identities of the matrix algebra of second order over a field of characteristic zero (1981) Serdica, 7, pp. 187-194Vasilovsky S., Yu., The basis of identities of a three-dimensional simple Lie algebra over an infinite field (1989) Algebra i Logika, 28, pp. 534-554Vasilovsky, S., A finite basis for polynomial identities of the Jordan algebras of bilinear form (1991) Siberian Adv. Math., 1, pp. 1-43Vaughan-Lee M., R., Varieties of Lie algebras (1970) Quart. J. Math. Oxford (2), 21, pp. 297-308Vaughan-Lee M., R., Abelian-by-nilpotent varieties of Lie algebras (1975) J. London Math. Soc. (2), 11, pp. 263-266Zhevlakov K., A., Slin'ko A., M., Shestakov I., P., Shirshov A., I., Rings that Are Nearly Associative (1982) Pure and Applied Mathematics, 104. , New York: Academic Pres

Graded polynomial identities for the Lie algebra sl(2)(K)

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)The Lie algebra sl(2)( K) over a field K has a natural grading by Z(2), the cyclic group of order 2. We describe the graded polynomial identities for this grading when the base. eld is infinite and of characteristic different from 2. We exhibit a basis of these identities that consists of one polynomial. In order to obtain this basis we employ methods and results from Invariant theory. As a by-product we obtain finite bases of the graded identities for sl(2)( K) graded by other groups such as Z(2) x Z(2), and by the integers Z.185825836Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)CNPq [302655/2005-0]FAPESP [2004/13766-2, 2005/60337-2

Identities and isomorphisms of graded simple algebras

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Let G be an arbitrary abelian group and let A and B be two finite dimensional G-graded simple algebras over an algebraically closed field F such that the orders of all finite subgroups of G are invertible in F. We prove that A and B are isomorphic if and only if they satisfy the same G-graded identities. We also describe all isomorphism classes of finite dimensional G-graded simple algebras. (C) 2010 Elsevier Inc. All rights reserved.4321231413148Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)RFBR [09-01-00303, SSC-1983.2008.1]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)CNPq [302651/2008-0]FAPESP [05/60337-2, 2008/02938-8]RFBR [09-01-00303, SSC-1983.2008.1

Gk Dimension Of The Relatively Free Algebra For Sl2

Let sl2(K) be the Lie algebra of the 2 × 2 traceless matrices over an infinite field K of characteristic different from 2, denote by Rm = Rm(sl2(K)) the relatively free (also called universal) algebra of rankm in the variety of Lie algebras generated by sl2(K). In this paper we compute the Gelfand-Kirillov dimension of the Lie algebra Rm(sl2(K)). It turns out that whenever m ≥ 2 one has GK dim Rm = 3(m − 1). In order to compute it we use the explicit form of the Hilbert series of Rm described by Drensky. This result is new for m &gt; 2; the case m = 2 was dealt with by Bahturin in 1979.1754543553Bahturin, Y.A., Two-variable identities in the Lie Algebra sl(2, k) (Russian). Trudy Sem. Petrovsk. 5 (1979), 205–208. Translation: Contemp. Soviet Math, Petrovski˘ Seminar 5, Topics in Modern Math, Plenum (1985) New York-London, pp. 259-263Berele, A., Homogeneous polynomial identities (1982) Israel J. Math., 42, pp. 258-272Berele, A., Generic verbally prime PI-algebras and their GK-dimensions (1993) Commun. Algebra, 21 (5), pp. 1487-1504Drensky, V., A minimal basis for the identities of a second-order matrix algebra over a field of characteristic 0 (Russian), Algebra i Logika 20, pp. 282–290 (1981) (1981) English Translation: Algebra Logic, 20, pp. 188-194Drensky, V., Codimensions of T-ideals and Hilbert series of relatively free algebras (1984) J. Algebra, 91 (1), pp. 1-17Drensky, V., (2000) Free algebras and PI-Algebras. Graduate course in algebra, , Springer, Singapore:Kemer, A.R., Ideals of identities of associative algebras (1991) Translations of Mathematical Monographs, 87. , American Mathematical Society, Providence, RI:Kirillov, A.A., Kontsevich, M.L., The growth of the Lie algebra generated by two generic vector fields on the line, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 4, pp. 15–20 (1983). (In Russian). English Transl.: Vestnik Moskov. Univ. Ser. I Mat (1983) Mekh, 38 (4), pp. 14-20Kirillov, A.A., Kontsevich, M.L., Molev, A.I., Algebras of intermediate growth (1990) Sel. Math. Soviet., 9 (2), pp. 137-153Koshlukov, P., Graded polynomial identities for the Lie algebra sl2(K) (2008) Int. J. Algebra Comput., 18 (5), pp. 825-836Krause, G.R., Lenagan, T.H., Growth of Algebras and Gelfand-Kirillov Dimension (2000) Graduate Studies in Mathematics, 22. , American Mathematical Society, Providence, RI:Markov, V.T., The Gelfand-Kirillov dimension: nilpotency, representability, non-matrix varieties, Siberian school on varieties of algebraic systems (1988) Barnaul, 1988Abstracts, pp, 43-45 (In Russian)Petrogradsky, V.M., Some type of intermediate growth in Lie algebras, Uspekhi Mat. Nauk 48, no. 5 (293) pp. 181–182 (1993)English Transl. Russian (1993) Math. Surveys, 48 (5), pp. 181-182Petrogradsky, V.M., Growth of finitely generated polynilpotent Lie algebras and groups, generalized partitions, and functions analytic in the unit circle (1999) Int. J. Algebra Comput., 9 (2), pp. 179-212Procesi, C., Non-commutative affine rings (1967) Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I, 8 (8), pp. 237-255Razmyslov, P.Y., Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero (Russian), Algebra i Logika 12, pp. 83–113 (1973) (1973) English Translation: Algebra Logic, 10, pp. 47-63Repin, D.V.: Graded identities of a simple three-dimensional Lie algebra, Vestn. Samar. Gos. Univ. Estestvennonauchn. Special Issue 2, 5–16 (2004) (Russian)Vasilovsky, S.Y., The basis of identities of a three-dimensional simple Lie algebra over an infinite field (Russian), Algebra i Logika 28(5):534–554 (1989) (1989) English translation: Algebra Logic, 28 (5), pp. 355-36

Polynomial identities for the Jordan algebra of upper triangular matrices of order 2

AbstractThe associative algebras UTn(K) of the upper triangular matrices of order n play an important role in PI theory. Recently it was suggested that the Jordan algebra UJ2(K) obtained by UT2(K) has an extremal behaviour with respect to its codimension growth. In this paper we study the polynomial identities of UJ2(K). We describe a basis of the identities of UJ2(K) when the field K is infinite and of characteristic different from 2 and from 3. Moreover we give a description of all possible gradings on UJ2(K) by the cyclic group Z2 of order 2, and in each of the three gradings we find bases of the corresponding graded identities. Note that in the graded case we need only an infinite field K, charK≠2

A Basis For The Graded Identities Of The Matrix Algebra Of Order Two Over A Finite Field Of Characteristics P ≠ 2

Let K be a finite field of characteristic p &gt; 2, and let M2(K) be the matrix algebra of order two over K. We describe up to a graded isomorphism the 2-gradings of M2(K). It turns out that there are only two nonisomorphic nontrivial such gradings. Furthermore, we exhibit finite bases of the graded polynomial identities for each one of these two gradings. One can distinguish these two gradings by means of the graded polynomial identities they satisfy. © 2002 Elsevier Science (USA).84597609de Concini, C., Procesi, C., A characteristic free approach to invariant theory (1976) Adv. in Math., 21, pp. 330-354Di Vincenzo, O.M., On the graded identities of M1,1 (E) (1992) Israel J. Math., 80, pp. 323-335Formanek, E., (1991) "The Polynomial Identities and Invariants of N × N Matrices", 78. , CBMS Regional Conference Series in Mathematics, Amer. Math. Soc. Providence, RIHerstein, I.N., (1968) "Noncommutative Rings", , Math. Assoc. Amer. New YorkKemer, A.R., (1991) "Ideals of Identities of Associative Algebras", 87. , Translation of Mathematical Monographs, Amer. Math. Soc., Providence, RIKoshlukov, P., Basis of the identities of the matrix algebra of order two over a field of characteristic p ≠ 2 (2002) J. Algebra, 241, pp. 410-434Koshlukov, P., Azevedo, S.S., Graded identities for T-prime algebras over fields of positive characteristic (2002) Israel J. Math., , to appearKruse, R.L., Identities satisfied by a finite ring (1973) J. Algebra, 26, pp. 298-318Lvov, I.V., Varieties of associative rings (1973) Algebra Logic, 12, pp. 150-167Maltsev, Yu.N., Kuzmin, E.N., A basis for identities of the algebra of second order matrices over a finite field (1978) Algebra Logic, 17, pp. 17-21McDonald, B.R., (1974) "Finite Rings With Identities", , Marcel Dekker, New YorkRazmyslov, Yu.P., (1994) "Identities of Algebras and Their Representations", 38. , Translation of Mathematical Monographs, Amer. Math. Soc., Providence, RIWall, C.T.C., Graded Brauer groups (1963) J. Reine Angew. Math., 213, pp. 187-19