*-polynomial identities of matrices

with the transpose involution:

The low degrees

Authors:
Alain D'Amour and Michel Racine

Journal:
Trans. Amer. Math. Soc. **351** (1999), 5089-5106

MSC (1991):
Primary 16R10, 16R50

DOI:
https://doi.org/10.1090/S0002-9947-99-02301-6

Published electronically:
May 21, 1999

MathSciNet review:
1603886

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we investigate -polynomial identities of minimal degree for the algebra of matrices over a field, where and is the transpose involution. We first present some basic generators, and then proceed to show that all other minimal degree identities can be derived from those.

**[1]**Amitsur, A. S. and Levitzki, J.,*Minimal identities for algebras*, Proc. Amer. Math. Soc.**1**(1950), 449-463. MR**12:155d****[2]**Amitsur, A. S.,*Identities in rings with involution*, Israel J. Math.**7**(1969), 63-68. MR**39:4216****[3]**Drensky, Vesselin, and Giambruno, Antonio,*Cocharacters, codimensions and Hilbert series of the polynomial identities for matrices with involution*, Canad. J. Math.**46**(1994), 718-733. MR**95d:16028****[4]**Giambruno, Antonio,*On -polynomial identities for matrices*, J. Algebra**133**(1990) 433-438. MR**91g:16015****[5]**Kostant, B.,*A theorem of Frobenius, a theorem of Amitsur-Levitzki, and cohomology theory*, Indiana J. Math.**7**(1958), 237-264. MR**19:1153e****[6]**Kostant, B.,*A Lie algebra generalization of the Amitsur-Levitzki theorem*, Adv. in Math.**40**(1980), 155-175. MR**82h:17008****[7]**Levchenko, Diana V.,*Finite basis property of identities with involution of a second-order matrix algebra*(Russian), Serdica**8**(1982), 42-56. MR**84e:16008****[8]**Levchenko, Diana V.,*Identity with involution of a second-order matrix algebra*(Russian), C.R. Acad. Bulgare Sci.**33**(1980), 1043-1045. MR**82h:16011****[9]**Ma, W.,*A decomposition of elements of the free algebra*, Proc. Amer. Math. Soc.**118**(1993), 37-45. MR**93g:16040****[10]**Ma, W., Racine, M. L.,*Minimal identities of symmetric matrices*, Trans. Amer. Math. Soc.**320**(1990), 171-192. MR**91k:16018****[11]**Neher, E.,*Jordan Triple Systems by the Grid Approach*, Lecture Notes in Mathematics 1280, Springer, New York, 193 pp. MR**89b:17024****[12]**Racine, M. L.,*Central polynomials for Jordan algebras I*, J. Algebra**41**(1976) 224-237. MR**54:5303****[13]**Racine, M. L.,*Minimal identities for Jordan algebras of degree 2*, Comm. Alg.**5**(1985), 2493-2506. MR**87e:17026****[14]**Razmyslov, Y. P.,*Finite basing of the identities of a matrix algebra of second order over a field of characteristic 0*, Algebra and Logic**12**(1973), 47-63. MR**49:5103****[15]**Rowen, L. H.,*Standard polynomials in matrix algebras*, Trans. Amer. Math. Soc.**190**(1974), 253-284. MR**50:2208****[16]**Rowen, L. H.,*Polynomial Identities in Ring Theory*, Academic Press, New York, 1980, 365 pp. MR**82a:16021****[17]**Rowen, L. H.,*A simple proof of Kostant's theorem and an analogue for the symplectic involution*, Contemp. Math., vol. 13, Amer. Math. Soc. (1982), 207-215. MR**84d:16024**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
16R10,
16R50

Retrieve articles in all journals with MSC (1991): 16R10, 16R50

Additional Information

**Alain D'Amour**

Affiliation:
Department of Mathematics & Computer Science, Denison University, Granville, Ohio 43023

Email:
damour@cc.denison.edu

**Michel Racine**

Affiliation:
Department of Mathematics, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada

Email:
me@mathstat.uottawa.ca

DOI:
https://doi.org/10.1090/S0002-9947-99-02301-6

Received by editor(s):
May 18, 1997

Published electronically:
May 21, 1999

Additional Notes:
The second author’s research is supported in part by a grant from NSERC

Article copyright:
© Copyright 1999
American Mathematical Society