Identities With Involution For The Matrix Algebra Of Order Two In Characteristic

Let M2(K) be the matrix algebra of order two over an infinite field K of characteristic p ≠ 2. If K is algebraically closed then, up to isomorphism, there are two involutions of first kind on M2(K), namely the transpose and the symplectic. If K is not algebraically closed, studying *-identities it is still sufficient to consider only these two involutions. We describe bases of the polynomial identities with involution in each of these cases.146337355Anisimov, N., Zp-codimensions of Zp-identities of Grassmann algebra (2001) Communications in Algebra, 29, pp. 4211-4230. , Special issue dedicated to Alexei Ivanovich KostrikinColombo, J., Koshlukov, P., Central polynomials in the matrix algebra of order two (2004) Linear Algebra and Its Applications, 377, pp. 53-67D'Amour, A., Racine, M., *-polynomial identities of matrices with the transpose involution: The low degrees (1999) Transactions of the American Mathematical Society, 351, pp. 5089-5106D'Amour, A., Racine, M., *-polynomial identities of matrices with the symplectic involution: The low degrees (2004) Communications in Algebra, 32, pp. 895-918De Concini, C., Procesi, C., A characteristic free approach to invariant theory (1976) Advances in Mathematics, 21, pp. 330-354Drensky, V., Giambruno, A., Cocharacters, codimensions and Hilbert series of the polynomial identities for 2 × 2 matrices with involution (1994) Canadian Journal of Mathematics, 46, pp. 718-733Drensky, V., (1999) Free Algebras and PI Algebras: Graduate Course in Algebra, , Springer, SingaporeGiambruno, A., Koshlukov, P., On the identities of the Grassmann algebras in characteristic p > 0 (2001) Israel Journal of Mathematics, 122, pp. 305-316Iltyakov, A.V., Specht ideals of identities of certain simple nonassociative algebras (1985) Algebra and Logic, 24, pp. 210-228Koshlukov, P., Weak polynomial identities for the matrix algebra of order two (1997) Journal of Algebra, 188, pp. 610-625Koshlukov, P., Finitely based ideals of weak polynomial identities (1998) Communications in Algebra, 26, pp. 3335-3359Koshlukov, P., Basis of the identities of the matrix algebra of order two over a field of characteristic p ≠ 2 (2001) Journal of Algebra, 241, pp. 410-434Levchenko, D., Finite basis property of identities with involution of a second-order matrix algebra (1982) Serdica, 8 (1), pp. 42-56. , RussianLevchenko, D., Bases of identities with involution of second-order matrix algebras over finite fields (1984) Serdica, 10 (1), pp. 55-67. , RussianRazmyslov, Yu.P., Identities of Algebras and their (1994) Representations, Translations of Mathematical Monographs, 138. , American Mathematical Society, Providence, RIRowen, L.H., Polynomial Identities in Ring Theory (1980) Pure and Applied Mathematics Series, 84. , Academic Press, New YorkVasilovsky, S., The basis of identities of a three-dimensional simple Lie algebra over an infinite field (1989) Algebra and Logic, 28, pp. 355-36