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Cocharacters for the weak polynomial identities of the Lie algebra of \(3 \times 3\) skew-symmetric matrices. (English) Zbl 07258198
Let \(so_3\) be the Lie algebra of the skew-symmetric matrices of order 3 over a fixed field \(K\) of characteristic 0, and let \(M_3\) be the (associative) full matrix algebra of order 3 over \(K\). The authors study the weak polynomial identities for the pair \((M_3, so_3)\). If \(A\) is an associative algebra denote by \(A^-\) the Lie algebra on the vector space of \(A\) defined by the usual bracket \([a,b]=ab-ba\). Let us recall that if \(L\) is a Lie algebra and \(A\) an associative algebra such that \(L\subseteq A^-\) is a subalgebra and \(L\) generates \(A\), then an associative polynomial is a weak identity for the pair \((A,L)\) whenever \(f\) vanishes on \(L\). Clearly the set \(I(A,L)\) of all weak identities for the pair \((A,L)\) is an ideal in the free associative algebra; this ideal is closed under Lie substitutions of the variables. Weak identities were introduced by J. P. Razmyslov [Algebra Logic 12, 47–63 (1974; Zbl 0282.17003); translation from Algebra Logika 12, 83–113 (1973)], see also the monograph [Yu. P. Razmyslov, Identities of algebras and their representations. Transl. from the Russian by A. M. Shtern. Transl. ed. by Simeon Ivanov. Providence, RI: American Mathematical Society (1994; Zbl 0827.17001)]. They were essential in establishing the finite basis property of the identities satisfied by \(sl_2\) and by \(M_2\). The weak identities were used by Razmyslov in order to construct central polynomials for the full matrix algebras as well. Weak identities have a wide field of applications in studying polynomial identities for non-associative algebras other than Lie algebras. Recall that, in a sense, the weak identities (in the Lie case) can be considered as identities of representations of Lie algebras. A basis (that is a generating set) of the ideal of weak identities for \((M_3, so_3)\) was described by Razmyslov, see the above cited monograph.
By using the isomorphism \(so_3\cong sl_2\) one can interpret the pair \((M_3, so_3)\) as \((M_3, ad(sl_2))\) where \(ad(sl_2)\) stands for the adjoint representation of \(sl_2\). When dealing with polynomial identities in characteristic 0 it is enough to consider the multilinear ones. If \(P_n\) is the vector space of all multilinear polynomials of degree \(n\) in the first \(n\) variables, in the free associative algebra then \(P_n\) is naturally a left module over the symmetric group \(S_n\); it acts by permuting the variables. If \(I\) is an ideal which is invariant under permutations from \(S_n\) then \(I\cap P_n\) is a submodule. Clearly ideals of (weak) identities are such ideals. When studying polynomial identities it is more convenient to work with the quotient \(P_n/(P_n\cap I)\). Its \(S_n\)-character is the \(n\)-th cocharacter of the ideal \(I\). Thus knowing the generators of \(I\) and its cocharacter sequence yields a very precise description of the identities in \(I\).
The paper under review studies the weak identities of the pair \((M_3, so_3)\). The authors determine explicitly the cocharacter of \(I(M_3, so_3)\) (see Lemma 3.3 and Theorem 3.7 of the paper). Although this is very significant contribution to the quantitative theory of polynomial identities the methods and results used to achieve it seem more important. The authors make use of the duality of the representations of \(S_n\) with the polynomial representations of the general linear group \(\mathrm{GL}_p\). Thus consider \(p\)-tuples of generic \(3\times 3\) skew-symmetric matrices \(t_1\), …, \(t_p\), acted on by the special orthogonal group \(\mathrm{SO}_3\) by simultaneous conjugation. Let \(T_p\) be the polynomial algebra in the \(3p\) variables that are the entries of the generic matrices \(t_i\), \(1\le i\le p\). Put \(\mathcal{F}_p\) the (associative and unitary) subalgebra of \(M_3(K[T_p])\) generated by the \(t_i\), and let \(\mathcal{E}_p\) be the subalgebra of \(M_3(K[T_p])\) consisting of the \(\mathrm{SO}_3(K)\)-equivariant polynomial maps from \(p\) copies of \(so_3\) to \(M_3\). The generators of \(\mathcal{E}_p\) are well known from classical Invariant theory. The group \(\mathrm{GL}_p\) acts on the right on \(so_3^{\oplus p}\) in a natural way, and this action gives rise to one from the left on \(K[T_p]\) and on \(M_3(K[T_p])\). Clearly the actions of \(\mathrm{GL}_p\) and of \(\mathrm{SO}_3\) commute hence \(\mathcal{E}_p\) and \(\mathcal{F}_p\) both become \(\mathrm{GL}_p\)-modules, \(\mathcal{F}_p\subset\mathcal{E}_p\). The authors obtain descriptions of the \(\mathrm{GL}_p\)-module structure of both \(\mathcal{F}_p\) and \(\mathcal{E}_p\), that is their decompositions into irreducibles, and compute the corresponding multiplicities. It should be noted that they compute the multiplicities in the case of \(\mathcal{E}_p\) by means of classical Invariant theory. These clearly give upper bounds for the corresponding multiplicities of \(\mathcal{F}_p\). Afterwards by means of direct and heavy (CAS-aided) computations they find the differences for \(\mathcal{F}_p\). In a sense the study can be reduced to that of 3 generic matrices.
The paper will be of interest to people working on PI, on Invariant theory, and Representation theory of groups.
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
16R30 Trace rings and invariant theory (associative rings and algebras)
17B01 Identities, free Lie (super)algebras
17B20 Simple, semisimple, reductive (super)algebras
17B45 Lie algebras of linear algebraic groups
20C30 Representations of finite symmetric groups
Full Text: DOI
[1] Aslaksen, H.; Tan, E.-C.; Zhu, C.-B., Invariant theory of special orthogonal groups, Pac. J. Math., 168, 2, 207-215 (1995) · Zbl 0830.15027
[2] Berele, A., Homogeneous polynomial identities, Isr. J. Math., 42, 258-272 (1982) · Zbl 0522.16015
[3] CoCoATeam, : a system for doing Computations in Commutative Algebra, available at
[4] Domokos, M.; Drensky, V., Gröbner bases for the rings of invariants of special orthogonal and \(2 \times 2\) matrix invariants, J. Algebra, 243, 706-716 (2001) · Zbl 1031.16016
[5] Drensky, V., Representations of the symmetric group and varieties of linear algebras, Mat. Sb.. Mat. Sb., Math. USSR Sb., 43, 85-101 (1981), (in Russian); translation:
[6] Drensky, V., Free Algebras and PI-Algebras (2000), Springer-Verlag: Springer-Verlag Singapore · Zbl 0936.16001
[7] Drensky, V.; Formanek, E., Polynomial Identity Rings, Advanced Courses in Mathematics, CRM Barcelona (2004), Birkhäuser: Birkhäuser Basel-Boston · Zbl 1077.16025
[8] Drensky, V.; Koshlukov, P. E., Weak polynomial identities for a vector space with a symmetric bilinear form, (Math. and Education in Math., Proc. of the 16-th Spring Conf. of the Union of Bulgar. Mathematicians. Math. and Education in Math., Proc. of the 16-th Spring Conf. of the Union of Bulgar. Mathematicians, Sunny Beach, April 6-10, 1987 (1987), Publishing House of the Bulgarian Academy of Sciences: Publishing House of the Bulgarian Academy of Sciences Sofia), 213-219 · Zbl 0658.16012
[9] Formanek, E., Central polynomials for matrix rings, J. Algebra, 23, 129-132 (1972) · Zbl 0242.15004
[10] Kaplansky, I., Problems in the theory of rings (June, 1956), National Acad. of Sci.-National Research Council: National Acad. of Sci.-National Research Council Washington, Publ. 502 (1957) 1-3
[11] Kaplansky, I., Problems in the theory of rings revised, Am. Math. Mon., 77, 445-454 (1970) · Zbl 0208.29701
[12] Le Bruyn, L., Trace Rings of Generic 2 by 2 Matrices, Mem. Amer. Math. Soc., vol. 66 (1987), No. 363 · Zbl 0675.16009
[13] Macdonald, I. G., Symmetric Functions and Hall Polynomials (1995), Oxford Univ. Press (Clarendon): Oxford Univ. Press (Clarendon) Oxford · Zbl 0899.05068
[14] Procesi, C., The invariant theory of \(n \times n\) matrices, Adv. Math., 198, 306-381 (1976) · Zbl 0331.15021
[15] Procesi, C., Computing with \(2 \times 2\) matrices, J. Algebra, 87, 342-359 (1984) · Zbl 0537.16013
[16] Procesi, C., Lie Groups (an Approach Through Invariants and Representations) (2007), Springer: Springer New York · Zbl 1154.22001
[17] Razmyslov, Yu. P., Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero, Algebra Log.. Algebra Log., Algebra Log., 12, 47-63 (1973), (in Russian); translation: · Zbl 0282.17003
[18] Razmyslov, Yu. P., On a problem of Kaplansky, Izv. Akad. Nauk SSSR, Ser. Mat.. Izv. Akad. Nauk SSSR, Ser. Mat., Math. USSR, Izv., 7, 479-496 (1973), (in Russian); translation: · Zbl 0314.16016
[19] Razmyslov, Yu. P., Finite basis property for identities of representations of a simple three-dimensional Lie algebra over a field of characteristic zero, (Algebra, Work Collect., Dedic. O.Yu. Shmidt. Algebra, Work Collect., Dedic. O.Yu. Shmidt, Moskva (1982)). (Algebra, Work Collect., Dedic. O.Yu. Shmidt. Algebra, Work Collect., Dedic. O.Yu. Shmidt, Moskva (1982)), Transl. Am. Math. Soc. (2), 140, 101-109 (1988), (in Russian); translation: · Zbl 0658.17014
[20] Razmyslov, Yu. P., Identities of Algebras and Their Representations, Translations of Math. Monographs, vol. 138 (1994), Sovremennaya Algebra, Nauka: Sovremennaya Algebra, Nauka Moscow: AMS: Sovremennaya Algebra, Nauka: Sovremennaya Algebra, Nauka Moscow: AMS Providence, RI, (in Russian); translation: · Zbl 0827.17001
[21] Regev, A., Algebras satisfying a Capelli identity, Isr. J. Math., 33, 149-154 (1979) · Zbl 0422.16008
[22] Sibirskii, K. S., Unitary and orthogonal invariants of matrices, Dokl. Akad. Nauk SSSR, 172, 40-43 (1967), (in Russian)
[23] Weyl, H., The Classical Groups, Their Invariants and Representations (1997), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 1024.20501
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