Cocharacters for the weak polynomial identities of the Lie algebra of \(3 \times 3\) skew-symmetric matrices.

*(English)*Zbl 07258198Let \(so_3\) 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.

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.

Reviewer: Plamen Koshlukov (Campinas)

##### MSC:

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 |

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\textit{M. Domokos} and \textit{V. Drensky}, Adv. Math. 374, Article ID 107343, 40 p. (2020; Zbl 07258198)

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