Invariant theory for matrices over the Grassmann algebra.
The finite dimensional complex \(\mathbb Z_2\)-graded simple algebras form three families: (1) the \(n\times n\) matrix algebra \(M_n(\mathbb C)\) with trivial grading; (2) the algebra \(M_{k,\ell}(\mathbb C)=M_{k+\ell}(\mathbb C)\) with even part \(Ae_{11}+De_{22}\) and odd part \(Be_{12}+Ce_{21}\), where the blocks \(A,B,C\), and \(D\) are, respectively, \(k\times k\), \(k\times\ell\), \(\ell\times k\), and \(\ell\times\ell\) matrices; (3) the algebra \(M_n(\mathbb C[u]/(u^2-1))\) where \(M_n(\mathbb C)\) is of even degree and \(u\) is of odd degree. This algebra is isomorphic to the subalgebra of \(M_{n,n}(\mathbb C)\) spanned by \(A(e_{11}+e_{22})\) and \(B(e_{12}+e_{21})\), \(A,B\in M_n(\mathbb C)\). To each \(\mathbb Z_2\)-graded simple algebra one associates a Lie superalgebra, a Lie supergroup, and a verbally prime PI-algebra. (Verbally prime PI-algebras are the key objects in the structure theory of T-ideals developed by Kemer.) The corresponding Lie superalgebras are constructed from the graded algebras using the super Lie bracket, and are equal, respectively, to: (1) the general linear Lie algebra \(gl_n(\mathbb C)\); (2) the general linear superalgebra \(pl(k,\ell)\); (3) the queer superalgebra \(q(n)\). The corresponding verbally prime PI-algebra in the case (1) is \(M_n(\mathbb C)\). In the cases (2) and (3) one has to replace the even and the odd entries of the matrices by even and odd elements of the infinite dimensional Grassmann algebra \(E\). The corresponding verbally prime PI-algebras are \(M_{k,\ell}(E)\) and \(SM_{n,n}(E)\) which is isomorphic to \(M_n(E)\). The related supergroups are: (1) the general linear group \(GL_n(\mathbb C)\); (2) the general linear supergroup \(PL(k,\ell)\) of the invertible matrices in \(pl(k,\ell)\); (3) the queer supergroup \(Q(n)\) of the invertible matrices in \(SM_n(E)\) which is isomorphic to \(GL_n(E)\). Each of the Lie superalgebras has a basic representation \(U\) and there is a double centralizer theorem describing the centralizer of the action of the superalgebra on \(U^{\otimes d}\). In the cases of \(gl_n(\mathbb C)\) and \(pl(k,\ell)\) this double centralizer theory has been applied to invariant theory, generic matrices and trace identities. The paper under review contains important contributions to the topic. Its main results describe the invariant theory and the related theory of generic matrix objects in the only left case of \(q(n)\). The author studies the algebras of polynomial maps \(M_n(E)^d\to E\) and \(M_n(E)^d\to M_n(E)\) which are invariant under simultaneous conjugation by matrices from \(GL_n(E)\). The results are stated in terms of the queer trace of matrices with Grassmann entries defined as the odd component of the sum of the diagonal entries of the matrix. The main results are: The algebra of invariant functions \(M_n(E)^d\to E\) is generated by the queer trace polynomials. The algebra of invariant functions \(M_n(E)^d\to M_n(E)\) is generated by the coordinate functions \((A_1,\dots,A_d)\to A_i\) together with the even number of queer trace monomials. Finally, working with the superanalogues of the generic matrix algebra and the generic trace algebra, the author establishes the relations of these objects with the corresponding invariant theory.
scientific article; zbMATH DE number 6154616