Extensions of the Tensor Algebra and Their Applications

This article presents a natural extension of the tensor algebra. This extended algebra is based on a vector space as the ordinary tensor algebra is. In addition to “left multiplications” by vectors, we can consider “derivations” by covectors as fundamental operators on this algebra. These two types of operators satisfy an analogue of the canonical commutation relations, and we can regard the algebra generated by these operators as an analogue of the Weyl algebra and the Clifford algebra (actually this operator algebra contains these algebras naturally as quotient algebras). These extensions of the tensor algebra have some applications: (i) applications to invariant theory related to tensor products, and (ii) applications to immanants. The latter one includes a new method to study the quantum immanants in the universal enveloping algebras of the general linear Lie algebras and their Capelli type identity (higher Capelli identity). Introduction In this article, we introduce some extensions of the tensor algebra. The most basic one is constructed as a vector space as follows: T̄ (V ) = ⊕ p≥0 V ⊗p ⊗CSp CS∞. For this T̄ (V ), we can naturally define an associative algebra structure. The ordinary tensor algebra T (V ) can be regarded as a subalgebra of this algebra. This extended algebra T̄ (V ) is remarkable, because we can consider a natural “derivation” L(v) determined from any covector v ∈ V ∗ as an operator on T̄ (V ). An analogue of the canonical commutation relations holds between these derivations and the left multiplications L(v) by vectors v ∈ V (Theorem 2.3). It is also natural to call these multiplications and derivations “creation operators” and “annihilation operators,” respectively (namely, we can regard this T̄ (V ) as an analogue of the Boson and Fermion Fock spaces). The algebra L(V ) generated by these two types of operators is naturally isomorphic to ⊕ p,q≥0 V ⊗p ⊗CSp CS∞ ⊗CSq V ∗⊗q as vector spaces, and we can regard this operator algebra as an analogue of the Weyl algebra and the Clifford algebra (actually L(V ) contains the Weyl algebra and the Clifford algebra naturally as quotient algebras). This framework has some applications to invariant theory related to tensor products. First, we can describe the commutants of some fundamental classes of operators on tensor 2000 Mathematics Subject Classification. Primary 15A72; Secondary 15A15, 17B35, 20C30.