A qq-analogue of derivations on the tensor algebra and the qq-Schur-Weyl duality

This paper presents a $q$-analogue of an extension of the tensor algebra given by the same author. This new algebra naturally contains the ordinary tensor algebra and the Iwahori-Hecke algebra type $A$ of infinite degree. Namely this algebra can be regarded as a natural mix of these two algebras. Moreover, we can consider natural "derivations" on this algebra. Using these derivations, we can easily prove the $q$-Schur-Weyl duality (the duality between the quantum enveloping algebra of the general linear Lie algebra and the Iwahori-Hecke algebra of type $A$).Comment: 10 pages; revised version; to appear in Lett. Math. Phy

Invariant theory in exterior algebras and Amitsur-Levitzki type theorems

This article discusses invariant theories in some exterior algebras, which are closely related to Amitsur-Levitzki type theorems. First we consider the exterior algebra on the vector space of square matrices of size $n$, and look at the invariants under conjugations. We see that the algebra of these invariants is isomorphic to the exterior algebra on an $n$-dimensional vector space. Moreover we give a Cayley-Hamilton type theorem for these invariants (the anticommutative version of the Cayley-Hamilton theorem). This Cayley-Hamilton type theorem can also be regarded as a refinement of the Amitsur-Levitzki theorem. We discuss two more Amitsur-Levitzki type theorems related to invariant theories in exterior algebras. One is a famous Amitsur-Levitzki type theorem due to Kostant and Rowen, and this is related to $O(V)$-invariants in $\Lambda(\Lambda_2(V))$. The other is a new Amitsur-Levitzki type theorem, and this is related to $GL(V)$-invariants in $\Lambda(\Lambda_2(V) \oplus S_2(V^*))$.Comment: 18 pages; minor revision; to appear in Adv. Mat

Two determinants in the universal enveloping algebras of the orthogonal Lie algebras

AbstractThis paper gives a direct proof for the coincidence of the following two central elements in the universal enveloping algebra of the orthogonal Lie algebra: an element recently given by A. Wachi in terms of the column-determinant in a way similar to the Capelli determinant, and an element given by T. Umeda and the author in terms of the symmetrized determinant. The fact that these two elements actually coincide was shown by A. Wachi, but his observation was based on the following two non-trivial results: (i) the centrality of the first element, and (ii) the calculation of the eigenvalue of the second element. The purpose of this paper is to prove this coincidence of two central elements directly without using these (i) and (ii). Conversely this approach provides us new proofs of (i) and (ii). A similar discussion can be applied to the symplectic Lie algebras

Valence-Band Structures of Lead Halides by Ultraviolet Photoelectron Spectroscopy

Article信州大学工学部紀要 80: 19-28 (1998)departmental bulletin pape

Is Auger-free luminescence present in CeF₃?

It is well known that Auger-free luminescence (AFL) is observable when the condition E-g>E-vc is satisfied, where E-g is the band-gap energy between the lowest unoccupied band and the highest occupied band and E-vc the energy difference between the top of the highest occupied band and the top of the next lower occupied band. From measurements of reflection and X-ray photoelectron spectra, CeF₃ is demonstrated to really satisfy this condition. No evidence for AFL is found, nevertheless. The absence of AFL in CeF₃ is related to a characteristic nature of its highest and next lower occupied bands, which are quite different from those of previously studied AFL-materials.ArticleJOURNAL OF LUMINESCENCE. 129(9):984-987 (2009)journal articl