Related papers: Commutation relations among generic quantum minors…
Complete sets of commutation relations for arbitrary pairs of quantum minors are computed, with explicit coefficients in closed form.
We produce neccessary and sufficient conditions for pairs of quantum minors in the quantized coordinate algebra $\Bbb{C}_q[Mat_{k \times m}]$ to quasi-commute. In addition we study the combinatorics of maximal (by inclusion) families of…
We calculate commutation relations of vertex operators for the spin representation of $U_q(D_n^{(1)})$ by using recursive formulae of R-matrices. In quantum symmetry approach, we obtain the energy and momentum spectrum of the quantum spin…
The model of generalized quons is described in a purely algebraic way. Commutation relations and corresponding consistency conditions for our generalized quons system are studied in terms of quantum Weyl algebras. Fock space representation…
The notion of quantum matrix pairs is defined. These are pairs of matrices with non-commuting entries, which have the same pattern of internal relations, q-commute with each other under matrix multiplication, and are such that products of…
We give a complete combinatorial characterization of homogeneous quadratic relations of "universal character" valid for minors of quantum matrices (more precisely, for minors in the quantized coordinate ring $O_q(M_{m,n}(K))$ of $m\times n$…
The meaning of quantum group transformation properties is discussed in some detail by comparing the (co)actions of the quantum group with those of the corresponding Lie group, both of which have the same algebraic (matrix) form of the…
For the family of the orthogonal quantum matrix algebras we investigate the structure of their characteristic subalgebras -- special commutative subalgebras, which for the subfamily of the reflection equation algebras appear to be central.…
The traces of the quantum powers of a generic quantum matrix pairwise commute. This was conjectured by Kaoru Ikeda, in connection with certain Hamiltonian systems. The proof involves Newton's formulae for quantum matrices, relating traces…
We present several identities involving quasi-minors of noncommutative generic matrices. These identities are specialized to quantum matrices, yielding q-analogues of various classical determinantal formulas.
We give a detailed derivation of the commutation relations for the Poincar\'e--Birkhoff--Witt generators of the quantum superalgebra $\mathrm U_q(\mathfrak{gl}_{M|N})$.
We define a "quantum relation" on a von Neumann algebra M \subset B(H) to be a weak* closed operator bimodule over its commutant M'. Although this definition is framed in terms of a particular representation of M, it is effectively…
A simple exposition of the rarely discussed fact that a set of free boson fields describing different, i.e. kinematically different particle types can be quantized with mutual anticommutation relations is given by the explicit construction…
We consider quantum integrable models with $\mathfrak{gl}(2|1)$ symmetry. We derive a set of multiple commutation relations between the monodromy matrix entries. These multiple commutation relations allow us to obtain different…
The main theorem of the paper allows to generalize a class of identities among the quantum minors for quantum linear groups to similar identities but with the row labels of the quantum minors involved permuted.
To any complex Hadamard matrix we associate a quantum permutation group. The correspondence is not one-to-one, but the quantum group encapsulates a number of subtle properties of the matrix. We investigate various aspects of the…
A standard bicovariant differential calculus on a quantum matrix space ${\tt Mat}(m,n)_q$ is considered. The principal result of this work is in observing that the $U_q\frak{s}(\frak{gl}_m\times \frak{gl}_n))_q$ is in fact a…
We complete the rules of translation between standard complex quantum mechanics (CQM) and quaternionic quantum mechanics (QQM) with a complex geometry. In particular we describe how to reduce ($2n$+$1$)-dimensional complex matrices to {\em…
We carry out a generalization of quantum group co-representations in order to encode in this structure those cases where non-commutativity between endomorphism matrix entries and quantum space coordinates happens.
It is shown how formulas of the author for general operator transmutation can be adapted to a quantum group context