Mathematics

We answer a question of A. Skalski and P.M. Sołtan (2016) about inner faithfulness of the S.~Curran's map of extending a quantum increasing sequence to a quantum permutation in full generality. To do so, we exploit some novel techniques introduced by Banica (2018) and Brannan, Chirvasitu, Freslon (2018) concerned with the Banica's conjecture regarding quantum permutation groups. Roughly speaking, we find a inductive setting in which the inner faithfulness of Curran's map can be boiled down to inner faithfulness of similar map for smaller algebras and then rely on inductive generation result for quantum permutation groups of Brannan, Chirvasitu and Freslon.

The free complex sphere S N?? C,+ is the noncommutative manifold defined by the equations ??i x i x ??i = ??i x ??i x i =1 . Certain submanifolds X??S N?? C,+ , related to the quantum groups, are known to have Riemannian features, including an integration functional. We review here the known facts on the subject.

The permutation group S N has a free analogue S + N , which is non-classical and infinite at N?? . We review here the known basic facts on S + N , with emphasis on algebraic and probabilistic aspects. We discuss as well the structure of the closed subgroups G??S + N , with particular attention to the quantum reflection groups.

The purpose of this paper is to propose a sheaf theoretic approach to the theory of quantum principal bundles over non affine bases. We study the quantization of principal bundles G -> G/P, where G is a semisimple group and P a parabolic subgroup.

By using certain quantum differential operators, we construct a super representation for the quantum queer supergroup U_v(q_n). The underlying space of this representation is a deformed polynomial superalgebra in 2n^2 variables whose homogeneous components can be used as the underlying spaces of queer q-Schur superalgebras. We then extend the representation to its formal power series algebra which contains a (super) submodule isomorphic to the regular representation of U_v(q_n). A monomial basis M for U_v(q_n) plays a key role in proving the isomorphism. In this way, we may present the quantum queer supergroup U_v(q_n) by another new basis L together with some explicit multiplication formulas by the generators. As an application, similar presentations are obtained for queer q-Schur superalgebras via the above mentioned homogeneous components. The existence of the bases M and L and the new presentation show that the seminal construction of quantum gl_n established by Beilinson-Lusztig-MacPherson thirty years ago extends to this "queer" quantum supergroup via a completely different approach.

We explain the notion of " q -deformed real numbers" introduced in our previous work and overview their main properties. We will also introduce q -deformed Conway-Coxeter friezes.

We introduce the notion of nonlocal symmetry of a graph G , defined as a winning quantum correlation for the G -automorphism game that cannot be produced classically. Recent connections between quantum group theory and quantum information show that quantum correlations for this game correspond to tracial states on C(Qut(G)) -- the algebra of functions on the quantum automorphism group of G . This allows us to also define nonlocal symmetry for any quantum permutation group. We investigate the differences and similarities between this and the notion of quantum symmetry, defined as non-commutativity of C(Qut(G)) . Roughly speaking, quantum symmetry vs nonlocal symmetry can be viewed respectively as non-classicality of our model of reality vs non-classicality of our observation of reality. We show that quantum symmetry is necessary but not sufficient for nonlocal symmetry. In particular, we show that the complete graph on five vertices is the only connected graph on five or fewer vertices with nonlocal symmetry, despite a dozen others having quantum symmetry. In particular this shows that the quantum symmetric group on four points, S + 4 , does not exhibit nonlocal symmetry, answering a question from the literature. In contrast to quantum symmetry, we show that two disjoint classical automorphisms do not guarantee nonlocal symmetry. However, three disjoint automorphisms do suffice. We also give a construction of quantum permutation matrices built from a finite abelian group ? and a permutation ? on |?| elements. Computational evidence suggests that for cyclic groups of increasing size almost all permutations ? result in nonlocal symmetry. Surprisingly, the construction never results in nonlocal symmetry when Z 3 2 is used. We also investigate under what conditions nonlocal symmetry arises when taking unions or products of graphs.

We introduce and study the quantum toroidal algebra E m|n ( q 1 , q 2 , q 3 ) associated with the superalgebra gl m|n with m≠n , where the parameters satisfy q 1 q 2 q 3 =1 . We give an evaluation map. The evaluation map is a surjective homomorphism of algebras E m|n ( q 1 , q 2 , q 3 )→ U ˜ q gl ˆ m|n to the quantum affine algebra associated with the superalgebra gl m|n at level c completed with respect to the homogeneous grading, where q 2 = q 2 and q m−n 3 = c 2 . We also give a bosonic realization of level one E m|n ( q 1 , q 2 , q 3 ) -modules.

We consider fundamental facts from the theory of Hopf superalgebras. We use them to construct the quantum double of the quantum superalgebra sl(2|1) at roots of unity. Thus we obtain a multiplicative formula for universal R -matrix. Next we construct an R -matrix to investigate parametrized family of centralizer algebras. We give multiplication laws in particular case and describe a structure of such algebras in the general case.

The generalized quantum group U(ϵ) of type A is an affine analogue of quantum group associated to a general linear Lie superalgebra gl M|N . We prove that there exists a unique R matrix on tensor product of fundamental type representations of U(ϵ) for arbitrary parameter sequence ϵ corresponding to a non-conjugate Borel subalgebra of gl M|N . We give an explicit description of its spectral decomposition, and then as an application, construct a family of finite-dimensional irreducible U(ϵ) -modules which have subspaces isomorphic to the Kirillov-Reshetikhin modules of usual affine type A (1) M−1 or A (1) N−1 .