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Kazuyuki Oshima

Publications and source records attributed to Kazuyuki Oshima.

8 recordsLinked to original sources
arXiv2024

Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations

We introduce a new elliptic quantum toroidal algebra U_{q,κ,p}(g_tor) associated with an arbitrary toroidal algebra g_tor. We show that U_{q,κ,p}(g_tor) contains two elliptic quantum algebras associated with a corresponding affine Lie algebra bg as subalgebras. They are analogue of the horizontal and the vertical subalgebras in the quantum toroidal algebra U_{q,κ}(g_tor). A Hopf algebroid structure is introduced as a co-algebra structure of U_{q,κ,p}(g_tor) using the Drinfeld comultiplication. We also investigate the Z-algebra structure of U_{q,κ,p}(g_tor) and show that the Z-algebra governs the irreducibility of the level (k(\not=0),l)-infinite dimensional U_{q,κ,p}(g_tor)-modules in the same way as in the elliptic quantum group U_{q,p}(g). As an example, we construct the level (1,l) irreducible representation of U_{q,κ,p}(g_tor) for the simply laced gtor. We also construct the level (0,1) representation of U_{q,κ,p}(g_N,tor) and discuss a conjecture on its geometric interpretation as an action on the torus equivariant elliptic cohomology of the affine A_{N-1} quiver variety.

math.RT↗
arXiv2023

Elliptic Quantum Toroidal Algebra $U_{q,t,p}(gl_{1,tor})$ and Affine Quiver Gauge Theories

We introduce a new elliptic quantum toroidal algebra $U_{q,t,p}(gl_{1,tor})$. Various representations in the quantum toroidal algebra $U_{q,t}(gl_{1,tor})$ are extended to the elliptic case including the level (0,0) representation realized by using the elliptic Ruijsenaars difference operator. Intertwining operators of $U_{q,t,p}(gl_{1,tor})$-modules w.r.t. the Drinfeld comultiplication are also constructed. We show that $U_{q,t,p}(gl_{1,tor})$ gives a realization of the affine quiver $W$-algebra $W_{q,t}(Γ(\widehat{A}_0))$ proposed by Kimura-Pestun. This realization turns out to be useful to derive the Nekrasov instanton partition functions, i.e. the $χ_{y^-}$ and elliptic genus, of the 5d and 6d lifts of the 4d $\mathcal{N}=2^*$ theories and provide a new Alday-Gaiotto-Tachikawa correspondence.

math.QA↗
arXiv2014

Elliptic Algebra U_{q,p}(g^) and Quantum Z-algebras

A new definition of the elliptic algebra U_{q,p}(g^) associated with an untwisted affine Lie algebra g^ is given as a topological algebra over the ring of formal power series in p. We also introduce a quantum dynamical analogue of Lepowsky-Wilson's Z-algebras. The Z-algebra governs the irreducibility of the infinite dimensional U_{q,p}(g^)-modules. Some level-1 examples indicate a direct connection of the irreducible U_{q,p}(g^)-modules to those of the W-algebras associated with the coset g^ \oplus g^ \supset (g^)_{diag} with level (r-g-1,1) (g:the dual Coxeter number), which includes Fateev- Lukyanov's WB_l-algebra.

math.QA↗
arXiv2014

Elliptic Quantum Group U_{q,p}(B_N^{(1)}) and Vertex Operators

Assuming the existence of the L-operators, we study the Hopf algebroid structure of U_{q,p}(B_N^{(1)}). As an application, we derive the type I and II vertex operators, which intertwine the U_{q,p}(B_N^{(1)})-modules of generic level, by assuming some analytic properties of the L-operators. For the level-1 case, we construct their free field realizations and show that the results satisfy the desired commutation relations with coefficients given by the elliptic dynamical R-matrices of the B_N^{(1)} type.

math.QA↗
arXiv2009

Symmetry of $osp(m|n)$ spin Calogero-Sutherland models

We introduce osp(m|n) spin Calogero-Sutherland models and find that the models have the symmetry of osp(m|n) half-loop algebra or Yangian of osp(m|n) if and only if the coupling constant of the model equals to 2/(m-n-4).

nlin.SI↗
arXiv2006

The Lie algebraic structure of extended Sutherland models

We disclose the Lie algebraic structure of two extended Sutherland models. Their Hamiltonians are BC_N, and A_N Sutherland Hamiltonians with some additional terms. We show that both Hamiltonians can be written in the quadratic forms of generators of the Lie algebra gl(N+1).

nlin.SI↗
arXiv1996

Spectral Decomposition of Path Space in Solvable Lattice Model

We give the {\it spectral decomposition} of the path space of the $U_q(\hatsl)$ vertex model with respect to the local energy functions. The result suggests the hidden Yangian module structure on the $\hatsl$ level $l$ integrable modules, which is consistent with the earlier work [1] in the level one case. Also we prove the fermionic character formula of the $\hatsl$ level $l$ integrable representations in consequence.

q-alg↗