# Quadratic relations of the deformed W-superalgebra

@inproceedings{Kojima2020QuadraticRO, title={Quadratic relations of the deformed W-superalgebra}, author={Takeo Kojima}, year={2020} }

This paper is a continuation of the study by Ding and Feigin, Contemp.Math. 248, 83 (1998). We find a bosonization of the deformed W -superalgebras Wq,t(sl(2|1)) that commute up to the total difference with deformed screening currents. Using the bosonization, we derive a set of quadratic relations of generators for the deformed W -superalgebra Wq,t(sl(2|1)), which is independent of the choice of Dynkin-diagram for the superalgebra sl(2|1), though the deformed screening currents depend on it.

## 6 Citations

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## References

SHOWING 1-10 OF 40 REFERENCES

Deformations of W-algebras associated to simple Lie algebras

- Mathematics
- 1997

Deformed W-algebra Wq,t(g) associated to an arbitrary simple Lie alge- bra g is defined together with its free field realizations and the screening operators. Explicit formulas are given for…

Drinfeld–Sokolov Reduction for Difference
Operators and Deformations of W-Algebras¶ II. The General Semisimple Case

- Mathematics
- 1998

Abstract:The paper is the sequel to [9]. We extend the Drinfeld--Sokolov reduction procedure to q-difference operators associated with arbitrary semisimple Lie algebras. This leads to a new elliptic…

Integrals of motion from quantum toroidal algebras

- Mathematics
- 2017

We identify the Taylor coefficients of the transfer matrices corresponding to quantum toroidal algebras with the elliptic local and non-local integrals of motion introduced by Kojima, Shiraishi,…

Deformed Virasoro Algebras from Elliptic Quantum Algebras

- Mathematics
- 2016

We revisit the construction of deformed Virasoro algebras from elliptic quantum algebras of vertex type, generalizing the bilinear trace procedure proposed in the 1990s. It allows us to make contact…

Commutation relations of vertex operators for Uq(sl^(M|N))

- MathematicsJournal of Mathematical Physics
- 2018

We consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra $U_q(\widehat{sl}(M|N))$ by using bosonization. We show that vertex operators…

Quantized W-algebra of sl(2,1) : a construction from the quantization of screening operators

- Mathematics
- 1998

Starting from bosonization, we study the operator that commute or commute up-to a total difference with of any quantized screen operator of a free field. We show that if there exists a operator in…

Drinfeld–Sokolov Reduction for Difference Operators and Deformations of -Algebras¶I. The Case of Virasoro Algebra

- Mathematics
- 1998

Abstract:We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical -algebras by reduction from Poisson-Lie loop groups. We consider in…

Integrals of motion and quantum groups

- Mathematics
- 1993

A homological construction of integrals of motion of the classical and quantum Toda field theories is given. Using this construction, we identify the integrals of motion with cohomology classes of…

On the defining relations of the affine Lie superalgebras and their quantized universal enveloping superalgebras

- Mathematics
- 1996

In this paper, we give defining relations of the affine Lie superalgebras an and defining relations of a super-version of the Drinfeld[D]-Jimbo[J] affine quantized universal enveloping algebras. As a…

A Level-one Representation of the Quantum Affine Superalgebra U Q ( Sl(m + 1|n + 1))

- Mathematics
- 1996

A level-one representation of the quantum affine superalgebra U q (sl(M + 1|N + 1)) and vertex operators associated with the fundamental representations are constructed in terms of free bosonic…