On the structure of quantum affine superalgebra Uv(A(0,2)(4))
We investigate the algebra Uv(A(0, 2)(4))+ defined by quantum Serre relations, when v is not a root of unity. We prove that Uv(A(0, 2)(4))+ is isomorphic to the Nichols algebra of its degree one part. In other words, Uv(A(0, 2)(4))+ can equivalently be defined by the radical of the standard bilinear...
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| Format: | Dissertation |
| Sprache: | Englisch |
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Philipps-Universität Marburg
2025
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| Zusammenfassung: | We investigate the algebra Uv(A(0, 2)(4))+ defined by quantum Serre relations, when v is not a root of unity. We prove that Uv(A(0, 2)(4))+ is isomorphic to the Nichols algebra of its degree one part. In other words, Uv(A(0, 2)(4))+ can equivalently be defined by the radical of the standard bilinear form. We determine all the root multiplicities and give a PBW basis of Uv(A(0, 2)(4))+. We construct new distinguished imaginary root vectors and determine all the commutation relations among all the real root vectors and distinguished imaginary root vectors, i.e. Drinfeld second realization is obtained for Uv(A(0, 2)(4))+.
The thesis contains five chapters. In the first chapter, we introduce some basic notation and theories such as the theory of affine Kac-Moody algebras, the theory of quantum affine algebras and quantum affine superalgebras, the theory of Nichols algebras, including the definitions, the Cartan graphs, the Weyl groupoids and the root systems of Nichols algebras of diagonal type, the Lyndon word theory, the affine Nichols algebras, and an important subquotient of Nichols algebra of diagonal type etc. The first chapter also contains some elementary results on the main research object of this thesis, i.e. Uv(A(0, 2)(4))+.
In the second chapter, we discuss the Nichols algebra B(V). First we discuss some basic properties of B(V) and the root system of B(V). Then we derive the main relations in B(V), especially in the imaginary roots space and then write these relations in the form of series. We also get information about some central elements and the comultiplication in the subquotient, which will be used in the third chapter.
In the third chapter, we determine all the root multiplicities completely by using Lyndon word theory and the subquotient. The key step is finding the primitive elements at certain degrees in the subquotient and determine they are not zero in B(V). For the primitivities, we use the exponential and arctangent series to construct new distinguished imaginary root vectors, which is a generation of the use of the exponential series in the classical Drinfeld second realizations of the affine quantum groups.
In the fourth chapter, we prove that B(V) is isomorphic to Uv(A(0,2)(4))+. The main step is to verify all the commutation relations between the root vectors in B(V) come from the quantum Serre relations.
In the fifth chapter, we determine all the commutation relations among all the real root vectors and distinguished imaginary root vectors concretely, that is, an analog of the Drinfeld second realization is obtained for Uv(A(0, 2)(4))+. |
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| DOI: | 10.17192/z2025.0067 |