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A homomorphism from the affine Yangian $Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n))$ to the affine Yangian $Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n+1))$

Mamoru Ueda

Abstract

We construct a homomorphism from the affine Yangian $Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n))$ to the affine Yangian $Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n+1))$. We also give the relationship between this homomorphism and the one from the affine Yangian $Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n))$ to the universal enveloping algebra of the rectangular algebra $\mathcal{W}^k(\mathfrak{gl}(2n),(2^n))$ constructed by the author.

A homomorphism from the affine Yangian $Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n))$ to the affine Yangian $Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n+1))$

Abstract

We construct a homomorphism from the affine Yangian to the affine Yangian . We also give the relationship between this homomorphism and the one from the affine Yangian to the universal enveloping algebra of the rectangular algebra constructed by the author.
Paper Structure (12 sections, 8 theorems, 85 equations)

This paper contains 12 sections, 8 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Affine Yangian
  3. A homomorphism from the affine Yangian $Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n))$ to the affine Yangian $Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n+1))$
  4. Compatibility with \ref{['Eq2.3']}
  5. Compatibility with \ref{['Eq2.5']}
  6. Compatibility with \ref{['Eq2.6']}
  7. Compatibility with \ref{['Eq2.7']}
  8. Compatibility with \ref{['Eq2.8']}
  9. Compatibility with \ref{['Eq2.9']}
  10. Compatibility with \ref{['Eq2.1']}
  11. The rectangular $W$-algebra $\mathcal{W}^k(\mathfrak{gl}(2n),(2^n))$
  12. The relationship between homomorphism $\Psi$ and the rectangular $W$-algebra $\mathcal{W}^k(\mathfrak{gl}(2n),(2^n))$

Key Result

Proposition 2.8

Suppose that $n\geq3$. The affine Yangian $Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}}(n))$ is isomorphic to the associative algebra $Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n))$ generated by $X_{i,r}^{+}, X_{i,r}^{-}, H_{i,r}$$(i \in \{0,1,\cdots, n-1\}, r = 0,1)$ subject to the f where $\tilde{H}_{i,1}=H_{i,1}-\dfrac{\hbar}{2}H_{i,0}^2$, $\hbar=\varepsilon_1+\varepsilon_2$ and

Theorems & Definitions (14)

  • Definition 2.1
  • Proposition 2.8
  • Remark 2.21
  • Lemma 2.22: Proposition 3.21 in GNW
  • Theorem 3.1
  • Corollary 3.2
  • proof
  • Theorem 4.2: Theorem 3.1 and Corollary 3.2 in AM
  • Remark 4.3
  • Theorem 4.4
  • ...and 4 more