本文選題：量子頂點(diǎn)代數(shù) + 量子仿射代數(shù)　； 參考：《中國(guó)科學(xué):數(shù)學(xué)》2017年11期

【摘要】：在廣義的頂點(diǎn)代數(shù)領(lǐng)域中,一個(gè)基本的公開(kāi)問(wèn)題是,建立一個(gè)適當(dāng)?shù)牧孔禹旤c(diǎn)代數(shù)理論使得量子仿射代數(shù)和量子頂點(diǎn)代數(shù)自然地聯(lián)系起來(lái).部分地受Etingof和Kazhdan的量子頂點(diǎn)算子代數(shù)理論的啟發(fā),自2005年,作者系統(tǒng)地發(fā)展和研究了一個(gè)(弱)量子頂點(diǎn)代數(shù)及其擬模和φ-坐標(biāo)擬模理論,建立了一些經(jīng)典代數(shù)(如雙楊氏代數(shù))同量子頂點(diǎn)代數(shù)的自然聯(lián)系,特別是最終給出了量子仿射代數(shù)同該意義下的弱量子頂點(diǎn)代數(shù)的一個(gè)自然聯(lián)系.在此聯(lián)系中,相對(duì)應(yīng)的弱量子頂點(diǎn)代數(shù)在理論上存在,但其具體結(jié)構(gòu)仍需要進(jìn)一步去確定,并需證明它們是量子頂點(diǎn)代數(shù).在某種程度上講,這給所提的公開(kāi)問(wèn)題提供了一個(gè)初步答案.另一方面,這個(gè)理論在其發(fā)展的同時(shí)已被用來(lái)建立一些重要的代數(shù)同量子頂點(diǎn)代數(shù)的聯(lián)系,顯示了該理論的實(shí)用價(jià)值.本篇綜述概括總結(jié)作者在這方面的主要結(jié)果,其中包括Zamolodchikov-Faddeev代數(shù)、無(wú)中心雙楊氏代數(shù)、量子βγ-系統(tǒng)和量子仿射代數(shù)同(弱)量子頂點(diǎn)代數(shù)的聯(lián)系.
[Abstract]:In the field of generalized vertex algebra, a basic open problem is to establish an appropriate quantum vertex algebra theory so that the quantum affine algebra and the quantum vertex algebra are connected naturally. Inspired in part by Etingof and Kazhdan's theory of quantum vertex operator algebra, since 2005, the author has systematically developed and studied a (weak) quantum vertex algebra and its quasi-module and 蠁 -coordinate pseudomodule theory. In this paper, we establish the natural relation between some classical algebras (such as double Young algebras) and quantum vertex algebras, especially, we give a natural relation between quantum affine algebras and weak quantum vertex algebras in this sense. In this connection, the corresponding weak quantum vertex algebras exist in theory, but their specific structures still need to be further determined, and they need to be proved to be quantum vertex algebras. To some extent, this provides a preliminary answer to the open question. On the other hand, this theory has been used to establish the relation between some important algebras and quantum vertex algebras, which shows the practical value of the theory. In this paper, the author's main results in this field are summarized, including Zamolodchikov-Faddeev algebra, non-centroid double Young algebra, quantum 尾 緯 -system and quantum affine algebra and the relation between (weak) quantum vertex algebra and (weak) quantum vertex algebra.
【作者單位】： Department
【基金】：國(guó)家自然科學(xué)基金(批準(zhǔn)號(hào):11471268和11571391)資助項(xiàng)目
【分類號(hào)】：O152.5

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