2012/02/29 by Zhaojia Tong, Naihong Hu, Xiuling Wang · 6 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Base (topology) #Combinatorics #Dimension (graph theory) #Discrete mathematics #Hopf algebra #Lie algebra #Mathematical analysis #Mathematics #Modulo #Noncommutative geometry #Nonlinear Waves and Solitons #Polynomial #Prime (order theory) #Product (mathematics) #Pure mathematics #Reduction (mathematics) #Type (biology) #Universal enveloping algebra #math.QA #msc:17B37 #msc:17B50 #msc:17B62
paper · pdf · doi:10.1090/conm/652/12980
published in Contemporary mathematics - American Mathematical Society, 173-206 (American Mathematical Society) · 33 pages
openalex publication_date 2015/01/01 · arxiv created 2015/12/20 · arxiv updated 2015/12/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We construct explicit Drinfel’d twists for the Lie algebras of generalized Cartan type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and also obtain the corresponding quantizations and their integral forms. By using modular reduction and base changes, we derive certain quantizations of the restricted universal enveloping algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold u left-parenthesis bold upper H left-parenthesis 2 n semicolon ModifyingBelow 1 With bar right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">u</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">H</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>;</mml:mo> <mml:munder> <mml:mn>1</mml:mn> <mml:mo> _ </mml:mo> </mml:munder> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf u(\mathbf H(2n;\underline 1))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the restricted Hamiltonian algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper H left-parenthesis 2 n semicolon ModifyingBelow 1 With bar right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">H</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>;</mml:mo> <mml:munder> <mml:mn>1</mml:mn> <mml:mo> _ </mml:mo> </mml:munder> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf H(2n;\underline 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in prime characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . These quantizations are new non-pointed Hopf algebras of prime-power dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Superscript p Super Superscript 2 n Superscript minus 1"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">p^p2n-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and contain the well-known Radford algebras as Hopf subalgebras. As a by-product we also obtain some Jordanian quantizations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German s German p Subscript 2 n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">p</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak sp2n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .