%!PS-Adobe-3.0 %%BoundingBox: 54 72 558 720 %%Creator: Mozilla (NetScape) HTML->PS %%DocumentData: Clean7Bit %%Orientation: Portrait %%Pages: 15 %%PageOrder: Ascend %%Title: http://msfac1.math.yorku.ca/Web/FPSAC98/Benkart/Benkart.tex %%EndComments %%BeginProlog [ /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis] /isolatin1encoding exch def /c { matrix currentmatrix currentpoint translate 3 1 roll scale newpath 0 0 1 0 360 arc setmatrix } bind def /F0 /Times-Roman findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding isolatin1encoding def currentdict end definefont pop /f0 { /F0 findfont exch scalefont setfont } bind def /F1 /Times-Bold findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding isolatin1encoding def currentdict end definefont pop /f1 { /F1 findfont exch scalefont setfont } bind def /F2 /Times-Italic findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding isolatin1encoding def currentdict end definefont pop /f2 { /F2 findfont exch scalefont setfont } bind def /F3 /Times-BoldItalic findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding isolatin1encoding def currentdict end definefont pop /f3 { /F3 findfont exch scalefont setfont } bind def /F4 /Courier findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding isolatin1encoding def currentdict end definefont pop /f4 { /F4 findfont exch scalefont setfont } bind def /F5 /Courier-Bold findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding isolatin1encoding def currentdict end definefont pop /f5 { /F5 findfont exch scalefont setfont } bind def /F6 /Courier-Oblique findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding isolatin1encoding def currentdict end definefont pop /f6 { /F6 findfont exch scalefont setfont } bind def /F7 /Courier-BoldOblique findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding isolatin1encoding def currentdict end definefont pop /f7 { /F7 findfont exch scalefont setfont } bind def /rhc { { currentfile read { dup 97 ge { 87 sub true exit } { dup 48 ge { 48 sub true exit } { pop } ifelse } ifelse } { false exit } ifelse } loop } bind def /cvgray { % xtra_char npix cvgray - (string npix long) dup string 0 { rhc { cvr 4.784 mul } { exit } ifelse rhc { cvr 9.392 mul } { exit } ifelse rhc { cvr 1.824 mul } { exit } ifelse add add cvi 3 copy put pop 1 add dup 3 index ge { exit } if } loop pop 3 -1 roll 0 ne { rhc { pop } if } if exch pop } bind def /smartimage12rgb { % w h b [matrix] smartimage12rgb - /colorimage where { pop { currentfile rowdata readhexstring pop } false 3 colorimage } { exch pop 8 exch 3 index 12 mul 8 mod 0 ne { 1 } { 0 } ifelse 4 index 6 2 roll { 2 copy cvgray } image pop pop } ifelse } def /cshow { dup stringwidth pop 2 div neg 0 rmoveto show } bind def /rshow { dup stringwidth pop neg 0 rmoveto show } bind def %%EndProlog %%Page: 1 1 %%BeginPageSetup /pagelevel save def 54 0 translate %%EndPageSetup newpath 0 72 moveto 504 0 rlineto 0 648 rlineto -504 0 rlineto closepath clip newpath 0 711.9 moveto 10 f4 (%\\input amstex) show 0 701.4 moveto 10 f4 (\\documentstyle{gen-j}) show 0 690.9 moveto 10 f4 (\\NoBlackBoxes) show 0 680.4 moveto 10 f4 (\\TagsOnLeft) show 0 669.9 moveto 10 f4 (%\\magnification = \\magstep1 ) show 0 659.4 moveto 10 f4 (\\overfullrule0pt) show 0 648.9 moveto 10 f4 (%\\hsize=35 true pc) show 0 638.4 moveto 10 f4 (%\\vsize=54 true pc) show 0 627.9 moveto 10 f4 (%\\voffset=0.1 true in) show 0 617.4 moveto 10 f4 (%\\hoffset=1.0 true cm ) show 0 606.9 moveto 10 f4 (\\def \\n {\\noindent}) show 0 596.4 moveto 10 f4 (\\def \\s {\\smallskip}) show 0 585.9 moveto 10 f4 (\\def \\m {\\medskip}) show 0 575.4 moveto 10 f4 (\\def \\b {\\bigskip}) show 0 564.9 moveto 10 f4 (\\def \\a {\\alpha}) show 0 554.4 moveto 10 f4 (\\def \\ad {{\\text {ad}}} ) show 0 543.9 moveto 10 f4 (\\def \\be {\\beta} ) show 0 533.4 moveto 10 f4 (\\def \\d {\\Delta}) show 0 522.9 moveto 10 f4 (\\def \\del {\\delta} ) show 0 512.4 moveto 10 f4 (\\def \\e {\\epsilon} ) show 0 501.9 moveto 10 f4 (\\def \\End {{\\text {End}}} ) show 0 491.4 moveto 10 f4 (\\def\\fsl{\\frak s\\frak l_2} ) show 0 480.9 moveto 10 f4 (\\def\\fst{\\frak s\\frak l_3}) show 0 470.4 moveto 10 f4 (\\def\\fss{\\frak s\\frak l\(1,1\)}) show 0 459.9 moveto 10 f4 (\\def \\fb {{\\frak b}}) show 0 449.4 moveto 10 f4 (\\def \\ga {\\gamma} ) show 0 438.9 moveto 10 f4 (\\def \\g {{\\frak g}} ) show 0 428.4 moveto 10 f4 (\\def \\h {{\\frak h}}) show 0 417.9 moveto 10 f4 (\\def \\k {\\kappa} ) show 0 407.4 moveto 10 f4 (\\def \\Hom {{\\text {Hom}}}) show 0 396.9 moveto 10 f4 (\\def \\l {\\lambda}) show 0 386.4 moveto 10 f4 (\\def \\la {\\langle}) show 0 375.9 moveto 10 f4 (\\def \\o {\\overline}) show 0 365.4 moveto 10 f4 (\\def \\oc {{\\Cal O}}) show 0 354.9 moveto 10 f4 (\\def \\osp{\\frak o\\frak s\\frak p} ) show 0 344.4 moveto 10 f4 (\\def \\ot {\\otimes}) show 0 333.9 moveto 10 f4 (\\def \\p {\\frak p}) show 0 323.4 moveto 10 f4 (\\def \\ra {\\rangle} ) show 0 312.9 moveto 10 f4 (\\def \\sgn {{\\text {sgn}}}) show 0 302.4 moveto 10 f4 (\\def \\t {\\theta}) show 0 291.9 moveto 10 f4 (\\def \\tr {{\\text {tr}}}) show 0 281.4 moveto 10 f4 (\\def \\u {\\underline}) show 0 270.9 moveto 10 f4 (\\def \\w {\\omega} ) show 0 260.4 moveto 10 f4 (\\def \\z {\\zeta}) show 0 249.9 moveto 10 f4 (\\def \\C {{\\text {\\bf C}}}) show 0 239.4 moveto 10 f4 (\\def \\K {{\\text {\\bf K}}} ) show 0 228.9 moveto 10 f4 (\\def \\Z {{\\text {\\bf Z}}} ) show 0 207.9 moveto 10 f4 (\\topmatter) show 0 197.4 moveto 10 f4 (\\title Differential Posets and Down-up Algebras ) show 0 186.9 moveto 10 f4 (\\endtitle) show 0 176.4 moveto 10 f4 (\\author Georgia Benkart \\endauthor) show 0 165.9 moveto 10 f4 (\\thanks The author gratefully acknowledges ) show 0 155.4 moveto 10 f4 (support from National Science ) show 0 144.9 moveto 10 f4 (Foundation Grant \\#{}DMS--9622447. \\endthanks ) show 0 134.4 moveto 10 f4 (\\subjclass ) show 0 123.9 moveto 10 f4 (Primary 16S15, 16S30, 17B35, 17B10 ) show 0 113.4 moveto 10 f4 (Secondary 81R50, 06A07 \\endsubjclass ) show 0 92.4 moveto 10 f4 (\\abstract) show 0 81.9 moveto 10 f4 (Down-up algebras originated in the study of differential posets. ) show pagelevel restore showpage %%Page: 2 2 %%BeginPageSetup /pagelevel save def 54 0 translate %%EndPageSetup newpath 0 72 moveto 504 0 rlineto 0 648 rlineto -504 0 rlineto closepath clip newpath 0 711.9 moveto 10 f4 (In this paper we discuss their combinatorial origins, ) show 0 701.4 moveto 10 f4 (representations, and structure. Down-up algebras) show 0 690.9 moveto 10 f4 (exhibit many of the important features of the universal enveloping) show 0 680.4 moveto 10 f4 (algebra $U\(\\fsl\)$ of the Lie algebra $\\fsl$ including a) show 0 669.9 moveto 10 f4 (Poincar\\'e-Birkhoff-Witt type basis and a well-behaved representation theory.) show 0 659.4 moveto 10 f4 (They have many interesting connections with Weyl algebras, quantum groups, ) show 0 648.9 moveto 10 f4 (and Witten's ) show 0 638.4 moveto 10 f4 (deformations of $U\(\\fsl\)$. ) show 0 627.9 moveto 10 f4 (\\endabstract) show 0 606.9 moveto 10 f4 (\\endtopmatter) show 0 585.9 moveto 10 f4 (\\document) show 0 564.9 moveto 10 f4 (\\subhead {Differential posets} \\endsubhead ) show 0 554.4 moveto 10 f4 (\\m) show 0 543.9 moveto 10 f4 (In [St2], Stanley introduced a class of partially ordered sets,) show 0 533.4 moveto 10 f4 (which he termed {\\it differential posets}. Many of the) show 0 522.9 moveto 10 f4 (remarkable enumerative and combinatorial properties of these posets ) show 0 512.4 moveto 10 f4 (involve counting saturated chains $y_1 \\prec y_2 \\prec \\dots) show 0 501.9 moveto 10 f4 (\\prec y_k$ or Hasse walks $y_1, y_2, \\dots, y_k$, \(where either $y_{i+1}$) show 0 491.4 moveto 10 f4 (covers $y_i$ or $y_i$ covers $y_{i+1}$\). ) show 0 480.9 moveto 10 f4 (Essential in the computations are two operators,) show 0 470.4 moveto 10 f4 ($d$ \(down\) and $u$ \(up\), which are defined on the complex vector space ) show 0 459.9 moveto 10 f4 ($\\C P$ having basis the elements of the poset $P$. ) show 0 449.4 moveto 10 f4 (If $y \\in P$, then $d\(y\)$ is the sum of all elements that $y$ covers ) show 0 438.9 moveto 10 f4 (and $u\(y\)$ is the) show 0 428.4 moveto 10 f4 (sum of all elements that cover) show 0 417.9 moveto 10 f4 ($y$. For many posets the down and up operators give well-defined) show 0 407.4 moveto 10 f4 (linear transformations of $\\C P$. Precursors) show 0 396.9 moveto 10 f4 (of the operators $d$ and $u$ appeared in [St1] and [P], where) show 0 386.4 moveto 10 f4 (they were used to show posets are Sperner or rank unimodal. ) show 0 375.9 moveto 10 f4 (\\m) show 0 365.4 moveto 10 f4 (The characterizing property of an $r$-differential poset is) show 0 354.9 moveto 10 f4 (that the down and up) show 0 344.4 moveto 10 f4 (operators satisfy) show 0 333.9 moveto 10 f4 ($du - u d = r I$ for some positive integer $r$ \(see [St2, Thm. 2.2]\),) show 0 323.4 moveto 10 f4 (where $I$ is the identity transformation on $\\C P$. ) show 0 312.9 moveto 10 f4 (Thus, the poset affords a representation of the Weyl algebra, \(the associative) show 0 302.4 moveto 10 f4 (algebra with generators) show 0 291.9 moveto 10 f4 ($y,x$ subject to the relation $y x - x y = 1$\), via the mapping $y \\mapsto d/r$,) show 0 281.4 moveto 10 f4 (and $x \\mapsto u$. Since the Weyl algebra also can be realized) show 0 270.9 moveto 10 f4 (as differential operators $y \\mapsto d/dx$ and $x \\mapsto x$ \(multiplication) show 0 260.4 moveto 10 f4 (by $x$\) on $\\C[x]$, Stanley referred to the ) show 0 249.9 moveto 10 f4 (posets satisfying $du-ud = rI$) show 0 239.4 moveto 10 f4 (as $r$-{\\it differential}) show 0 228.9 moveto 10 f4 (or simply {\\it differential} when $r = 1$. Fomin [F] studied) show 0 218.4 moveto 10 f4 (essentially the same class of posets for $r = 1$, calling them) show 0 207.9 moveto 10 f4 (``$Y$-graphs''. This terminology comes from the fact that Young's) show 0 197.4 moveto 10 f4 (lattice $Y$ of all partitions of all nonnegative integers ) show 0 186.9 moveto 10 f4 (is the prototypical example. ) show 0 176.4 moveto 10 f4 (\\m) show 0 155.4 moveto 10 f4 (A partition $\\mu$ of a nonnegative integer $m$ can be) show 0 144.9 moveto 10 f4 (regarded as a descending sequence ) show 0 134.4 moveto 10 f4 ($\\mu = \(\\mu_1 \\geq \\mu_2 \\geq \\dots\)$ of parts whose sum $|\\mu| = \\sum_i) show 0 123.9 moveto 10 f4 (\\mu_i$ equals $m$. If $\\nu = \(\\nu_1 \\geq \\nu_2 \\geq) show 0 113.4 moveto 10 f4 (\\dots\)$ is a second partition, then) show 0 102.9 moveto 10 f4 ($\\mu \\leq \\nu$ when $\\mu_i \\leq \\nu_i$ for all $i$. The partition) show 0 92.4 moveto 10 f4 ($\\nu$ covers $\\mu$ \(written here as $\\mu \\prec \\nu$\) if $\\mu < \\nu$) show 0 81.9 moveto 10 f4 (and $|\\nu| = 1 + |\\mu|$. Thus, $\\mu \\prec \\nu$ if the partition $\\mu$ is) show pagelevel restore showpage %%Page: 3 3 %%BeginPageSetup /pagelevel save def 54 0 translate %%EndPageSetup newpath 0 72 moveto 504 0 rlineto 0 648 rlineto -504 0 rlineto closepath clip newpath 0 711.9 moveto 10 f4 (obtained from $\\nu$ by subtracting 1 from exactly one of the parts of $\\nu$,) show 0 701.4 moveto 10 f4 (and $d\(\\nu\)$ is the sum of all such $\\mu$. Analogously,) show 0 690.9 moveto 10 f4 ($u\(\\nu\)$ is the sum of all partitions $\\pi$ obtained from $\\nu$ by) show 0 680.4 moveto 10 f4 (adding 1 to one part of $\\nu$. Young's lattice $Y$ is) show 0 669.9 moveto 10 f4 (a 1-differential poset, and $Y^r$ is $r$-differential ) show 0 659.4 moveto 10 f4 (\([St2, Cor. 1.4]\). ) show 0 648.9 moveto 10 f4 (\\m) show 0 638.4 moveto 10 f4 (The down and up operators on Young's lattice have a representation) show 0 627.9 moveto 10 f4 (theoretic significance. The simple modules of the symmetric group) show 0 617.4 moveto 10 f4 ($S_n$ are indexed by the partitions $\\nu$ of $n$. Upon restriction to) show 0 606.9 moveto 10 f4 ($S_{n-1}$, the representation labelled by $\\nu$) show 0 596.4 moveto 10 f4 (decomposes into a direct sum of simple $S_{n-1}$-modules) show 0 585.9 moveto 10 f4 (indexed by the) show 0 575.4 moveto 10 f4 (partitions) show 0 564.9 moveto 10 f4 ($\\mu \\prec \\nu$, so it is given by $d\(\\nu\)$. When the) show 0 554.4 moveto 10 f4 (simple module labelled by $\\nu$ is induced to a representation) show 0 543.9 moveto 10 f4 (of $S_{n+1}$, it decomposes into a sum of simple $S_{n+1}$-modules indexed by) show 0 533.4 moveto 10 f4 (the partitions $\\pi$ of $n+1$ such that $\\nu \\prec \\pi$, which is just) show 0 522.9 moveto 10 f4 ($u\(\\nu\)$. ) show 0 512.4 moveto 10 f4 (\\m ) show 0 501.9 moveto 10 f4 (In his study [T1] of) show 0 491.4 moveto 10 f4 (uniform posets, Terwilliger considered finite ranked posets $P$) show 0 480.9 moveto 10 f4 (whose down and up operators satisfy the) show 0 470.4 moveto 10 f4 (following relation ) show 0 449.4 moveto 10 f4 ($$ d_{i}d_{i+1}u_{i}=\\alpha _{i}d_{i}u_{i-1}d_{i}+\\beta) show 0 438.9 moveto 10 f4 (_{i}u_{i-2}d_{i-1}d_{i} +\\gamma _{i}d_{i}, $$) show 0 417.9 moveto 10 f4 (\\n where $d_{i}$ and $u_{i}$ denote the restriction of $d$ and $u$ to the) show 0 407.4 moveto 10 f4 (elements of rank $i$. \(There is an analogous second relation,) show 0 386.4 moveto 10 f4 ($$ d_{i+1}u_{i}u_{i-1}=\\alpha _{i}u_{i-1}d_{i}u_{i-1}+\\beta) show 0 375.9 moveto 10 f4 (_{i}u_{i-1}u_{i-2}d_{i-1} +\\gamma _{i}u_{i-1}, $$) show 0 365.4 moveto 10 f4 ( ) show 0 354.9 moveto 10 f4 (\\n which holds automatically in this case because $d_{i+1}$ and $u_i$ are) show 0 344.4 moveto 10 f4 (adjoint operators relative to a certain bilinear form.\)) show 0 333.9 moveto 10 f4 (In many examples the constants in these relations do not) show 0 323.4 moveto 10 f4 (depend on the rank $i$. In particular, a poset whose down and up) show 0 312.9 moveto 10 f4 (operators satisfy) show 0 291.9 moveto 10 f4 ($$) show 0 281.4 moveto 10 f4 (\\aligned) show 0 270.9 moveto 10 f4 (d^{2}u & = q\(q+1\)dud -q^{3}ud^{2}+rd \\\\) show 0 260.4 moveto 10 f4 (du^2 & = q\(q+1\)udu - q^3 u^2 d + r u \\\\) show 0 249.9 moveto 10 f4 (\\endaligned $$) show 0 228.9 moveto 10 f4 (\\n where $q$ and $r$ are fixed complex numbers is ) show 0 218.4 moveto 10 f4 (said to be {\\it $\(q,r\)$-differential}. Many interesting) show 0 207.9 moveto 10 f4 (examples of $\(q,r\)$-differential posets in [T1] arise) show 0 197.4 moveto 10 f4 (from considering certain subspaces of a vector space over the) show 0 186.9 moveto 10 f4 (field $GF\(q\)$ of $q$ elements:) show 0 176.4 moveto 10 f4 (\\m) show 0 165.9 moveto 10 f4 (\\item{\(1\)} Assume $W$ is an $n$-dimensional vector space over) show 0 155.4 moveto 10 f4 ($GF\(q\)$ and consider the set of pairs $P = \\{\(U,f\) \\mid U$ is) show 0 144.9 moveto 10 f4 (a subspace of $W$ and $f$ is an alternating bilinear form on $U\\}$) show 0 134.4 moveto 10 f4 (with the ordering: $\(U,f\) \\leq \(V,g\)$ if $U$ is a subspace of $V$) show 0 123.9 moveto 10 f4 (and $g|_U = f$. Then $P$ is a $\(q,r\)$-differential poset) show 0 113.4 moveto 10 f4 (with $r = -q^n\(q+1\)$. ) show 0 102.9 moveto 10 f4 (\\m) show 0 92.4 moveto 10 f4 (\\item{\(2\)} In example \(1\), replace ``an alternating bilinear form'' with) show 0 81.9 moveto 10 f4 (``a quadratic form''. The resulting poset $P$ is $\(q,) show pagelevel restore showpage %%Page: 4 4 %%BeginPageSetup /pagelevel save def 54 0 translate %%EndPageSetup newpath 0 72 moveto 504 0 rlineto 0 648 rlineto -504 0 rlineto closepath clip newpath 0 711.9 moveto 10 f4 (-q^{n+1}\(q+1\)\)$-differential.) show 0 701.4 moveto 10 f4 (\\m) show 0 690.9 moveto 10 f4 (\\item{\(3\)} In this example assume $W$ is an $n$-dimensional space over) show 0 680.4 moveto 10 f4 ($GF\(q^2\)$ and the bilinear forms are Hermitian. The poset) show 0 669.9 moveto 10 f4 ($P$ is $\(q^{2},-q^{2n+1}\(q^{2}+1\)\)$-differential in this case.) show 0 648.9 moveto 10 f4 (\\b ) show 0 638.4 moveto 10 f4 (\\subhead {Down-up algebras} \\endsubhead ) show 0 627.9 moveto 10 f4 (\\m) show 0 617.4 moveto 10 f4 ( ) show 0 606.9 moveto 10 f4 (To better understand the algebra generated by the down and up operators of a) show 0 596.4 moveto 10 f4 (poset and its action on the poset, we introduced the notion of) show 0 585.9 moveto 10 f4 (a down-up algebra in our joint work with Roby \(see [BR]\). Although the initial) show 0 575.4 moveto 10 f4 (motivation for our investigations came from posets, we made no assumptions) show 0 564.9 moveto 10 f4 (about the existence of posets whose down and up operators satisfy our) show 0 554.4 moveto 10 f4 (relations. However, when such a poset exists, it affords a representation) show 0 543.9 moveto 10 f4 (of the down-up algebra, so our primary focus in [BR] was on determining) show 0 533.4 moveto 10 f4 (explicit information about the) show 0 522.9 moveto 10 f4 (representations of down-up algebras. ) show 0 512.4 moveto 10 f4 (\\m ) show 0 501.9 moveto 10 f4 (\\proclaim {Definition 1} Let $\\a,\\be,\\ga$ be fixed but arbitrary complex numbers. ) show 0 491.4 moveto 10 f4 (The unital associative algebra $A\(\\a,\\be,\\ga\)$ over $\\C$ with generators) show 0 480.9 moveto 10 f4 ($d,u$ and defining relations) show 0 470.4 moveto 10 f4 (\\s) show 0 459.9 moveto 10 f4 (\\item {}{\(R1\)} $d^2u = \\a dud + \\be ud^2 + \\ga d,$) show 0 449.4 moveto 10 f4 (\\s ) show 0 438.9 moveto 10 f4 (\\item {}{\(R2\)} $du^2 = \\a udu + \\be u^2 d + \\ga u,$ ) show 0 428.4 moveto 10 f4 (\\m) show 0 417.9 moveto 10 f4 (\\n is a down-up algebra. \\endproclaim ) show 0 407.4 moveto 10 f4 (\\m) show 0 396.9 moveto 10 f4 (It is easy to see that when $\\ga \\neq 0$ the down-up algebra $A\(\\a,\\be,\\ga\)$) show 0 386.4 moveto 10 f4 (is isomorphic to $A\(\\a,\\be,1\)$ by the map, $d \\mapsto d'$, $u \\mapsto \\ga u'$.) show 0 375.9 moveto 10 f4 (Therefore, it would suffice to treat just two cases $\\ga = 0,1$, but to) show 0 365.4 moveto 10 f4 (avoid dividing considerations into these two cases, we retain) show 0 354.9 moveto 10 f4 (the notation $\\ga$. ) show 0 344.4 moveto 10 f4 (\\b) show 0 333.9 moveto 10 f4 (\\subhead Examples of down-up algebras \\endsubhead) show 0 323.4 moveto 10 f4 (\\m) show 0 312.9 moveto 10 f4 (\\n {\\bf Example \(i\)}. \\ If $B$ is the associative algebra generated by the down) show 0 302.4 moveto 10 f4 (and up operators $d,u$ of) show 0 291.9 moveto 10 f4 (a $\(q,r\)$-differential poset, ) show 0 281.4 moveto 10 f4 (then relations \(R1\) and \(R2\) hold with ) show 0 270.9 moveto 10 f4 ($\\a = q\(q+1\)$, $\\be = -q^{3}$, and $\\ga = r$. Thus, $B$ is a homomorphic image) show 0 260.4 moveto 10 f4 (of the algebra $A\(\\a,\\be,\\ga\)$ with these parameters, and the action of) show 0 249.9 moveto 10 f4 ($B$ on the poset gives a representation of $A\(\\a,\\be,\\ga\)$. ) show 0 239.4 moveto 10 f4 (\\m ) show 0 218.4 moveto 10 f4 (\\n {\\bf Example \(ii\)}. \\ The relation $du - ud = rI$ of an $r$-differential poset) show 0 207.9 moveto 10 f4 (can be multiplied on the left by $d$ and on the right by $d$ and) show 0 197.4 moveto 10 f4 (the resulting equations can be added to get the relation) show 0 186.9 moveto 10 f4 ($d^2 u - ud^2 = 2rd$ of a $\(-1,2r\)$-differential poset.) show 0 176.4 moveto 10 f4 (Thus, the Weyl algebra is a homomorphic image \(by the ideal generated) show 0 165.9 moveto 10 f4 (by $du-ud - r1$\) of the algebra $A\(0,1,2r\)$. ) show 0 155.4 moveto 10 f4 (Similarly, the $q$-Weyl algebra is a homomorphic image of the algebra ) show 0 144.9 moveto 10 f4 ($A\(0,q^2,\(q+1\)\)$ by the ideal generated by $du - qud - 1$. ) show 0 134.4 moveto 10 f4 (The skew polynomial) show 0 123.9 moveto 10 f4 (algebra $\\C_q[d,u]$, or quantum plane \(see [M]\), is the associative algebra with) show 0 113.4 moveto 10 f4 (generators $d,u$ which satisfy the relation $du = qud$. Therefore,) show 0 102.9 moveto 10 f4 ($\\C_q[d,u]$ is a homomorphic image \(by the ideal generated by $du-qud$\)) show 0 92.4 moveto 10 f4 (of the algebra $A\(2q,-q^2,0\)$. ) show 0 81.9 moveto 10 f4 (\\m) show pagelevel restore showpage %%Page: 5 5 %%BeginPageSetup /pagelevel save def 54 0 translate %%EndPageSetup newpath 0 72 moveto 504 0 rlineto 0 648 rlineto -504 0 rlineto closepath clip newpath 0 711.9 moveto 10 f4 (\\n) show 0 701.4 moveto 10 f4 ({\\bf Example \(iii\)}. \\ Consider the poset ${\\Cal L}\(2,2\) = \\{ {\\u a} = \(a_1,a_2\)) show 0 690.9 moveto 10 f4 (\\mid 2 \\geq a_1 \\geq a_2$ and $a_1,a_2 \\in \\Z_{\\geq 0}\\}$ with the order relation) show 0 680.4 moveto 10 f4 ($\\u a \\leq \\u b$ if $a_i \\leq b_i$ for $i = 1,2$. This is just) show 0 669.9 moveto 10 f4 (the set of partitions which fit into a $2 \\times 2$ box. By direct calculation) show 0 659.4 moveto 10 f4 (it is easy to verify that the down and up operators on this poset satisfy) show 0 648.9 moveto 10 f4 ($d^2u = dud - ud^2 +d$ and $du^2 = udu - u^2 d + u$, so the algebra they ) show 0 638.4 moveto 10 f4 (generate is a homomorphic image of the down-up algebra $A\(1,-1,1\)$. ) show 0 627.9 moveto 10 f4 (\\m ) show 0 617.4 moveto 10 f4 (\\n {\\bf Example \(iv\)}. \\ Suppose $\\g$ is a 3-dimensional Lie algebra over $\\C$) show 0 606.9 moveto 10 f4 (with basis) show 0 596.4 moveto 10 f4 ($x,y,[x,y]$ such that $[x[x,y]] = \\ga x$ and $[[x,y],y] = \\ga y$. ) show 0 585.9 moveto 10 f4 (In the universal enveloping) show 0 575.4 moveto 10 f4 (algebra $U\(\\g\)$ of $\\g$ where $[x,y] = xy-yx$, these relations become) show 0 554.4 moveto 10 f4 ($$\\aligned) show 0 543.9 moveto 10 f4 (& x^2 y -2 xyx + yx^2 = \\ga x \\\\) show 0 533.4 moveto 10 f4 (& xy^2 -2yxy + y^2x = \\ga y. \\\\) show 0 522.9 moveto 10 f4 (\\endaligned$$) show 0 501.9 moveto 10 f4 (\\n Thus, $U\(\\g\)$ is a homomorphic image of the down-up algebra $A\(2,-1,\\ga\)$) show 0 491.4 moveto 10 f4 (via the mapping $\\phi: A\(2,-1,\\ga\) \\rightarrow U\(\\g\)$ with $\\phi: d \\mapsto x$,) show 0 480.9 moveto 10 f4 ($\\phi: u) show 0 470.4 moveto 10 f4 (\\mapsto y$. The) show 0 459.9 moveto 10 f4 (mapping $\\psi: \\g \\rightarrow A\(2,-1,\\ga\)$ with $\\psi: x \\mapsto d$, $) show 0 449.4 moveto 10 f4 (\\psi: y \\mapsto u$,) show 0 438.9 moveto 10 f4 (and $\\psi: [x,y] \\mapsto du - ud$ extends, by the universal property of $U\(\\g\)$,) show 0 428.4 moveto 10 f4 (to an algebra homomorphism $\\psi: U\(\\g\) \\rightarrow A\(2,-1,\\ga\)$ which) show 0 417.9 moveto 10 f4 (is the inverse of $\\phi$. ) show 0 407.4 moveto 10 f4 (Consequently, $U\(\\g\)$ is isomorphic to $A\(2,-1,\\ga\)$. ) show 0 396.9 moveto 10 f4 (\\m ) show 0 375.9 moveto 10 f4 (The Lie algebra) show 0 365.4 moveto 10 f4 ($\\fsl$ of $2 \\times 2$ complex matrices of trace zero has a standard) show 0 354.9 moveto 10 f4 (basis $e = E_{1,2}, f = E_{2,1},$ and $h = E_{1,1}-E_{2,2}$ of) show 0 344.4 moveto 10 f4 (matrix units, which satisfies $[e,f] = h, \\; [h,e] = 2e$, and $[h,f] = -2 f$. From) show 0 333.9 moveto 10 f4 (this we see that $U\(\\fsl\) \\cong A\(2,-1,-2\)$. The Heisenberg Lie algebra $\\frak H$ has) show 0 323.4 moveto 10 f4 (a basis $x,y,z$ where $[x,y] = z$, and $[z,{\\frak H}] = 0$,) show 0 312.9 moveto 10 f4 (so $U\({\\frak H}\) \\cong) show 0 302.4 moveto 10 f4 (A\(2,-1,0\)$. ) show 0 291.9 moveto 10 f4 (\\m) show 0 281.4 moveto 10 f4 (\\n {\\bf Example \(v\)}. \\ The $2 \\times 2$ complex matrices ) show 0 270.9 moveto 10 f4 ($y = \\left \(\\matrix y_1 & y_2 \\\\) show 0 260.4 moveto 10 f4 ( y_3 & y_4 \\endmatrix \\right \)$) show 0 249.9 moveto 10 f4 (with supertrace $y_1-y_4 = 0$ is the special linear) show 0 239.4 moveto 10 f4 (Lie superalgebra $L = \\fss = L_{\\o 0} \\oplus L_{\\o 1}$ ) show 0 228.9 moveto 10 f4 (under the supercommutator $[x,y] =) show 0 218.4 moveto 10 f4 (xy-\(-1\)^{ab}yx$ for $x\\in L_{\\o a},\\;y\\in L_{\\o b}$. It has a ) show 0 207.9 moveto 10 f4 (presentation by generators $e,f$ \(which belong to $L_{\\o 1}$ and can be identified) show 0 197.4 moveto 10 f4 (with the matrix units $e = E_{1,2}$, $f = E_{2,1}$\) and relations) show 0 186.9 moveto 10 f4 ($[e,[e,f]] = 0$, $[[e,f],f] = 0$, $[e,e] = 0$, $[f,f] = 0$.) show 0 176.4 moveto 10 f4 (The universal enveloping algebra $U\(\\fss\)$ of $\\fss$ has generators $e,f$ and) show 0 165.9 moveto 10 f4 (relations $e^2 f - f e^2 = 0$, $e f^2 - f^2 e = 0$, $e^2 = 0$, $f^2 = 0$.) show 0 155.4 moveto 10 f4 (Thus, $U\(\\fss\)$ is a homomorphic image of the down-up algebra) show 0 144.9 moveto 10 f4 ($A\(0,1,0\)$ by the ideal generated by the elements $e^2$ and $f^2$, which are) show 0 134.4 moveto 10 f4 (central in $A\(0,1,0\)$. ) show 0 123.9 moveto 10 f4 (\\m) show 0 113.4 moveto 10 f4 (\\n {\\bf Example \(vi\)}. \\ The orthosymplectic Lie superalgebra $\\osp\(1,2\) = L_{\\o 0}) show 0 102.9 moveto 10 f4 (\\oplus L_{\\o 1}$ has generators $x,y \\in L_{\\o 1}$ which satisfy) show 0 81.9 moveto 10 f4 ($$xy + yx = t \\in L_{\\o 0} \\quad \\quad tx - xt = x \\quad \\quad yt -ty = y. $$) show pagelevel restore showpage %%Page: 6 6 %%BeginPageSetup /pagelevel save def 54 0 translate %%EndPageSetup newpath 0 72 moveto 504 0 rlineto 0 648 rlineto -504 0 rlineto closepath clip newpath 0 708.9 moveto 10 f4 (\\n By combining these relations, we see that its universal enveloping algebra) show 0 698.4 moveto 10 f4 (\\break $U\(\\osp\(1,2\)\)$ is a homomorphic image of $A\(0,1,1\)$. ) show 0 687.9 moveto 10 f4 (\\m) show 0 677.4 moveto 10 f4 (\\n {\\bf Example \(vii\)}. \\ Consider the field $\\C\(q\)$ of rational functions in the) show 0 666.9 moveto 10 f4 (indeterminate $q$ over the complex numbers, and let) show 0 656.4 moveto 10 f4 ($U_q\(\\g\)$ be the quantized enveloping algebra \(quantum group\) of) show 0 645.9 moveto 10 f4 (a finite-dimensional simple complex Lie algebra $\\g$ corresponding) show 0 635.4 moveto 10 f4 (to the Cartan matrix $\\frak A = \(a_{i,j}\)_{i,j= 1}) show 0 624.9 moveto 10 f4 (^n$. There are relatively prime integers $\\ell_i$ so that) show 0 614.4 moveto 10 f4 (the matrix $\(\\ell_i a_{i,j}\)$ is symmetric. Let ) show 0 593.4 moveto 10 f4 ($$q_i = q^{\\ell_i},\\quad \\quad \\text {and} \\quad \\quad ) show 0 582.9 moveto 10 f4 ([m]_i = \\frac {q_i^{m} - q_i^{-m}} {q_i -) show 0 572.4 moveto 10 f4 (q_i^{-1}}$$) show 0 551.4 moveto 10 f4 (\\n for all $m \\in \\Z_{\\geq 0}$. When $m \\geq 1$, let ) show 0 540.9 moveto 10 f4 ($[m]_i! = \\prod_{j = 1}^m [j]_i.$ Set $[0]_i! = 1$ and define) show 0 519.9 moveto 10 f4 ($$\\left [\\matrix m \\\\ n \\\\ \\endmatrix) show 0 509.4 moveto 10 f4 (\\right]_i = \\frac{[m]_i!}) show 0 498.9 moveto 10 f4 ({[n]_i! [m-n]_i!}.$$ ) show 0 477.9 moveto 10 f4 (\\n Then $U = U_q\(\\g\)$ is the unital associative algebra over $\\C\(q\)$ with generators ) show 0 467.4 moveto 10 f4 ($E_i, F_i, K_{i}, K_{i}^{-1}$ \($i = 1, \\dots, n$\) subject to the relations) show 0 456.9 moveto 10 f4 (\\b) show 0 446.4 moveto 10 f4 (\\item {}{\(Q1\)} $K_i K_i^{-1} = K_i^{-1} K_i,$) show 0 435.9 moveto 10 f4 (\\quad \\quad $K_i K_j = K_jK_i$) show 0 425.4 moveto 10 f4 (\\m) show 0 414.9 moveto 10 f4 (\\item {}{\(Q2\)} $K_i E_j K_i^{-1} =) show 0 404.4 moveto 10 f4 (q_i^{a_{i,j}}E_j$ \\quad \\quad $K_i F_j K_i^{-1} =) show 0 393.9 moveto 10 f4 (q_i^{-a_{i,j}}F_j$ ) show 0 383.4 moveto 10 f4 (\\m) show 0 372.9 moveto 10 f4 (\\item {}{\(Q3\)} $E_i F_j - F_j E_i) show 0 362.4 moveto 10 f4 (= \\displaystyle \\delta_{i,j} \\frac {K_i - K_i^{-1}}) show 0 351.9 moveto 10 f4 ({q_i - q_i^{-1}}$ ) show 0 341.4 moveto 10 f4 (\\m) show 0 330.9 moveto 10 f4 (\\item {}{\(Q4\)}$\\displaystyle \\sum_{k = 0}^{1-a_{i,j}} \(-1\)^k) show 0 320.4 moveto 10 f4 (\\left [\\matrix 1-a_{i,j}\\\\ k \\\\ \\endmatrix) show 0 309.9 moveto 10 f4 (\\right]_i E_i^{1-a_{i,j}-k} E_j E_i^k = 0$ \\quad \\quad for \\quad $i \\neq j$) show 0 299.4 moveto 10 f4 (\\m) show 0 288.9 moveto 10 f4 (\\item {}{\(Q5\)}$\\displaystyle \\sum_{k = 0}^{1-a_{i,j}}) show 0 278.4 moveto 10 f4 (\(-1\)^k\\left [\\matrix 1-a_{i,j}\\\\ k \\\\ \\endmatrix) show 0 267.9 moveto 10 f4 (\\right]_i F_i^{1-a_{i,j}-k} F_j F_i^k = 0$ \\quad \\quad for \\quad $i \\neq j.$) show 0 257.4 moveto 10 f4 (\\b) show 0 246.9 moveto 10 f4 (Suppose $a_{i,j} = -1 = a_{j,i}$ for some $i \\neq j$, and consider the subalgebra) show 0 236.4 moveto 10 f4 ($U_{i,j}$ generated by $E_i, E_j$. In this special case, the quantum Serre) show 0 225.9 moveto 10 f4 (relation \(Q4\) reduces to) show 0 204.9 moveto 10 f4 ($$\\aligned & E_i^2 E_j - [2]_i E_i E_j E_i +) show 0 194.4 moveto 10 f4 (E_j E_i^2 = 0\\quad \\quad \\text {and} \\\\) show 0 183.9 moveto 10 f4 (& E_j^2 E_i - [2]_j E_j E_i E_j +) show 0 173.4 moveto 10 f4 (E_i E_j^2 = 0.\\\\) show 0 162.9 moveto 10 f4 (\\endaligned $$) show 0 141.9 moveto 10 f4 (\\n Since $-\\ell_i = \\ell_ia_{i,j} = \\ell_ja_{j,i} = -\\ell_j$, the coefficients) show 0 131.4 moveto 10 f4 ($[2]_i$ and $[2]_j$ are equal. ) show 0 120.9 moveto 10 f4 (The algebra $U_{i,j}$ \(with $q$ specialized to) show 0 110.4 moveto 10 f4 (a complex number which is not a root of unity\) is isomorphic to) show 0 99.9 moveto 10 f4 ($A\([2]_i,-1,0\)$ by the mapping $E_i \\mapsto d$,) show 0 89.4 moveto 10 f4 ($E_j \\mapsto u$. The same result is true if) show 0 78.9 moveto 10 f4 (the corresponding $F$'s are used in place of the $E$'s. ) show pagelevel restore showpage %%Page: 7 7 %%BeginPageSetup /pagelevel save def 54 0 translate %%EndPageSetup newpath 0 72 moveto 504 0 rlineto 0 648 rlineto -504 0 rlineto closepath clip newpath 0 711.9 moveto 10 f4 (In particular, when $\\g = \\fst$ \($3 \\times 3$ matrices of trace 0\), the ) show 0 701.4 moveto 10 f4 (algebra $U_{i,j}$ is just the subalgebra of $U_q\(\\fst\)$) show 0 690.9 moveto 10 f4 (generated by the $E$'s. ) show 0 680.4 moveto 10 f4 (\\m ) show 0 669.9 moveto 10 f4 (\\n {\\bf Example \(viii\)}. \\ To provide an explanation of the existence of quantum) show 0 659.4 moveto 10 f4 (groups, Witten \([W1], [W2]\) introduced a 7-parameter deformation of the) show 0 648.9 moveto 10 f4 (universal enveloping algebra $U\(\\fsl\)$.) show 0 638.4 moveto 10 f4 (Witten's deformation is a unital associative algebra) show 0 627.9 moveto 10 f4 (over a field $\\K$ \(which is algebraically closed of) show 0 617.4 moveto 10 f4 (characteristic zero and which could be) show 0 606.9 moveto 10 f4 (assumed to be $\\C$\) and depends on a 7-tuple $\\u \\xi =) show 0 596.4 moveto 10 f4 (\(\\xi_1, \\dots, \\xi_7\)$ of elements of $\\K$. It has) show 0 585.9 moveto 10 f4 (a presentation by generators $x,y,z$ and defining) show 0 575.4 moveto 10 f4 (relations) show 0 564.9 moveto 10 f4 ($$\\align ) show 0 554.4 moveto 10 f4 (& xz - \\xi_1 z x = \\xi_2 x \\tag 2\\\\) show 0 543.9 moveto 10 f4 (& zy - \\xi_3 yz = \\xi_4 y \\tag 3\\\\) show 0 533.4 moveto 10 f4 (& yx - \\xi_5 xy = \\xi_6 z^2 + \\xi_7 z. \\tag 4 \\\\) show 0 522.9 moveto 10 f4 (\\endalign$$) show 0 501.9 moveto 10 f4 (\\n We denote the resulting algebra by $ {\\frak W}\(\\u \\xi\) $. ) show 0 491.4 moveto 10 f4 (In applications of these deformation algebras, the parameters ) show 0 480.9 moveto 10 f4 (depend on the coupling constant of the particular physical theory,) show 0 470.4 moveto 10 f4 (and Witten [W2] gives an evaluation of them in the special) show 0 459.9 moveto 10 f4 (case of the three-dimensional Chern-Simons gauge theory. ) show 0 449.4 moveto 10 f4 (\\m ) show 0 438.9 moveto 10 f4 (Let us assume $\\xi_6 = 0$ and $\\xi_7 \\neq 0$. Then) show 0 428.4 moveto 10 f4 (substituting expression \(4\) into \(2\) and \(3\) and rearranging shows that) show 0 417.9 moveto 10 f4 ( ) show 0 407.4 moveto 10 f4 ($$\\aligned & -\\xi_5 x^2y + \(1 + \\xi_1\\xi_5\)xyx - \\xi_1yx^2 = \\xi_2 \\xi_7 x) show 0 396.9 moveto 10 f4 (\\\\) show 0 386.4 moveto 10 f4 (& -\\xi_5 xy^2 + \(1+\\xi_3\\xi_5\)yxy -\\xi_3 y^2 x = \\xi_4 \\xi_7 y. \\\\) show 0 375.9 moveto 10 f4 (\\endaligned $$) show 0 365.4 moveto 10 f4 (\\n In particular, when $\\xi_5 \\neq 0$, $\\xi_1 = \\xi_3$,) show 0 354.9 moveto 10 f4 (and $\\xi_2 = \\xi_4$ we obtain) show 0 344.4 moveto 10 f4 ($$) show 0 333.9 moveto 10 f4 (\\aligned) show 0 323.4 moveto 10 f4 (& x^2 y = \\frac{1+\\xi_1 \\xi_5}{\\xi_5} xyx - \\frac{\\xi_1}{\\xi_5}yx^2 - ) show 0 312.9 moveto 10 f4 (\\frac{\\xi_2 \\xi_7}{\\xi_5} x \\\\) show 0 302.4 moveto 10 f4 (& x y^2 = \\frac{1+\\xi_1 \\xi_5}{\\xi_5} yxy - \\frac{\\xi_1}{\\xi_5}y^2x - ) show 0 291.9 moveto 10 f4 (\\frac{\\xi_2 \\xi_7}{\\xi_5}y. \\\\) show 0 281.4 moveto 10 f4 (\\endaligned$$) show 0 260.4 moveto 10 f4 (\\n From this it is easy to see that ) show 0 249.9 moveto 10 f4 (a Witten deformation algebra ${\\frak W}\(\\u \\xi\) $ with) show 0 239.4 moveto 10 f4 ($\\xi_6 = 0$, $\\xi_5\\xi_7 \\neq 0$, $\\xi_1 = \\xi_3$,) show 0 228.9 moveto 10 f4 (and $\\xi_2 = \\xi_4$ is a homomorphic image) show 0 218.4 moveto 10 f4 (of the down-up algebra $A\(\\a,\\be,\\gamma\)$ ) show 0 207.9 moveto 10 f4 (with) show 0 197.4 moveto 10 f4 ($$\\a = \\frac{1+\\xi_1 \\xi_5}{\\xi_5}, \\quad \\quad ) show 0 186.9 moveto 10 f4 (\\be = - \\frac{\\xi_1}{\\xi_5}, \\quad \\quad \\gamma = -\\frac{\\xi_2) show 0 176.4 moveto 10 f4 (\\xi_7}{\\xi_5}. \\tag 5$$) show 0 155.4 moveto 10 f4 (\\n In fact, in [B, Thm. 2.6] we proved the following) show 0 144.9 moveto 10 f4 (\\m) show 0 134.4 moveto 10 f4 (\\proclaim{Proposition 6} A Witten deformation algebra ${\\frak W}\(\\u \\xi\) $) show 0 123.9 moveto 10 f4 (with) show 0 113.4 moveto 10 f4 ($$\\xi_6 = 0, \\;\\; \\xi_5\\xi_7 \\neq 0, \\; \\xi_1 = \\xi_3, \\; \\;\\text{and}) show 0 102.9 moveto 10 f4 (\\;\\;\\xi_2 = \\xi_4 \\tag 7$$) show 0 81.9 moveto 10 f4 (\\n is isomorphic to the down-up algebra $A\(\\a,\\be,\\ga\)$) show pagelevel restore showpage %%Page: 8 8 %%BeginPageSetup /pagelevel save def 54 0 translate %%EndPageSetup newpath 0 72 moveto 504 0 rlineto 0 648 rlineto -504 0 rlineto closepath clip newpath 0 711.9 moveto 10 f4 (with $\\a,\\be,\\ga$ given by \(5\). Conversely, any) show 0 701.4 moveto 10 f4 (down-up algebra $A\(\\a,\\be,\\ga\)$ with not both $\\a$ and $\\be$) show 0 690.9 moveto 10 f4 (equal to 0 is isomorphic to a Witten deformation algebra ${\\frak W}\(\\u \\xi\) $) show 0 680.4 moveto 10 f4 (whose parameters satisfy \(7\). \\endproclaim ) show 0 669.9 moveto 10 f4 (\\m) show 0 659.4 moveto 10 f4 (A deformation algebra ${\\frak W}\(\\u \\xi\) $) show 0 648.9 moveto 10 f4 (has a filtration, and ) show 0 638.4 moveto 10 f4 (Le Bruyn \([L1], [L2]\) investigated the algebras ${\\frak W}\(\\u \\xi\) $ ) show 0 627.9 moveto 10 f4 (whose associated graded algebras are Auslander regular. ) show 0 617.4 moveto 10 f4 (They determine a 3-parameter family of deformation algebras which) show 0 606.9 moveto 10 f4 (are called {\\it conformal $\\fsl$ algebras} and whose defining relations) show 0 596.4 moveto 10 f4 (are ) show 0 585.9 moveto 10 f4 ($$xz - a zx = x, \\quad \\quad ) show 0 575.4 moveto 10 f4 (zy - a yz = y, \\quad \\quad yx - c xy = b z^2 + z \\tag 8$$) show 0 554.4 moveto 10 f4 (\\n When $c \\neq 0$ and $b = 0$, the conformal $\\fsl$ algebra) show 0 543.9 moveto 10 f4 (with defining relations given by \(8\) is isomorphic to) show 0 533.4 moveto 10 f4 (the down-up algebra $A\(\\a,\\be,\\ga\)$ with) show 0 522.9 moveto 10 f4 ($\\a = c^{-1}\(1+ac\), \\be = -ac^{-1}$ and $\\ga = -c^{-1}$.) show 0 512.4 moveto 10 f4 (If $b= c = 0$ and $a \\neq 0$, then the conformal $\\fsl$ algebra) show 0 501.9 moveto 10 f4 (is isomorphic to the down-up algebra $A\(\\a,\\be,\\ga\)$) show 0 491.4 moveto 10 f4 (with $\\a = a^{-1}$, $\\be = 0$ and $\\ga = -a^{-1}$. ) show 0 480.9 moveto 10 f4 (\\m) show 0 470.4 moveto 10 f4 (In a recent paper [K1], Kulkarni has shown that under certain) show 0 459.9 moveto 10 f4 (assumptions on the parameters a Witten deformation algebra is isomorphic to) show 0 449.4 moveto 10 f4 (a conformal $\\fsl$ algebra or to a double skew polynomial extension.) show 0 438.9 moveto 10 f4 (Kulkarni studies the simple modules of) show 0 428.4 moveto 10 f4 (the conformal $sl_2$ algebras and of the skew polynomial) show 0 417.9 moveto 10 f4 (algebras. Critical to the investigations in [K1] is the) show 0 407.4 moveto 10 f4 (observation that the conformal $\\fsl$ algebra of \(8\) can be realized as a ) show 0 396.9 moveto 10 f4 ({\\it hyperbolic) show 0 386.4 moveto 10 f4 (ring}. Kulkarni then applies results of Rosenberg [R]) show 0 375.9 moveto 10 f4 (on noncommutative algebraic geometry ) show 0 365.4 moveto 10 f4 (to describe the left ideals in the left spectrum of the algebra and to) show 0 354.9 moveto 10 f4 (determine the maximal left ideals for the conformal $sl_2$ algebras.) show 0 344.4 moveto 10 f4 (\\m ) show 0 333.9 moveto 10 f4 (\\n {\\bf Example \(ix\)}. \\ The quadratic Askey-Wilson algebras studied in) show 0 323.4 moveto 10 f4 ([GLZ] can be regarded as having generators $a,b$ and defining relations) show 0 312.9 moveto 10 f4 (which depend on fixed parameters $\(\\a,\\gamma,\\delta,\\epsilon, \\zeta,\\eta,\\mu,\\nu\)$) show 0 302.4 moveto 10 f4 (according to:) show 0 281.4 moveto 10 f4 ($$\\aligned a^2b & = \\a aba - ba^2 + \\zeta\(ab+ba\) + \\eta a^2 + \\gamma a + \\delta b) show 0 270.9 moveto 10 f4 (+ \\mu 1 \\\\) show 0 260.4 moveto 10 f4 (ab^2 & = \\a bab - b^2a + \\eta\(ab+ba\) +\\zeta b^2 + \\gamma b + \\epsilon a) show 0 249.9 moveto 10 f4 (+ \\nu 1. \\\\ \\endaligned$$) show 0 228.9 moveto 10 f4 (\\n It is apparent that when $\\delta=\\epsilon=\\zeta=\\eta=\\mu=\\nu = 0$, this) show 0 218.4 moveto 10 f4 (algebra is just the down-up algebra $A\(\\a,-1,\\ga\)$. Askey-Wilson algebras) show 0 207.9 moveto 10 f4 (are related to the Leonard systems introduced in [T2] as abstract ) show 0 197.4 moveto 10 f4 (algebraic generalizations) show 0 186.9 moveto 10 f4 (of $q$-Racah polynomials and of families of orthogonal polynomials that) show 0 176.4 moveto 10 f4 (include the quantum $q$-Krawtchouk, Racah, Hahn, dual Hahn, and) show 0 165.9 moveto 10 f4 (Krawtchouk polynomials. ) show 0 144.9 moveto 10 f4 ( ) show 0 134.4 moveto 10 f4 (\\b ) show 0 123.9 moveto 10 f4 (\\subhead {Highest weight modules} \\endsubhead) show 0 113.4 moveto 10 f4 (\\m ) show 0 102.9 moveto 10 f4 (Down-up algebras have a rich representation theory \(see [BR, Sec. 2]\). In particular,) show 0 92.4 moveto 10 f4 (they have highest weight modules and weight modules ) show 0 81.9 moveto 10 f4 (which mimic those of $\\fsl$. ) show pagelevel restore showpage %%Page: 9 9 %%BeginPageSetup /pagelevel save def 54 0 translate %%EndPageSetup newpath 0 72 moveto 504 0 rlineto 0 648 rlineto -504 0 rlineto closepath clip newpath 0 711.9 moveto 10 f4 (\\m ) show 0 701.4 moveto 10 f4 (A module $V$ for $A = A\(\\a,\\be,\\ga\)$) show 0 690.9 moveto 10 f4 (is said to be a {\\it highest weight module of weight $\\l$}) show 0 680.4 moveto 10 f4 (if $V$ has a vector $y_0$ such that $d \\cdot y_0 = 0$,) show 0 669.9 moveto 10 f4 ($\(d u\) \\cdot y_0 = \\l y_0$, and $V = Ay_0$. The vector) show 0 659.4 moveto 10 f4 ($y_0$ is a {\\it maximal vector} or {\\it highest weight vector} of) show 0 648.9 moveto 10 f4 ($V$. ) show 0 638.4 moveto 10 f4 (\\m ) show 0 627.9 moveto 10 f4 (\\proclaim {Proposition 9} \(See [BR, Sec. 2]\) Set $\\l_{-1} = 0$) show 0 617.4 moveto 10 f4 (and let) show 0 606.9 moveto 10 f4 ($\\l_0 = \\l \\in \\C$ be arbitrary. For $n \\geq 1$, ) show 0 596.4 moveto 10 f4 ( define $\\l_n$ inductively by the ) show 0 585.9 moveto 10 f4 (recurrence relation, ) show 0 575.4 moveto 10 f4 ($$\\lambda_n = \\a \\lambda_{n-1} + \\be \\l_{n-2} + \\ga. \\tag 10$$) show 0 554.4 moveto 10 f4 (\\n The $\\C$-vector space $V\(\\l\)$ with basis $\\{v_n \\mid n = 0,1,2, \\dots\\}$) show 0 543.9 moveto 10 f4 (and with $A\(\\a,\\be,\\ga\)$-action given by) show 0 533.4 moveto 10 f4 ($$\\aligned) show 0 522.9 moveto 10 f4 (d \\cdot v_n & = \\l_{n-1} v_{n-1}, \\quad n \\geq 1, \\quad \\text {and}) show 0 512.4 moveto 10 f4 (\\quad d) show 0 501.9 moveto 10 f4 (\\cdot v_0 = 0 \\\\) show 0 491.4 moveto 10 f4 (u \\cdot v_n & = v_{n+1}. \\\\) show 0 480.9 moveto 10 f4 (\\endaligned \\tag 11$$) show 0 459.9 moveto 10 f4 (\\n is a highest weight) show 0 449.4 moveto 10 f4 (module for $A\(\\a,\\be,\\ga\)$. Every $A\(\\a,\\be,\\ga\)$-module of) show 0 438.9 moveto 10 f4 (highest weight $\\l$ is a homomorphic image of $V\(\\l\)$. The module) show 0 428.4 moveto 10 f4 ($V\(\\l\)$ is simple) show 0 417.9 moveto 10 f4 (if and only if $\\l_n \\neq 0$ for any $n$. ) show 0 407.4 moveto 10 f4 (\\endproclaim) show 0 396.9 moveto 10 f4 (\\m ) show 0 386.4 moveto 10 f4 (Because it shares the) show 0 375.9 moveto 10 f4 (same universal property and) show 0 365.4 moveto 10 f4 (many of the same features as Verma modules for finite-dimensional) show 0 354.9 moveto 10 f4 (semisimple complex Lie algebras, the module $V\(\\l\)$ is) show 0 344.4 moveto 10 f4 (said to be a {\\it Verma module} for $A\(\\a,\\be,\\ga\)$. ) show 0 333.9 moveto 10 f4 (\\b ) show 0 323.4 moveto 10 f4 (\\subhead {Weight modules} \\endsubhead ) show 0 312.9 moveto 10 f4 (\\m) show 0 302.4 moveto 10 f4 (If we multiply the relation $d^2 u - \\a dud - \\be u d^2 = \\ga d$) show 0 291.9 moveto 10 f4 (on the left by $u$ and the relation $d u^2 - \\a udu - \\be u^2 d = \\ga u$) show 0 281.4 moveto 10 f4 (on the right by $d$ and subtract the second from the first, the resulting) show 0 270.9 moveto 10 f4 (equation is ) show 0 249.9 moveto 10 f4 ($$0 = ud^2u - du^2d \\quad \\quad \\text {or} \\quad \\quad \(du\)\(ud\) = \(ud\)\(du\).$$) show 0 228.9 moveto 10 f4 (\\n Therefore, the elements $du$ and $ud$ commute in $A = A\(\\a,\\be,\\ga\)$. For) show 0 218.4 moveto 10 f4 (any basis element $v_n \\in V\(\\l\)$, we have $du \\cdot v_n = \\l_n v_n$ and $ud \\cdot v_n) show 0 207.9 moveto 10 f4 (= \\l_{n-1}v_n$. Using that with $n = 0$ and $\\l \\neq 0$, it is easy to see that $du$ and $ud$) show 0 197.4 moveto 10 f4 (are linearly independent. Let $\\h = \\C du \\oplus \\C ud$. ) show 0 186.9 moveto 10 f4 (\\b) show 0 176.4 moveto 10 f4 (We say an $A$-module $V$ is a {\\it weight module} if $V = \\bigoplus_{\\nu \\in \\h^*}) show 0 165.9 moveto 10 f4 (V_\\nu$, where $V_\\nu = \\{v \\in V \\mid h \\cdot v = \\nu\(h\)v$ for all $h \\in \\h\\}$, and) show 0 155.4 moveto 10 f4 (the sum is over elements in the dual space $\\h^*$ of $\\h$. ) show 0 144.9 moveto 10 f4 (Any submodule of a weight module is a weight module. ) show 0 134.4 moveto 10 f4 (If $V_\\nu \\neq \(0\)$, then $\\nu$ is a {\\it weight} and $V_\\nu$ is the) show 0 123.9 moveto 10 f4 (corresponding {\\it weight space}. Each weight $\\nu$) show 0 113.4 moveto 10 f4 (is determined by the pair $\(\\nu',\\nu''\)$ of complex numbers, $\\nu' = \\nu\(du\)$) show 0 102.9 moveto 10 f4 (and $\\nu'' = \\nu\(ud\)$. In particular, highest weight modules are weight modules in) show 0 92.4 moveto 10 f4 (this sense. The basis vector $v_n$ of $V\(\\l\)$ is a weight vector whose weight) show 0 81.9 moveto 10 f4 (is given by the pair $\(\\l_n,\\l_{n-1}\)$. Finding these weights explicitly involves) show pagelevel restore showpage %%Page: 10 10 %%BeginPageSetup /pagelevel save def 54 0 translate %%EndPageSetup newpath 0 72 moveto 504 0 rlineto 0 648 rlineto -504 0 rlineto closepath clip newpath 0 711.9 moveto 10 f4 (solving the linear recurrence relation in \(10\), which can be done) show 0 701.4 moveto 10 f4 (by standard methods as in [Br, Chap.7] for example. ) show 0 690.9 moveto 10 f4 ( ) show 0 680.4 moveto 10 f4 (\\proclaim { Proposition 12} Assume $\\l_{-1} = 0$, $\\l_0 = \\l\\in \\C$,) show 0 669.9 moveto 10 f4 (and $\\l_{n}$ for) show 0 659.4 moveto 10 f4 ($n \\geq 1$ is given by the recurrence relation) show 0 648.9 moveto 10 f4 ($\\l_{n}-\\a\\l_{n-1} -\\be \\l_{n-2} = \\ga$. Fix $t \\in \\C$ such) show 0 638.4 moveto 10 f4 (that) show 0 627.9 moveto 10 f4 ($$t^2 = \\frac {\\a^2 + 4\\be}{4}.$$) show 0 617.4 moveto 10 f4 (\\m) show 0 606.9 moveto 10 f4 (\\item {}{\(i\)} If $\\a^2 + 4 \\be\\neq 0$, then ) show 0 585.9 moveto 10 f4 ($$\\l_n = c_1 r_1^n) show 0 575.4 moveto 10 f4 (+ c_2 r_2^n + x_n, \\quad \\quad \\text{where}$$ ) show 0 554.4 moveto 10 f4 ($$) show 0 543.9 moveto 10 f4 (\\aligned & r_1 = \\frac {\\a} {2} + t, \\quad) show 0 533.4 moveto 10 f4 ( \\quad ) show 0 522.9 moveto 10 f4 ( r_2 = \\frac {\\a} {2} - t, \\\\) show 0 512.4 moveto 10 f4 (& x_n = \\cases ) show 0 501.9 moveto 10 f4 (\(1-\\a-\\be\)^{-1}\\ga \\ \\ \\text {\\quad if \\quad $\\a+\\be \\neq 1$} \\\\) show 0 491.4 moveto 10 f4 (\(2-\\a\)^{-1}\\ga n \\ \\ \\text {\\quad if \\quad $\\a+\\be = 1$ \\quad \(necessarily \\quad $\\a) show 0 480.9 moveto 10 f4 (\\neq 2\),$} ) show 0 470.4 moveto 10 f4 (\\\\) show 0 459.9 moveto 10 f4 (\\endcases \\\\) show 0 449.4 moveto 10 f4 (& \\text {and} \\quad \\quad \\left \(\\matrix c_1 \\\\ c_2 \\\\ \\endmatrix \\right \)) show 0 438.9 moveto 10 f4 ( = \\frac {1}{r_2-r_1}) show 0 428.4 moveto 10 f4 (\\left \(\\matrix r_2 & -1 \\\\ -r_1 & 1 \\\\ \\endmatrix \\right \)) show 0 417.9 moveto 10 f4 (\\left \(\\matrix \\l-x_0 \\\\ \\a\\l+\\ga-x_1 \\\\ \\endmatrix \\right \). ) show 0 407.4 moveto 10 f4 (\\endaligned $$) show 0 396.9 moveto 10 f4 ( \\m) show 0 386.4 moveto 10 f4 (\\item {}{\(ii\)} If $\\a^2 + 4 \\be = 0$ and $\\a \\neq 0$, then ) show 0 365.4 moveto 10 f4 ($$\\l_n = c_1 s^n + c_2 n s^n + x_n \\quad \\text {where}$$) show 0 354.9 moveto 10 f4 ( ) show 0 333.9 moveto 10 f4 ($$\\aligned & s = \\displaystyle \\frac {\\a}{2} \\\\) show 0 323.4 moveto 10 f4 (& x_n = \\cases ) show 0 312.9 moveto 10 f4 (\(1-\\a-\\be\)^{-1}\\ga \\ \\ \\text {\\quad if \\quad $\\a+\\be \\neq 1$} \\\\) show 0 302.4 moveto 10 f4 ( 2^{-1} n^2 \\ga \\ \\ \\text {\\quad if \\quad $\\a + \\be = 1$ ) show 0 291.9 moveto 10 f4 (\\quad i.e. \\quad if \\quad $\\a = 2,\\; \\be = -1$,} ) show 0 281.4 moveto 10 f4 (\\\\ \\endcases \\\\ ) show 0 270.9 moveto 10 f4 (& \\text {and} \\quad \\quad \\left \(\\matrix c_1 \\\\ c_2 \\\\ \\endmatrix \\right\)) show 0 260.4 moveto 10 f4 ( = ) show 0 249.9 moveto 10 f4 (\\left \(\\matrix 1 & 0 \\\\ -1 & 2\\a^{-1} \\\\ \\endmatrix \\right \)) show 0 239.4 moveto 10 f4 (\\left \(\\matrix \\l-x_0 \\\\ \\a\\l+\\ga-x_1 \\\\ \\endmatrix \\right \). ) show 0 228.9 moveto 10 f4 (\\endaligned $$) show 0 218.4 moveto 10 f4 (\\m) show 0 207.9 moveto 10 f4 (\\item {}{\(iii\)} If $\\a^2 + 4 \\be = 0$ and) show 0 197.4 moveto 10 f4 ($\\a = 0$, then $\\be = 0$ and $\\l_n = \\ga$ for all $n \\geq 1$.) show 0 186.9 moveto 10 f4 (\\endproclaim) show 0 176.4 moveto 10 f4 (\\m) show 0 165.9 moveto 10 f4 (If $\\a,\\be$ are) show 0 155.4 moveto 10 f4 (real, then it is natural to take $t = \\displaystyle\\frac{\\sqrt{\\a^2+4\\be}}{2}$) show 0 144.9 moveto 10 f4 (in the above calculations.) show 0 134.4 moveto 10 f4 (\\m ) show 0 123.9 moveto 10 f4 (Let us consider several special cases. ) show 0 113.4 moveto 10 f4 (\\m) show 0 102.9 moveto 10 f4 (\\n {\\bf Example \(a\)}. \\ Recall that ) show 0 92.4 moveto 10 f4 (the universal enveloping algebra $U\(\\fsl\)$ of $\\fsl$ is) show 0 81.9 moveto 10 f4 (isomorphic to the algebra $A\(2,-1,-2\)$, and the universal enveloping) show pagelevel restore showpage %%Page: 11 11 %%BeginPageSetup /pagelevel save def 54 0 translate %%EndPageSetup newpath 0 72 moveto 504 0 rlineto 0 648 rlineto -504 0 rlineto closepath clip newpath 0 711.9 moveto 10 f4 (algebra $U\({\\frak H}\)$ of the Heisenberg Lie algebra $\\frak H$ is isomorphic) show 0 701.4 moveto 10 f4 (to $A\(2,-1,0\)$. ) show 0 690.9 moveto 10 f4 (Applying \(ii\) with $s = \\a/2 = 1$ and $x_n = n^2 \\ga/2$ for any) show 0 680.4 moveto 10 f4 (algebra) show 0 669.9 moveto 10 f4 ($A\(2,-1,\\ga\)$, we have that ) show 0 659.4 moveto 10 f4 ( ) show 0 648.9 moveto 10 f4 ($$\\l_n = \\l + \(\\l+\\frac{\\ga}{2}\)n + \\frac{\\ga n^2}{2} = ) show 0 638.4 moveto 10 f4 (\(n+1\)\(\\l+\\frac {\\ga n}{2}\).$$) show 0 627.9 moveto 10 f4 ( ) show 0 617.4 moveto 10 f4 (In the $\\fsl$-case, it is customary to use the operator $h = du-ud$ rather) show 0 606.9 moveto 10 f4 (than $du$. The eigenvalues of $h$ are $\\l_n -\\l_{n-1} = \\l+n\\ga = \\l-2n$,) show 0 596.4 moveto 10 f4 ($n = 0,1, \\dots$. The analogous) show 0 585.9 moveto 10 f4 (computation in the Heisenberg Lie algebra shows that the central element $z = du-ud$) show 0 575.4 moveto 10 f4 (has constant eigenvalue $\\l_n = \\l$. ) show 0 564.9 moveto 10 f4 (\\m) show 0 554.4 moveto 10 f4 (\\n {\\bf Example \(b\)}. \\ Recall that the quantum case discussed earlier involves) show 0 543.9 moveto 10 f4 (the down-up algebra $A\([2]_i,-1,0\)$. ) show 0 533.4 moveto 10 f4 (In the particular case of $U_q\(\\fst\)$, the subalgebra generated by) show 0 522.9 moveto 10 f4 (the $E_i$'s is isomorphic to $A\([2],-1,0\)$ where $[2] = ) show 0 512.4 moveto 10 f4 (\\displaystyle{\\frac{q^2-q^{-2}}{q-q^{-1}}}$, and ) show 0 501.9 moveto 10 f4 ($\\l_n = [n+1]\\l = \\Big\(\\displaystyle{\\frac{q^{n+1}-q^{-\(n+1\)}}{q-q^{-1}}}) show 0 491.4 moveto 10 f4 (\\Big\)\\l$ \\ for all $n \\geq 0$ ) show 0 480.9 moveto 10 f4 (in that case. ) show 0 470.4 moveto 10 f4 (\\m) show 0 459.9 moveto 10 f4 (\\n {\\bf Example \(c\)}. \\ For the algebra $A\(1,1,0\)$, ) show 0 449.4 moveto 10 f4 ( the solutions to the associated) show 0 438.9 moveto 10 f4 (linear recurrence $\\l_n = \\l_{n-1} + \\l_{n-2}$, $\\l_0 = \\l$, $\\l_{-1} = 0$,) show 0 428.4 moveto 10 f4 (\(hence the eigenvalues of $du$ and $ud$ on $V\(\\l\)$\) ) show 0 417.9 moveto 10 f4 (are given by the Fibonacci sequence) show 0 407.4 moveto 10 f4 ($\\l_0 = \\l$, $\\l_1 = \\l$, $\\l_2 = 2 \\l$, $\\l_3 = 3\\l$, $\\l_4 = 5 \\l$,) show 0 396.9 moveto 10 f4 ($\\dots$. In this case, the equations in Proposition 12 reduce to ) show 0 386.4 moveto 10 f4 ($\\displaystyle{\\l_n = \\l\\frac {\\sqrt 5}{5}\\left\( \\left \(\\frac {1 + \\sqrt 5}) show 0 375.9 moveto 10 f4 ({2}\\right\)^{n+1} -\\left\(\\frac {1 - \\sqrt 5} {2}\\right\)^{n+1}\\right\).}$ ) show 0 365.4 moveto 10 f4 ( ) show 0 354.9 moveto 10 f4 (\\m ) show 0 344.4 moveto 10 f4 (In [BR] we investigated in detail the weight space and submodule structure of the) show 0 333.9 moveto 10 f4 (Verma module $V\(\\l\)$. Roots of unity play a critical role in determining) show 0 323.4 moveto 10 f4 (the dimension of a weight space. We introduced ``category $\\Cal O$'' modules) show 0 312.9 moveto 10 f4 (in the spirit of [BGG] ) show 0 302.4 moveto 10 f4 (and showed the simple objects were highest weight modules, and we explored) show 0 291.9 moveto 10 f4 (a more general category $\\Cal O'$ of modules for down-up algebras.) show 0 281.4 moveto 10 f4 (We briefly summarize some of the main results.) show 0 270.9 moveto 10 f4 (\\b ) show 0 260.4 moveto 10 f4 (\\proclaim{Proposition 13}\([BR, Secs. 2, 4, 5]\) ) show 0 249.9 moveto 10 f4 (\\item {\(a\)} In $V\(\\l\)$ each weight space is either) show 0 239.4 moveto 10 f4 (one-dimensional or infinite-dimensional. If an infinite-dimensional) show 0 228.9 moveto 10 f4 (weight space occurs, there are only finitely many weights.) show 0 218.4 moveto 10 f4 (\\m) show 0 207.9 moveto 10 f4 (\\item{\(b\)} If each weight space of $V\(\\l\)$ is) show 0 197.4 moveto 10 f4 (one-dimensional, then the proper submodules of $V\(\\l\)$ have the) show 0 186.9 moveto 10 f4 (form $N =$ span$_\\C\\{v_j \\mid j \\geq n+1\\}$ for some $n \\geq 0$ with $\\l_n = 0$.) show 0 176.4 moveto 10 f4 (Hence they are contained in $M\(\\l\) =$ span$_\\C\\{v_j \\mid j \\geq m+1\\}$,) show 0 165.9 moveto 10 f4 (where $\\l_m = 0$ and $m$ is minimal with that property. ) show 0 155.4 moveto 10 f4 (\\m) show 0 144.9 moveto 10 f4 (\\item{\(c\)} If $\\gamma = 0 = \\lambda$, then) show 0 134.4 moveto 10 f4 ($V\(\\l\)$ has infinitely many maximal proper submodules, each) show 0 123.9 moveto 10 f4 (of the form ) show 0 113.4 moveto 10 f4 ($N^{\(\\tau\)} = \\text{span}_\\C\\{v_n -\\tau v_{n-1}\\mid n = 1,2, \\dots\\}$) show 0 102.9 moveto 10 f4 (for some $\\tau \\in \\C$, and infinitely many one-dimensional) show 0 92.4 moveto 10 f4 (simple modules, $L\(0,\\tau\) = V\(0\)/N^{\(\\tau\)}$. In) show 0 81.9 moveto 10 f4 (all other cases, $M\(\\l\)$ is the unique maximal submodule of $V\(\\l\)$,) show pagelevel restore showpage %%Page: 12 12 %%BeginPageSetup /pagelevel save def 54 0 translate %%EndPageSetup newpath 0 72 moveto 504 0 rlineto 0 648 rlineto -504 0 rlineto closepath clip newpath 0 711.9 moveto 10 f4 (and there is a unique simple highest weight module,) show 0 701.4 moveto 10 f4 ($L\(\\l\)= V\(\\l\)/M\(\\l\)$, of weight $\\l$ up to) show 0 690.9 moveto 10 f4 (isomorphism. \\endproclaim) show 0 680.4 moveto 10 f4 (\\m) show 0 669.9 moveto 10 f4 (\\n {\\bf Example}. \\ Recall that the poset $\\Cal L\(2,2\)$ affords a) show 0 659.4 moveto 10 f4 (representation of the down-up algebra $A\(1,-1,1\)$. It is easy) show 0 648.9 moveto 10 f4 (to see that the down and up operators satisfy $d\(\\u 0\) = 0$) show 0 638.4 moveto 10 f4 (and $du\(\\u 0\) = \\u 0$, where $\\u 0 = \(0,0\)$. Thus, the) show 0 627.9 moveto 10 f4 (element $\\u 0 \\in \\Cal L\(2,2\)$ generates a highest weight module) show 0 617.4 moveto 10 f4 (with $\\l = 1$. If we solve) show 0 606.9 moveto 10 f4 (the corresponding recurrence relation in Proposition 12, we get from) show 0 596.4 moveto 10 f4 (\(i\) that $r_1 = 1/2\(1 + \\sqrt{-3}\)$, $r_2 = 1/2\(1 - \\sqrt{-3}\)$,) show 0 585.9 moveto 10 f4 (and $\\l_n = 1+ \(r_2^n - r_1^n\)/\(r_2-r_1\)$. Since $r_1^3 = -1 = r_2^3$,) show 0 575.4 moveto 10 f4 (\(and hence $r_1,r_2$ are 6th roots of unity\), ) show 0 564.9 moveto 10 f4 (we see that the sequence $\\l_0 = \\l,\\l_1,\\l_2, \\dots,$ is given) show 0 554.4 moveto 10 f4 (by $1,2,2,1,0,0,1,2,2,1,0,0,\\dots$. Thus, in the Verma module) show 0 543.9 moveto 10 f4 ($V\(1\)$, the maximal submodule $M\(1\) =$ span$_\\C\\{v_j \\mid j \\geq 5\\}$.) show 0 533.4 moveto 10 f4 (The irreducible quotient $L\(1\) = V\(1\)/M\(1\)$ is 5-dimensional, and it is) show 0 522.9 moveto 10 f4 (spanned modulo $M\(1\)$ by $v_0,v_1,v_2,v_3, v_4$. As an) show 0 512.4 moveto 10 f4 ($A\(1,-1,1\)$-module, $\\Cal L\(2,2\)$ decomposes as $L\(1\) \\oplus L\(0\)$,) show 0 501.9 moveto 10 f4 (where we identify the copy of $L\(1\)$ with the span of the vectors) show 0 491.4 moveto 10 f4 ($v_0 = \(0,0\)$, $v_1 =) show 0 480.9 moveto 10 f4 (\(1,0\)$, $v_2 = \(2,0\) + \(1,1\)$,) show 0 470.4 moveto 10 f4 ($v_3 = 2 \\cdot \(2,1\)$, $v_4 = 2 \\cdot \(2,2\)$, and $L\(0\)$ with) show 0 459.9 moveto 10 f4 (the span of $\(2,0\) - \(1,1\)$. ) show 0 449.4 moveto 10 f4 (\\b) show 0 438.9 moveto 10 f4 (\\subhead The structure of down-up algebras \\endsubhead ) show 0 428.4 moveto 10 f4 (\\m) show 0 417.9 moveto 10 f4 (>From a ring theoretic viewpoint, down-up algebras exhibit many interesting) show 0 407.4 moveto 10 f4 (features. For example, it is apparent from the) show 0 396.9 moveto 10 f4 (defining relations that) show 0 386.4 moveto 10 f4 (the monomials $u^i\(du\)^jd^k,$ $i,j,k = 0,1, \\dots$ in) show 0 375.9 moveto 10 f4 (a down-up algebra $A = A\(\\a,\\be,\\ga\)$ ) show 0 365.4 moveto 10 f4 (determine a spanning set. ) show 0 354.9 moveto 10 f4 (In [BR, Thm. 3.1] we applied the Diamond Lemma \(see [Be]\) to prove a ) show 0 344.4 moveto 10 f4 (Poincar\\'e-Birkhoff-Witt type result for down-up algebras. There) show 0 333.9 moveto 10 f4 (is one essential ambiguity, $\(d^2u\)u = d\(du^2\)$, and the result of) show 0 323.4 moveto 10 f4 (resolving the ambiguity in the two possible ways is the same. ) show 0 312.9 moveto 10 f4 (\\m) show 0 302.4 moveto 10 f4 (\\proclaim {Theorem 14} \(Poincar\\'e-Birkhoff-Witt Theorem\) Assume $A =) show 0 291.9 moveto 10 f4 (A\(\\a,\\be,\\ga\)$ is a down-up algebra over $\\C$. Then) show 0 281.4 moveto 10 f4 ($\\{u^i\(du\)^jd^k \\mid i,j,k = 0,1,\\dots \\}$ is a basis of $A$.) show 0 270.9 moveto 10 f4 (\\endproclaim) show 0 260.4 moveto 10 f4 (\\m) show 0 249.9 moveto 10 f4 (The Gelfand-Kirillov dimension is a natural dimension to assign to an algebra $A$,) show 0 239.4 moveto 10 f4 (and in many cases \(such as when $A$ is a domain\), ) show 0 228.9 moveto 10 f4 (it provides important structural information. ) show 0 218.4 moveto 10 f4 (Theorem 14 enables us to compute the GK-dimension of any down-up algebra $A =) show 0 207.9 moveto 10 f4 (A\(\\a,\\be,\\ga\)$. The spaces $A^{\(n\)} =$ span$_\\C\\{u^i \(du\)^j d^k \\mid) show 0 197.4 moveto 10 f4 (i+2j+k \\leq n\\}$ afford a filtration) show 0 186.9 moveto 10 f4 ($\(0\) \\subset A^{\(0\)} \\subset A^{\(1\)} \\subset \\dots \\subset \\cup_n A^{\(n\)} =) show 0 176.4 moveto 10 f4 (A\(\\a,\\be,\\ga\)$ of the down-up algebra, and $A^{\(m\)} A^{\(n\)} \\subseteq A^{\(m+n\)}$ since the) show 0 165.9 moveto 10 f4 (defining relations replace the words $d^2 u$ and $d u^2$ by words of the) show 0 155.4 moveto 10 f4 (same or lower total degree. The number of monomials $u^i \(du\)^j d^k$) show 0 144.9 moveto 10 f4 (with $i+2j+k = \\ell$ is $\(m+1\)\(m+1\)$ if $\\ell = 2m$ and is $\(m+1\)\(m+2\)$) show 0 134.4 moveto 10 f4 (if $\\ell = 2m+1$. Thus, dim$A^{\(n\)}$ is a polynomial in $n$ with) show 0 123.9 moveto 10 f4 (positive coefficients of degree 3, and ) show 0 113.4 moveto 10 f4 (the Gelfand-Kirillov dimension is given by) show 0 102.9 moveto 10 f4 ($$ ) show 0 92.4 moveto 10 f4 (\\text{GKdim}\\big\(A\(\\a,\\be,\\ga\)\\big\) = \\limsup_{n \\rightarrow \\infty} \\log_n\(\\dim) show 0 81.9 moveto 10 f4 (A^{\(n\)}\)) show pagelevel restore showpage %%Page: 13 13 %%BeginPageSetup /pagelevel save def 54 0 translate %%EndPageSetup newpath 0 72 moveto 504 0 rlineto 0 648 rlineto -504 0 rlineto closepath clip newpath 0 711.9 moveto 10 f4 ( = \\lim_{n \\rightarrow \\infty}{\\frac {\\ln\\Big\(\\dim) show 0 701.4 moveto 10 f4 (A^{\(n\)}\\Big\)}{\\ln n}} = 3.$$) show 0 690.9 moveto 10 f4 (\\m) show 0 680.4 moveto 10 f4 (\\proclaim {Proposition 15}\([BR, Sec. 3]\) If $A\(\\a,\\be,\\ga\)$) show 0 669.9 moveto 10 f4 (has infinitely many simple Verma modules) show 0 659.4 moveto 10 f4 ($V\(\\l\)$, then the intersection of the) show 0 648.9 moveto 10 f4 (annihilators of the simple Verma modules is zero. \\endproclaim ) show 0 638.4 moveto 10 f4 (\\m) show 0 627.9 moveto 10 f4 (It follows immediately that for such a down-up algebra $A\(\\a,\\be,\\ga\)$) show 0 617.4 moveto 10 f4 (the Jacobson radical, which is the intersection) show 0 606.9 moveto 10 f4 (of the annihilators of all the simple modules, is zero.) show 0 596.4 moveto 10 f4 (\\m) show 0 585.9 moveto 10 f4 (When $\\be = 0$, then $d\(du - \\a ud -\\ga\) = 0$) show 0 575.4 moveto 10 f4 (so that $A\(\\a,\\be,\\ga\)$ has zero divisors for any choice of $\\a,\\ga \\in \\C$.) show 0 564.9 moveto 10 f4 (Necessary and sufficient conditions for $A\(\\a,\\be,\\ga\)$ to be a domain or to be) show 0 554.4 moveto 10 f4 (Noetherian have been proven recently ) show 0 543.9 moveto 10 f4 (using Theorem 14:) show 0 533.4 moveto 10 f4 (\\m) show 0 522.9 moveto 10 f4 (\\proclaim {Proposition 16}\([KMP], and compare also [K2].\) For a down-up algebra $A =) show 0 512.4 moveto 10 f4 (A\(\\a,\\be,\\ga\)$, the following are equivalent:) show 0 501.9 moveto 10 f4 (\\s) show 0 491.4 moveto 10 f4 (\\item{}{\(i\)} $\\beta \\neq 0$.) show 0 480.9 moveto 10 f4 (\\s) show 0 470.4 moveto 10 f4 (\\item{}{\(ii\)} $A$ is a domain. ) show 0 459.9 moveto 10 f4 (\\s) show 0 449.4 moveto 10 f4 (\\item{}{\(iii\)} $A$ is right and left Noetherian.) show 0 438.9 moveto 10 f4 (\\s) show 0 428.4 moveto 10 f4 (\\item{}{\(iv\)} $\\C[du,ud]$ is a polynomial ring in the two variables,) show 0 417.9 moveto 10 f4 (\(i.e. $du$ and $ud$ are algebraically independent\). \\endproclaim) show 0 407.4 moveto 10 f4 (\\m) show 0 396.9 moveto 10 f4 ( A down-up algebra $A = A\(\\a,\\be,\\ga\)$ is $\\Z$-graded by assigning deg$\(d\) = -1$ and) show 0 386.4 moveto 10 f4 (deg$\(u\) = 1$ and extending to all of $A$ by setting deg$\(ab\)$ = deg$\(a\)$+deg$\(b\)$.) show 0 375.9 moveto 10 f4 (Then $A = \\bigoplus_{n \\in \\Z} A_n$ where $A_n = \\{a \\in A \\mid$ deg$\(a\) = n\\}$,) show 0 365.4 moveto 10 f4 (and $A_0$ is a commutative subalgebra.) show 0 354.9 moveto 10 f4 (It is shown in [BR] that if $A$ has infinitely many) show 0 344.4 moveto 10 f4 (simple Verma modules, the center lies in $A_0$. The center of $A$ has) show 0 333.9 moveto 10 f4 (been completely described in recent work \([K2] and [Z]\). ) show 0 323.4 moveto 10 f4 ( ) show 0 302.4 moveto 10 f4 (\\Refs ) show 0 291.9 moveto 10 f4 (\\widestnumber\\key{BGG}) show 0 281.4 moveto 10 f4 (\\b) show 0 270.9 moveto 10 f4 (\\ref \\key B \\by G. Benkart, \\paper Down-up algebras and Witten's ) show 0 260.4 moveto 10 f4 (deformations of) show 0 249.9 moveto 10 f4 (the universal enveloping algebra of $\\fsl$, \\jour Contemp. Math.) show 0 239.4 moveto 10 f4 (Amer. Math. Soc. \\toappear \\endref ) show 0 228.9 moveto 10 f4 (\\m) show 0 218.4 moveto 10 f4 (\\ref \\key BR \\by G. Benkart and T. Roby \\paper Down-up algebras) show 0 207.9 moveto 10 f4 (\\jour J. Algebra \\toappear \\endref ) show 0 197.4 moveto 10 f4 (\\m) show 0 186.9 moveto 10 f4 (\\ref\\key Be \\by G.M. Bergman \\paper The diamond lemma for ring theory \\jour) show 0 176.4 moveto 10 f4 (Adv. in Math. \\vol 29 \\yr 1978 \\pages 178--218\\endref) show 0 165.9 moveto 10 f4 (\\m) show 0 155.4 moveto 10 f4 ( ) show 0 144.9 moveto 10 f4 (\\ref\\key BGG\\by J. Bernstein, I.M. Gelfand, and S.I. Gelfand \\paper A category) show 0 134.4 moveto 10 f4 (of $\\g$-modules \\jour) show 0 123.9 moveto 10 f4 (Func. Anal. Appl. \\vol 10 \\yr 1976 \\pages 87--92\\endref) show 0 113.4 moveto 10 f4 (\\m) show 0 102.9 moveto 10 f4 (\\ref \\key Br \\by R.A. Brualdi \\book Introductory Combinatorics ) show 0 92.4 moveto 10 f4 (\\bookinfo Second Edition \\publ North Holland, New York\\yr 1992 \\endref) show 0 81.9 moveto 10 f4 (\\m ) show pagelevel restore showpage %%Page: 14 14 %%BeginPageSetup /pagelevel save def 54 0 translate %%EndPageSetup newpath 0 72 moveto 504 0 rlineto 0 648 rlineto -504 0 rlineto closepath clip newpath 0 711.9 moveto 10 f4 (\\ref \\key F \\by S.V. Fomin \\paper Duality of graded graphs \\jour) show 0 701.4 moveto 10 f4 (J. Alg. Comb. \\vol 3 \\yr 1994 \\pages 357--404\\endref ) show 0 690.9 moveto 10 f4 (\\m) show 0 680.4 moveto 10 f4 (\\ref \\key GLZ \\by Ya. I. Granovski$\\check{\\text i}$, I.M. Lutzenko, and A.S.) show 0 669.9 moveto 10 f4 (Zhedanov \\paper Quadratic algebras and dynamical symmetry of the) show 0 659.4 moveto 10 f4 (Schr\\"odinger equation \\jour Soviet Phys. JETP \\vol 72 \(2\) \\yr 1991 ) show 0 648.9 moveto 10 f4 (\\pages 205-209 \\endref ) show 0 638.4 moveto 10 f4 (\\m) show 0 627.9 moveto 10 f4 (\\ref \\key K1 \\by R. Kulkarni \\paper Irreducible) show 0 617.4 moveto 10 f4 (representations of Witten's deformations of $U\(sl_2\)$ \\toappear \\endref) show 0 606.9 moveto 10 f4 (\\m ) show 0 596.4 moveto 10 f4 (\\ref \\key K2 \\by R. Kulkarni \\paper Down-up algebras) show 0 585.9 moveto 10 f4 (and their representations \\toappear \\endref) show 0 575.4 moveto 10 f4 (\\m) show 0 564.9 moveto 10 f4 (\\ref \\key KMP \\by E. Kirkman, I. Musson, and D. Passman ) show 0 554.4 moveto 10 f4 (\\paper Noetherian down-up algebras \\toappear \\endref) show 0 543.9 moveto 10 f4 (\\m ) show 0 533.4 moveto 10 f4 (\\ref \\key L1 \\by L. Le Bruyn \\paper) show 0 522.9 moveto 10 f4 (Two remarks on Witten's quantum enveloping algebra \\jour) show 0 512.4 moveto 10 f4 (Comm. Algebra \\vol 22 \\yr 1994 \\pages 865--876 \\endref) show 0 501.9 moveto 10 f4 (\\m) show 0 491.4 moveto 10 f4 (\\ref \\key L2 \\by L. Le Bruyn ) show 0 480.9 moveto 10 f4 (\\paper Conformal $\\fsl$ enveloping algebras ) show 0 470.4 moveto 10 f4 (\\jour Comm. Algebra \\vol 23 \\yr 1995 \\pages 1325--1362 \\endref ) show 0 459.9 moveto 10 f4 ( ) show 0 449.4 moveto 10 f4 (\\m) show 0 438.9 moveto 10 f4 (\\ref \\key M \\by Yu. Manin \\paper Some remarks on Koszul algebras and quantum) show 0 428.4 moveto 10 f4 (groups \\jour Ann. Inst. Fourier \\vol 37 \\yr 1987 \\pages 191--205 \\endref) show 0 417.9 moveto 10 f4 (\\m) show 0 407.4 moveto 10 f4 (\\ref \\key P \\by R.A. Proctor \\paper Representations of $sl\(2,\\C\)$ on posets) show 0 396.9 moveto 10 f4 (and the Sperner property, SIAM J. Algebraic and Discrete Methods) show 0 386.4 moveto 10 f4 (\\vol 3 \\yr 1982 \\pages 275--280 \\endref ) show 0 375.9 moveto 10 f4 (\\m) show 0 365.4 moveto 10 f4 (\\ref \\key R \\by A.L. Rosenberg \\book Noncommutative Algebraic Geometry and) show 0 354.9 moveto 10 f4 (Representations of Quantized Algebras \\publ Kluwer \\yr 1995 \\endref) show 0 344.4 moveto 10 f4 (\\m) show 0 333.9 moveto 10 f4 (\\ref\\key St1 \\by R.P. Stanley \\paper Weyl groups, the hard Lefschetz theorem,) show 0 323.4 moveto 10 f4 (and the Sperner property \\jour SIAM J. Algebraic and Discrete Methods \\vol) show 0 312.9 moveto 10 f4 (1 \\yr 1980 \\pages 168--184 \\endref) show 0 302.4 moveto 10 f4 (\\m) show 0 291.9 moveto 10 f4 (\\ref\\key St2 \\by R.P. Stanley \\paper ) show 0 281.4 moveto 10 f4 (Differential posets \\jour) show 0 270.9 moveto 10 f4 (J. of Amer. Math. Soc. \\vol 4 \\yr 1988 \\pages 919--961 \\endref) show 0 260.4 moveto 10 f4 ( ) show 0 249.9 moveto 10 f4 (\\m) show 0 239.4 moveto 10 f4 (\\ref\\key T1 \\by P. Terwilliger \\paper The incidence algebra of a uniform poset) show 0 228.9 moveto 10 f4 (\\pages 193-212 \\moreref \\book Coding Theory and Design Theory, Part I ) show 0 218.4 moveto 10 f4 (\\bookinfo D. Ray-Chaudhuri, ed., IMA Series \\vol 20 \\publ) show 0 207.9 moveto 10 f4 (Springer-Verlag, New York \\yr 1990 \\endref) show 0 197.4 moveto 10 f4 (\\m ) show 0 186.9 moveto 10 f4 (\\ref\\key T2 \\by P. Terwilliger \\book Leonard Systems, An Algebraic Approach) show 0 176.4 moveto 10 f4 (to the $q$-Racah Polynomials \\toappear \\endref) show 0 165.9 moveto 10 f4 (\\m) show 0 155.4 moveto 10 f4 (\\ref \\key W1 \\by E. Witten ) show 0 144.9 moveto 10 f4 (\\paper Gauge theories, vertex models, and quantum groups \\jour) show 0 134.4 moveto 10 f4 (Nuclear Phys. B \\vol 330 \\yr 1990 \\pages 285--346 ) show 0 123.9 moveto 10 f4 (\\endref) show 0 113.4 moveto 10 f4 (\\m) show 0 102.9 moveto 10 f4 (\\ref ) show 0 92.4 moveto 10 f4 (\\key W2 \\by E. Witten \\paper) show 0 81.9 moveto 10 f4 (Quantization of Chern-Simons gauge theory with complex gauge group ) show pagelevel restore showpage %%Page: 15 15 %%BeginPageSetup /pagelevel save def 54 0 translate %%EndPageSetup newpath 0 72 moveto 504 0 rlineto 0 648 rlineto -504 0 rlineto closepath clip newpath 0 711.9 moveto 10 f4 (\\jour Comm. Math. Phys. \\vol 137 \\yr 1991 \\pages 29--66 \\endref) show 0 701.4 moveto 10 f4 (\\m ) show 0 690.9 moveto 10 f4 (\\ref \\key Z \\by K. Zhao \\paper Centers of down-up algebras ) show 0 680.4 moveto 10 f4 (\\toappear \\endref) show 0 659.4 moveto 10 f4 (\\endRefs) show 0 638.4 moveto 10 f4 (\\vskip 3.5 mm) show 0 627.9 moveto 10 f4 ( ) show 0 617.4 moveto 10 f4 (\\address ) show 0 606.9 moveto 10 f4 (\\newline ) show 0 596.4 moveto 10 f4 (Department of Mathematics, University of Wisconsin, Madison,) show 0 585.9 moveto 10 f4 (Wisconsin 53706 USA \\newline) show 0 575.4 moveto 10 f4 (benkart\\@math.wisc.edu \\endaddress) show 0 554.4 moveto 10 f4 (\\enddocument \\end ) show pagelevel restore showpage %%EOF