{ "cells": [ { "cell_type": "markdown", "id": "531a4e97", "metadata": {}, "source": [ "# PBW Deformations of Smash Products" ] }, { "cell_type": "markdown", "id": "adee9e34", "metadata": {}, "source": [ "We present the current functionality of the $\\texttt{PBWDeformations.jl}$ package." ] }, { "cell_type": "code", "execution_count": 1, "id": "58be0e38", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " ___ ____ ____ _ ____\n", " / _ \\ / ___| / ___| / \\ | _ \\ | Combining ANTIC, GAP, Polymake, Singular\n", "| | | |\\___ \\| | / _ \\ | |_) | | Type \"?Oscar\" for more information\n", "| |_| | ___) | |___ / ___ \\| _ < | Manual: https://docs.oscar-system.org\n", " \\___/ |____/ \\____/_/ \\_\\_| \\_\\ | Version 1.0.4\n" ] } ], "source": [ "using Oscar, PBWDeformations" ] }, { "cell_type": "code", "execution_count": 2, "id": "11bd76f4", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\u001b[32m\u001b[1mStatus\u001b[22m\u001b[39m `~/code/julia/PBWDeformations.jl/examples/Project.toml`\n", " \u001b[90m[f1435218] \u001b[39mOscar v1.0.4\n", " \u001b[90m[5e7992ee] \u001b[39mPBWDeformations v0.3.0\n" ] } ], "source": [ "] status" ] }, { "cell_type": "markdown", "id": "c1e8ef62", "metadata": {}, "source": [ "## Smash products" ] }, { "cell_type": "markdown", "id": "b91b95de", "metadata": {}, "source": [ "One can create smash products of the form $$TV \\rtimes U(L)$$ for some finite-dimensional semisimple Lie algebra $L$ and $V$ some finite dimensional $L$-module." ] }, { "cell_type": "markdown", "id": "5689fd7d", "metadata": {}, "source": [ "As an ongoing example, we use $L = \\mathfrak{gl}_3(\\mathbb{Q})$ and the module $V = V_{\\mathrm{nat}} \\oplus V_{\\mathrm{nat}}^\\ast$, where $V_{\\mathrm{nat}}$ is the natural module." ] }, { "cell_type": "code", "execution_count": 3, "id": "a1d6a9b1", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "General linear Lie algebra of degree 3\n", " of dimension 9\n", "over rational field" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "L = general_linear_lie_algebra(QQ, 3)" ] }, { "cell_type": "code", "execution_count": 4, "id": "30ea30ef-bf34-4bcb-8bdd-f626950a075f", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Direct sum module\n", " of dimension 6\n", " direct sum with direct summands\n", " standard module\n", " dual of \n", " standard module\n", "over general linear Lie algebra of degree 3 over QQ" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "V = direct_sum(standard_module(L), dual(standard_module(L)))" ] }, { "cell_type": "code", "execution_count": 5, "id": "6b441f57-6b1f-4663-9ed3-13ddb53b525e", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Smash Product of General linear Lie algebra of degree 3 over QQ and Direct sum module of dimension 6 over gl_3" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp = smash_product(L, V)" ] }, { "cell_type": "markdown", "id": "56d1fe0a-5d2b-4bb0-8063-78b82ab53444", "metadata": {}, "source": [ "This objects satisfies the interface for non-commutative rings from `AbstractAlgebra.jl`." ] }, { "cell_type": "code", "execution_count": 6, "id": "a3f7532d-57c5-48ab-a65a-db6d4fa30bf5", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "true" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp isa NCRing" ] }, { "cell_type": "markdown", "id": "99f142ff", "metadata": {}, "source": [ "Arithmetics works as expected, with the exception that not after every arithmetic operation the expressions are normalized. This will only be done in some few functions like `==`, and when explicitly calling `simplify`:" ] }, { "cell_type": "code", "execution_count": 7, "id": "d4cb09e5-fbc1-4de3-a03f-1f8ec192d5c9", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "x_2_1*(v_1^(1)) - (v_1^(1))*x_2_1" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "some_elem = gen(sp, 4)*gen(sp, 10) - gen(sp, 10)*gen(sp, 4) # just a formal commutator" ] }, { "cell_type": "code", "execution_count": 8, "id": "7a27f2cd-8aed-48ca-9e94-26a0752a994f", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(v_2^(1))" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "simplify(some_elem)" ] }, { "cell_type": "markdown", "id": "6ec29e6c", "metadata": {}, "source": [ "## Deforming smash products" ] }, { "cell_type": "markdown", "id": "aa5f0556", "metadata": {}, "source": [ "A deformation of a smash product $TV \\rtimes U(L)$ is formally $$A := A_{L,V,\\kappa} := (TV \\rtimes U(L))/I_\\kappa$$ where $I_\\kappa := \\big([v_i,v_j] - \\kappa(v_i \\wedge v_j)\\big)$ for some morphism $\\kappa: V \\wedge V \\to U(L)$." ] }, { "cell_type": "markdown", "id": "64dcbca8", "metadata": {}, "source": [ "In the context of this package, $\\kappa$ is always represented as a matrix $M_\\kappa$ where $M_\\kappa[i,j] = \\kappa(v_i \\wedge v_j)$." ] }, { "cell_type": "markdown", "id": "91a435ec", "metadata": {}, "source": [ "As a first example, consider the symmetric deformation induced by $\\kappa = 0$ and denoted by $A_0 := A_{H,V,0}$." ] }, { "cell_type": "code", "execution_count": 9, "id": "bcce03ca", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Symmetric deformation of Smash Product of General linear Lie algebra of degree 3 over QQ and Direct sum module of dimension 6 over gl_3" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "symmetric_deformation(sp)" ] }, { "cell_type": "markdown", "id": "7ec4c6d6", "metadata": {}, "source": [ "This is in fact only a shorthand for the following, slightly longer code." ] }, { "cell_type": "code", "execution_count": 10, "id": "aa628ffd", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Symmetric deformation of Smash Product of General linear Lie algebra of degree 3 over QQ and Direct sum module of dimension 6 over gl_3" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "kappa = zero_matrix(underlying_algebra(sp), dim(base_module(sp)), dim(base_module(sp)))\n", "deform(sp, kappa)" ] }, { "cell_type": "markdown", "id": "7c48253b", "metadata": {}, "source": [ "## PBW deformations" ] }, { "cell_type": "markdown", "id": "30c1cc64", "metadata": {}, "source": [ "$A$ becomes a filtered algebra via $$F_i(A) = \\overline{T^{\\leq i}V \\rtimes H}.$$\n", "A deformation $A$ is called a *PBW deformation* of $A_0 = SV \\rtimes H$ if $\\mathop{gr} A \\cong A_0$ (as $\\mathbb{N}$-graded algebras)." ] }, { "cell_type": "markdown", "id": "8d3045b4", "metadata": {}, "source": [ "This package can check for any $\\kappa$ if this induces a PBW deformation. To achieve this, it uses Theorem 3.1 of [[WW14]](#References)." ] }, { "cell_type": "code", "execution_count": 11, "id": "2691d53d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "true" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "is_pbwdeformation(symmetric_deformation(sp))" ] }, { "cell_type": "markdown", "id": "d42b5fda", "metadata": {}, "source": [ "The symmetric deformation is *always* a PBW deformation." ] }, { "cell_type": "code", "execution_count": 12, "id": "219ff3b1", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "true" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "kappa2 = zero_matrix(underlying_algebra(sp), dim(base_module(sp)), dim(base_module(sp)))\n", "kappa2[1,2] = gen(underlying_algebra(sp), 4)\n", "kappa2[2,1] = -kappa2[1,2]\n", "deform2 = deform(sp, kappa)\n", "is_pbwdeformation(deform2)" ] }, { "cell_type": "markdown", "id": "395a4688", "metadata": {}, "source": [ "Most deformations are no PBW deformations." ] }, { "cell_type": "markdown", "id": "d763a0a3", "metadata": {}, "source": [ "## Computing *all* PBW deformations" ] }, { "cell_type": "markdown", "id": "3f71fc37", "metadata": {}, "source": [ "Using Theorem 3.1 of [[WW14]](#References), one can compute a basis of the matrix space of all $M_\\kappa$ (up to a fixed degree) that induce a PBW deformation.\n", "\n", "**WARNING**: This computation needs a lot of time and RAM, even for small examples." ] }, { "cell_type": "code", "execution_count": 13, "id": "016821a4", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3-element Vector{MatElem{<:FreeAssAlgElem{QQFieldElem}}}:\n", " [0 0 0 1 0 0; 0 0 0 0 1 0; 0 0 0 0 0 1; -1 0 0 0 0 0; 0 -1 0 0 0 0; 0 0 -1 0 0 0]\n", " [0 0 0 x_1_1+1//2*x_2_2+1//2*x_3_3 1//2*x_1_2 1//2*x_1_3; 0 0 0 1//2*x_2_1 1//2*x_1_1+x_2_2+1//2*x_3_3 1//2*x_2_3; 0 0 0 1//2*x_3_1 1//2*x_3_2 1//2*x_1_1+1//2*x_2_2+x_3_3; -x_1_1-1//2*x_2_2-1//2*x_3_3 -1//2*x_2_1 -1//2*x_3_1 0 0 0; -1//2*x_1_2 -1//2*x_1_1-x_2_2-1//2*x_3_3 -1//2*x_3_2 0 0 0; -1//2*x_1_3 -1//2*x_2_3 -1//2*x_1_1-1//2*x_2_2-x_3_3 0 0 0]\n", " [0 0 0 x_1_1^2+2//3*x_1_1*x_2_2+2//3*x_1_1*x_3_3+2//3*x_1_2*x_2_1+2//3*x_1_3*x_3_1+1//3*x_2_2^2+1//3*x_2_2*x_3_3+1//3*x_2_3*x_3_2+1//3*x_3_3^2-4//3*x_1_1-1//6*x_2_2+1//6*x_3_3 2//3*x_1_1*x_1_2+2//3*x_1_2*x_2_2+1//3*x_1_2*x_3_3+1//3*x_1_3*x_3_2-7//6*x_1_2 2//3*x_1_1*x_1_3+1//3*x_1_2*x_2_3+1//3*x_1_3*x_2_2+2//3*x_1_3*x_3_3-7//6*x_1_3; 0 0 0 2//3*x_1_1*x_2_1+2//3*x_2_1*x_2_2+1//3*x_2_1*x_3_3+1//3*x_2_3*x_3_1+1//6*x_2_1 1//3*x_1_1^2+2//3*x_1_1*x_2_2+1//3*x_1_1*x_3_3+2//3*x_1_2*x_2_1+1//3*x_1_3*x_3_1+x_2_2^2+2//3*x_2_2*x_3_3+2//3*x_2_3*x_3_2+1//3*x_3_3^2-5//6*x_1_1-2//3*x_2_2+1//6*x_3_3 1//3*x_1_1*x_2_3+1//3*x_1_3*x_2_1+2//3*x_2_2*x_2_3+2//3*x_2_3*x_3_3-5//6*x_2_3; 0 0 0 2//3*x_1_1*x_3_1+1//3*x_2_1*x_3_2+1//3*x_2_2*x_3_1+2//3*x_3_1*x_3_3+1//2*x_3_1 1//3*x_1_1*x_3_2+1//3*x_1_2*x_3_1+2//3*x_2_2*x_3_2+2//3*x_3_2*x_3_3+1//2*x_3_2 1//3*x_1_1^2+1//3*x_1_1*x_2_2+2//3*x_1_1*x_3_3+1//3*x_1_2*x_2_1+2//3*x_1_3*x_3_1+1//3*x_2_2^2+2//3*x_2_2*x_3_3+2//3*x_2_3*x_3_2+x_3_3^2-5//6*x_1_1-1//2*x_2_2; -x_1_1^2-2//3*x_1_1*x_2_2-2//3*x_1_1*x_3_3-2//3*x_1_2*x_2_1-2//3*x_1_3*x_3_1-1//3*x_2_2^2-1//3*x_2_2*x_3_3-1//3*x_2_3*x_3_2-1//3*x_3_3^2+4//3*x_1_1+1//6*x_2_2-1//6*x_3_3 -2//3*x_1_1*x_2_1-2//3*x_2_1*x_2_2-1//3*x_2_1*x_3_3-1//3*x_2_3*x_3_1-1//6*x_2_1 -2//3*x_1_1*x_3_1-1//3*x_2_1*x_3_2-1//3*x_2_2*x_3_1-2//3*x_3_1*x_3_3-1//2*x_3_1 0 0 0; -2//3*x_1_1*x_1_2-2//3*x_1_2*x_2_2-1//3*x_1_2*x_3_3-1//3*x_1_3*x_3_2+7//6*x_1_2 -1//3*x_1_1^2-2//3*x_1_1*x_2_2-1//3*x_1_1*x_3_3-2//3*x_1_2*x_2_1-1//3*x_1_3*x_3_1-x_2_2^2-2//3*x_2_2*x_3_3-2//3*x_2_3*x_3_2-1//3*x_3_3^2+5//6*x_1_1+2//3*x_2_2-1//6*x_3_3 -1//3*x_1_1*x_3_2-1//3*x_1_2*x_3_1-2//3*x_2_2*x_3_2-2//3*x_3_2*x_3_3-1//2*x_3_2 0 0 0; -2//3*x_1_1*x_1_3-1//3*x_1_2*x_2_3-1//3*x_1_3*x_2_2-2//3*x_1_3*x_3_3+7//6*x_1_3 -1//3*x_1_1*x_2_3-1//3*x_1_3*x_2_1-2//3*x_2_2*x_2_3-2//3*x_2_3*x_3_3+5//6*x_2_3 -1//3*x_1_1^2-1//3*x_1_1*x_2_2-2//3*x_1_1*x_3_3-1//3*x_1_2*x_2_1-2//3*x_1_3*x_3_1-1//3*x_2_2^2-2//3*x_2_2*x_3_3-2//3*x_2_3*x_3_2-x_3_3^2+5//6*x_1_1+1//2*x_2_2 0 0 0]" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "kappas = all_pbwdeformations(sp, 0:2)" ] }, { "cell_type": "markdown", "id": "21f9b314", "metadata": {}, "source": [ "The space of $M_\\kappa$ of degree at most 2 and inducing a PBW deformation has dimension 2.\n", "\n", "We can check that these indeed induce PBW deformations." ] }, { "cell_type": "code", "execution_count": 14, "id": "9bf73aea", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "true\n", "true\n", "true\n" ] } ], "source": [ "for kappa in kappas\n", " d = deform(sp, kappa)\n", " println(is_pbwdeformation(d))\n", "end" ] }, { "cell_type": "markdown", "id": "46d1e0e0-1c3c-4e6a-afb3-3b8a136244dd", "metadata": {}, "source": [ "## Computing *all* PBW deformations using Arc diagrams" ] }, { "cell_type": "markdown", "id": "26e2f14c-3438-4899-bd33-b51291849286", "metadata": {}, "source": [ "[[FM22]](#References) introduced techniques using interpolation categories to efficiently parameterize a generating set of the morphism space of eligible deformation maps $\\kappa: V \\wedge V \\to U(L)$ for the case of $\\mathfrak{so}_n$. This was achieved using arc diagrams. We extend their methods to $\\mathfrak{gl}_n$ and need directed arc diagrams." ] }, { "attachments": {}, "cell_type": "markdown", "id": "6be8de99-0f9f-4a93-b64b-94a8bd3870ac", "metadata": {}, "source": [ "We can enumerate all arc diagrams needed for a particular morphism space, here e.g. for $\\mathrm{Hom}(V_{\\mathrm{nat}} \\otimes V_{\\mathrm{nat}}^\\ast, S^2 L) \\cong \\mathrm{Hom}(V_{\\mathrm{nat}} \\otimes V_{\\mathrm{nat}}^\\ast, S^2 (V_{\\mathrm{nat}} \\otimes V_{\\mathrm{nat}}^\\ast))$, using the following *internal* function." ] }, { "cell_type": "code", "execution_count": 15, "id": "2435c34b-2661-4ef6-a279-ce714020d968", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "6-element Vector{ArcDiagramDirected}:\n", " aA,CcEe\n", " aA,CdDc\n", " aB,AbEe\n", " aB,AdDb\n", " aB,CbAc\n", " aB,CcAb" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "PBWDeformations.pbw_arc_diagrams(PBWDeformations.GL(), tensor_product(standard_module(L), dual(standard_module(L))), 2) |> collect" ] }, { "attachments": { "fedf7d9b-91ee-4e04-b04a-36f645f365bd.png": { "image/png": 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" } }, "cell_type": "markdown", "id": "2ecb4a29-ed04-446e-bc44-65c73caf1b50", "metadata": {}, "source": [ "We then specialize the orbit of **aB,AdDb** to $\\mathrm{Hom}(V_{\\mathrm{nat}} \\otimes V_{\\mathrm{nat}}^\\ast, S^2 L)$ using the following *internal* function. This arc diagram is\n", "