<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismOfSemisimpleLieAlgebras</code>( <var class="Arg">L1</var>, <var class="Arg">L2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">L1</var> and <var class="Arg">L2</var> are two semisimple Lie algebras that are known to be isomorphic (i.e., they have the same type). This function returns an isomorphism.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ApplyWeylPermToCartanElement</code>( <var class="Arg">L</var>, <var class="Arg">w</var>, <var class="Arg">h</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">L</var> is a semisimple Lie algebra, <var class="Arg">w</var> is a permutation which is an element of <var class="Arg">WeylGroupAsPermGroup( RootSystem(L) )</var>, and <var class="Arg">h</var> is an element of the Cartan subalgebra <var class="Arg">CartanSubalgebra( L )</var>. The Weyl groups naturally acts on this Cartan subalgebra and this function returns the result of applying <var class="Arg">w</var> to <var class="Arg">h</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AdmissibleLattice</code>( <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">V</var> is a <em>simple</em> module over a semisimple Lie algebra. This function returns a basis of <var class="Arg">V</var> that spans an admissible lattice in <var class="Arg">V</var>. This means that for a root vector <span class="SimpleMath">\(x\)</span> of the acting Lie algebra the matrix <span class="SimpleMath">\(exp( mx )\)</span> is integral, where <span class="SimpleMath">\(mx\)</span> denotes the matrix of <span class="SimpleMath">\(x\)</span> relative to the admissible lattice.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DirectSumDecomposition</code>( <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">V</var> is a module over a semisimple Lie algebra; this function computes a list of sub-modules such that <var class="Arg">V</var> is their direct sum.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= SimpleLieAlgebra("G",2,Rationals);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= HighestWeightModule( L, [1,0] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= TensorProductOfAlgebraModules( V, V );</span>
<49-dimensional left-module over <Lie algebra of dimension 14 over Rationals>>
<span class="GAPprompt">gap></span> <span class="GAPinput">DirectSumDecomposition( W );</span>
[ <left-module over <Lie algebra of dimension 14 over Rationals>>,
<left-module over <Lie algebra of dimension 14 over Rationals>>,
<left-module over <Lie algebra of dimension 14 over Rationals>>,
<left-module over <Lie algebra of dimension 14 over Rationals>> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( last, Dimension );</span>
[ 27, 7, 14, 1 ]
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HighestWeightVector</code>( <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">V</var> is an irreducible module over a semisimple Lie algebra. This function returns a highest weight vector <var class="Arg">v0</var> in <var class="Arg">V</var>. This means that it is a weight vector for the Cartan subalgebra of the acting Lie algebra, and all positive root vectors send it to zero.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HighestWeight</code>( <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">V</var> is an irreducible module over a semisimple Lie algebra. This function returns the highest weight of <var class="Arg">V</var>. That is, the list of eigenvalues of the Cartan elements in a canonical generating set of the Lie algebra, when acting on a highest weight vector.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayHighestWeight</code>( <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">V</var> is an irreducible module over a semisimple Lie algebra. This function displays its highest weight, that is, it shows the coordinates of the highest weight on the Dynkin diagram of the Lie algebra.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">r:= LieAlgebraAndSubalgebras( "E8" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= r.liealg;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">K:= r.subalgs[823];</span>
<Lie algebra of dimension 58 over CF(84)>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayDynkinDiagram(K);</span>
A1: 1
B5: 2---3---4---5=>=6
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= AdjointModule( L );</span>
<248-dimensional left-module over <Lie algebra of dimension 248 over CF(84)>>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= ModuleByRestriction( V, K );</span>
<248-dimensional left-module over <Lie algebra of dimension 58 over CF(84)>>
<span class="GAPprompt">gap></span> <span class="GAPinput">dW:= DirectSumDecomposition( W ); </span>
[ <left-module over <Lie algebra of dimension 58 over CF(84)>>,
<left-module over <Lie algebra of dimension 58 over CF(84)>>,
<left-module over <Lie algebra of dimension 58 over CF(84)>>,
<left-module over <Lie algebra of dimension 58 over CF(84)>>,
<left-module over <Lie algebra of dimension 58 over CF(84)>>,
<left-module over <Lie algebra of dimension 58 over CF(84)>>,
<left-module over <Lie algebra of dimension 58 over CF(84)>>,
<left-module over <Lie algebra of dimension 58 over CF(84)>>,
<left-module over <Lie algebra of dimension 58 over CF(84)>>,
<left-module over <Lie algebra of dimension 58 over CF(84)>> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( dW, Dimension );</span>
[ 33, 3, 3, 3, 64, 64, 11, 11, 55, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayHighestWeight( dW[5] );</span>
A1: 1
B5: 0---0---0---0=>=1
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayHighestWeight( dW[1] );</span>
A1: 2
B5: 1---0---0---0=>=0
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismOfIrreducibleHWModules</code>( <var class="Arg">V1</var>, <var class="Arg">V2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">V1</var>, <var class="Arg">V2</var> are two irreducible modules over the same semisimple Lie algebra with the same highest weights. This function returns an isomorphism between the two.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DualAlgebraModule</code>( <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">V</var> is a module over a Lie algebra. This function returns the dual module.</p>
<p>The basis elements of this module are printed as <var class="Arg">F@v</var> where <var class="Arg">v</var> is a basis element of <var class="Arg">v</var>. This represents the function which takes the value 1 on te basis element <var class="Arg">v</var> and 0 on all other basis elements. However, an element of the module is a module element and not a function. We can access the function by taking the <var class="Arg">ExtRepOfObj</var> of an element of the module, as illustrated by the example below.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CharacteristicsOfStrata</code>( <var class="Arg">L</var>, <var class="Arg">hw</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CharacteristicsOfStrata</code>( <var class="Arg">L</var>, <var class="Arg">B</var>, <var class="Arg">hw</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">L</var> is a semisimple Lie algebra over a field of characteristic 0. Secondly, <var class="Arg">hw</var> is a dominant weight, represented as a list of non-negative integers (where the ordering of the fundamantal weights is given by the Cartan matrix of the root system of <var class="Arg">L</var>). Let <span class="SimpleMath">\(G\)</span> denote the semisimple algebraic group acting on the irreducible representation with highest weight <var class="Arg">hw</var>. Alternatively, <var class="Arg">hw</var> can also be a list of highest weights, in which case the representation is the direct sum of the irreducible representations with highest weights in the list. Hesselink (<a href="chapBib_mj.html#biBhesselink">[Hes79]</a>) defined a stratification of the nullcone relative to the action of <span class="SimpleMath">\(G\)</span>. Popov and Vinberg (<a href="chapBib_mj.html#biBpovin">[VP89]</a>) have described this stratification in terms of characteristics, which are elements of a Cartan subalgebra of <var class="Arg">L</var>. To each characteristic there corresponds a stratum. This function is an implementation of an algorithm due to Popov (<a href="chapBib_mj.html#biBpopov">[Pop03]</a>), for computing the characteristics of the strata. It returns a list of two lists. The first list contains the characteristics. The second list contains the dimensions of the corresponding strata. If the highest weight <var class="Arg">hw</var> defines the adjoint representation, then the characteristics of the strata are exactly the characteristics of the nilpotent orbits in <var class="Arg">L</var>. This means the following: let <span class="SimpleMath">\(h\)</span> be a characteristic, then there are <span class="SimpleMath">\(e,f\)</span> in <var class="Arg">L</var> such that the triple <span class="SimpleMath">\(h,e,f\)</span> satisfies the commutation relations of <span class="SimpleMath">\(\mathfrak{sl}_2\)</span>, and the elements <span class="SimpleMath">\(e\)</span> thus obtained are the representatives of the nilpotent <span class="SimpleMath">\(G\)</span>-orbits in <var class="Arg">L</var>.</p>
<p>We remark that the characteristics depend on the choice of an invariant bilinear form. This form is unique if <var class="Arg">L</var> is simple. If we give just two arguments, <var class="Arg">L</var>, <var class="Arg">hw</var>, then Killing form is chosen. It is possible to use a different form using the three argument variant of the function.</p>
<p>In the three argument variant <var class="Arg">L</var> is a reductive Lie algebra and <var class="Arg">B</var> is the restriction of a non-degenerate invariant bilinear form on the Cartan subalgebra of <var class="Arg">L</var>. This bilinear form must be given with respect to a specific basis, which we now describe. Let <var class="Arg">K</var> denote the derived subalgebra of <var class="Arg">L</var> (which is semisimple). Let <var class="Arg">h</var> be the list <var class="Arg">CanonicalGenerators( RootSystem( K ) )[3]</var> (this is a basis of a Cartan subalgebra of <var class="Arg">K</var>). Let <var class="Arg">c</var> be the list <var class="Arg">BasisVectors( Basis( LieCentre(K) ) )</var>. Then the basis we require is the concatenation of <var class="Arg">h</var> and <var class="Arg">c</var>. Again <var class="Arg">hw</var> can be a highest weight, or a list of highest weights. These highest weights are lists of eigenvalues of the elements of the particular basis of a Cartan subalgebra of <var class="Arg">L</var> described above.</p>
<p>In the next example we compute the strata of a representation of a reductive subalgebra of the Lie algebra of type <span class="SimpleMath">\(E_6\)</span>, obtained as the set of fixed points of an inner automorphism. We compute the strata of the <span class="SimpleMath">\(\theta\)</span>-representation corresponding to the automorphism. For this we first need to work out the highest weights of the module. The bilinear form is the restriction of the Killing form to the subalgebra.</p>
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