<h3>2 <span class="Heading">Algebras and their Actions</span></h3>
<p>All the algebras considered in this package will be associative and commutative. Scalars belong to a commutative ring <strong class="button">k</strong> with <span class="SimpleMath">1 ≠ 0</span>.</p>
<p><em>(Why not a field? A group ring over the integers is not an algebra. [CDW])</em></p>
<p>A <em>multiplier</em> in a commutative algebra <span class="SimpleMath">A</span> is a function <span class="SimpleMath">μ : A -> A</span> such that</p>
<p class="pcenter">
\mu(ab) ~=~ (\mu a)b ~=~ a(\mu b) \quad \forall~ a,b \in A.
</p>
<p>The <em>regular multipliers</em> of <span class="SimpleMath">A</span> are the functions</p>
<p class="pcenter">
\mu_a : A \to A ~:~ \mu_ab = ab \quad \forall~ b \in A.
</p>
<p>When <span class="SimpleMath">A</span> has a one, it follows from the defining condition that <span class="SimpleMath">μ(b1) = (μ 1)b</span> and so <span class="SimpleMath">μ = μ_a</span> where <span class="SimpleMath">a = μ 1</span>. Since an ideal <span class="SimpleMath">I</span> of <span class="SimpleMath">A</span> is closed under multiplication, a multiplier <span class="SimpleMath">μ</span> may be restricted to <span class="SimpleMath">I</span>.</p>
<p><strong class="button">Question:</strong> Is there an example of an algebra <span class="SimpleMath">A</span> <em>without</em> a one which has multipliers <em>not</em> of the form <span class="SimpleMath">μ_a</span>?</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MultiplierHomomorphism</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <span class="SimpleMath">M</span> is a multiplier algebra with elements of a subalgebra <spanclass="SimpleMath">B</span> of an algebra <span class="SimpleMath">A</span> multiplying an ideal <span class="SimpleMath">I</span> then this operation returns the homomorphism from <span class="SimpleMath">B</span> to <span class="SimpleMath">M</span> mapping <span class="SimpleMath">b</span> to <span class="SimpleMath">μ_b</span>.</p>
</li>
<li><p><span class="SimpleMath">(rr') ⋅ s ~=~ r ⋅ (r' ⋅ s), qquad</span> (so <span class="SimpleMath">1_R ⋅ s = s ~∀~ s ∈ S</span> when <span class="SimpleMath">R</span> has a one),</p>
</li>
</ul>
<p>for all <span class="SimpleMath">k ∈</span><strong class="button">k</strong>, <span class="SimpleMath">r,r' ∈ R, and s,s' ∈ S</span>.</p>
<p>Notice in particular that, for fixed <span class="SimpleMath">r ∈ R</span>, the map <span class="SimpleMath">s ↦ r ⋅ s</span> is a vector space homomorphism, but not in general an algebra homomorphism.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraAction</code>( <var class="Arg">args</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This global function calls one of the following operations, depending on the arguments supplied.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraActionByMultipliers</code>( <var class="Arg">A</var>, <var class="Arg">I</var>, <var class="Arg">B</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>When <span class="SimpleMath">I</span> is an ideal in <span class="SimpleMath">A</span> and <span class="SimpleMath">B</span> is a subalgebra of <span class="SimpleMath">A</span>, we have seen that the multiplier homomorphism from <span class="SimpleMath">A</span> to <code class="code">MultiplierAlgebraOfIdealBySubalgebra(A,I,B)</code> is an action.</p>
<p>In the example the algebra is the group ring of the cyclic group <span class="SimpleMath">C_6</span> over the field <span class="SimpleMath">GF(5)</span>. The ideal is generated by <span class="SimpleMath">v = () + (1,3,5)(2,4,6) + (1,5,3)(2,6,4)</span>. The generator <span class="SimpleMath">r = (1,2,3,4,5,6)</span> acts on <span class="SimpleMath">v</span> by multiplication to give the vector <span class="SimpleMath">r ⋅ v = (1,2,3,4,5,6) + (1,4)(2,5)(3,6) + (1,6,5,4,3,2)</span>, as shown in <code class="func">AlgebraActionByHomomorphism</code> (<a href="chap2.html#X8530E1B27BC2FBB7"><span class="RefLink">2.2-4</span></a>)</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraActionBySurjection</code>( <var class="Arg">hom</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <span class="SimpleMath">θ : B -> A</span> be a surjective algebra homomorphism such that <span class="SimpleMath">kerθ</span> is contained in the annihilator of <span class="SimpleMath">B</span>. Then <span class="SimpleMath">A</span> acts on <span class="SimpleMath">B</span> by <span class="SimpleMath">a ⋅ b = pb</span> where <span class="SimpleMath">p ∈ (θ^-1a)</span>. Note that this action is well defined since <span class="SimpleMath">θ^-1a = { p+k ~|~ k ∈ kerθ }</span> and <span class="SimpleMath">(p+k)b = pb+kb = pb+0</span>.</p>
<p>Continuing with the previous example, we construct the quotient algebra <span class="SimpleMath">Q1 = A1/I1</span>, and the natural homomorphism <span class="SimpleMath">θ_1 : A1 -> Q1</span>. The kernel of <span class="SimpleMath">θ</span> is not contained in the annihilator of <span class="SimpleMath">A1</span>, so an attempt to form the action fails.</p>
<p>An alternative example involves a matrix algebra <span class="SimpleMath">A_2</span> with generator <span class="SimpleMath">m_2</span>, basis <span class="SimpleMath">{m_2,m_2^2,m_2^3}</span>, and where <span class="SimpleMath">m_2^4=0</span>. The ideal <span class="SimpleMath">I_2</span> is generated by <span class="SimpleMath">m_2^3</span> and the quotient <span class="SimpleMath">Q_2</span> has basis <span class="SimpleMath">{[m_2],[m_2^2]}</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraActionByHomomorphism</code>( <var class="Arg">hom</var>, <var class="Arg">alg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <span class="SimpleMath">α : A -> C</span> is an algebra homomorphism where <span class="SimpleMath">C</span> is an algebra of left module isomorphisms of an algebra <span class="SimpleMath">B</span>, then <code class="code">AlgebraActionByHomomorphism( alpha, B )</code> attempts to return an action of <span class="SimpleMath">A</span> on <span class="SimpleMath">B</span>.</p>
<p>In the example the matrix algebra <code class="code">A3</code> and the group algebra <code class="code">Rc3</code> are isomorphic algebras, so the resulting action is equivalent to the multiplier action of <code class="code">Rc3</code> on itself.</p>
<p>Recall that a module can be made into an algebra by defining every product to be zero. When we apply this construction to a (left) algebra module, we obtain an algebra action on an algebra.</p>
<p>Recall the construction of algebra modules from Chapter 62 of the <strong class="pkg">GAP</strong> reference manual. In the example, the vector space <span class="SimpleMath">V3</span> becomes an algebra module <span class="SimpleMath">M3</span> with a left action by <span class="SimpleMath">A3</span>. Conversion between vectors in <span class="SimpleMath">V3</span> and those in <span class="SimpleMath">M3</span> is achieved using the operations <code class="code">ObjByExtRep</code> and <code class="code">ExtRepOfObj</code>. These vectors are indistinguishable when printed.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModuleAsAlgebra</code>( <var class="Arg">leftmod</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>To form an algebra <span class="SimpleMath">B</span> from <span class="SimpleMath">M</span> with zero products we may construct an algebra with the correct dimension using an empty structure constants table, as shown below. In doing so, the remaining information about <span class="SimpleMath">M</span> is lost, so it is essential to form isomorphisms between the corresponding underlying vector spaces.</p>
<p>If the module <span class="SimpleMath">M</span> has been given a name, then the operation <code class="code">ModuleAsAlgebra</code> assigns a name to the resulting algebra. The operation <code class="code">AlgebraByStructureConstants</code> assigns names <span class="SimpleMath">v_i</span> to the basis vectors unless a list of names is provided. The operation <code class="code">ModuleAsAlgebra</code> converts the basis elements of <span class="SimpleMath">M</span> into strings, with additional brackets added, and uses these as the names for the basis vectors. Note that these <code class="code">[[i,j,k]]</code> are just strings, and not vectors.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsModuleAsAlgebra</code>( <var class="Arg">alg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This is the property acquired when a module is converted into an algebra.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModuleToAlgebraIsomorphism</code>( <var class="Arg">alg</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">‣ AlgebraToModuleIsomorphism</code>( <var class="Arg">alg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>These two algebra mappings are attributes of a module converted into an algebra. They are required for the process of converting the action of <span class="SimpleMath">A</span> on <span class="SimpleMath">M</span> into an action on <span class="SimpleMath">B</span>. Note that these left module homomorphisms have as source or range the underlying module <span class="SimpleMath">V</span>, not <span class="SimpleMath">M</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraActionByModule</code>( <var class="Arg">alg</var>, <var class="Arg">leftmod</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This operation converts the action of <span class="SimpleMath">A</span> on <span class="SimpleMath">M</span> into an action of <span class="SimpleMath">A</span> on <span class="SimpleMath">B</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DirectSumOfAlgebrasWithInfo</code>( <var class="Arg">A1</var>, <var class="Arg">A2</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">‣ DirectSumOfAlgebrasInfo</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This attribute for direct sums of algebras is missing from the main library, and is added here to be used in methods for <code class="code">Embedding</code> and <code class="code">Projection</code>. In order to construct a direct sum with this information attribute the operation <code class="code">DirectSumOfAlgebrasWithInfo</code> may be used. This just calls <code class="code">DirectSumOfAlgebras</code> and sets up the attribute.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Embedding</code>( <var class="Arg">A</var>, <var class="Arg">nr</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">‣ Projection</code>( <var class="Arg">A</var>, <var class="Arg">nr</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Methods for <code class="code">Embedding</code> and <code class="code">Projection</code> for direct sums of algebras are missing from the main library, and so are included here.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SemidirectProductOfAlgebras</code>( <var class="Arg">R</var>, <var class="Arg">act</var>, <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>When <span class="SimpleMath">R,S</span> are commutative algebras and <span class="SimpleMath">R</span> acts on <span class="SimpleMath">S</span> then we can form the semidirect product <span class="SimpleMath">R ⋉ S</span>, where the product is given by:</p>
<p>If <span class="SimpleMath">B_R, B_S</span> are the sets of basis vectors for <span class="SimpleMath">R</span> and <span class="SimpleMath">S</span> then <span class="SimpleMath">R ⋉ S</span> has basis</p>
<p class="pcenter">
\{(r,0_S) ~|~ r \in B_R\} ~\cup~ \{(0_R,s) ~|~ s \in B_S\}
</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllAlgebraHomomorphisms</code>( <var class="Arg">A</var>, <var class="Arg">B</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">‣ AllBijectiveAlgebraHomomorphisms</code>( <var class="Arg">A</var>, <var class="Arg">B</var> )</td><tdclass="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllIdempotentAlgebraHomomorphisms</code>( <var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>These three operations list all the homomorphisms from <span class="SimpleMath">A</span> to <span class="SimpleMath">B</span> of the specified type. These lists can get very long, so the operations should only be used with small algebras.</p>
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