<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">\(1 \neq 0\)</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">\(\mu : A \to A\)</span> such that</p>
<p class="center">\[
\mu(ab) ~=~ (\mu a)b ~=~ a(\mu b) \quad \forall~ a,b \in A.
\]</p>
<p>The <em>regular multipliers</em> of <span class="SimpleMath">\(A\)</span> are the functions</p>
<p class="center">\[
\mu_a : A \to A ~:~ \mu_ab = ab \quad \forall~ b \in A.
\]</p>
<p>When <span class="SimpleMath">\(A\)</span> has a one, it follows from the defining condition that <span class="SimpleMath">\(\mu(b1) = (\mu 1)b\)</span> and so <span class="SimpleMath">\(\mu = \mu_a\)</span> where <span class="SimpleMath">\(a = \mu 1\)</span>. Since an ideal <span class="SimpleMath">\(I\)</span> of <span class="SimpleMath">\(A\)</span> is closed under multiplication, a multiplier <span class="SimpleMath">\(\mu\)</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">\(\mu_a\)</span>?</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MultiplierAlgebraOfIdealBySubalgebra </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>The regular multipliers <span class="SimpleMath">\(\mu_b : I \to I\)</span> for all <span class="SimpleMath">\(b \in B\)</span>, where <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>, form an algebra with product <span class="SimpleMath">\(\mu_b \circ \mu_{b'} = \mu_{bb'}\)</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 <span class="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">\(\mu_b\)</span>.</p>
<p>If <span class="SimpleMath">\(S\)</span> and <span class="SimpleMath">\(R\)</span> are commutative <strong class="button">k</strong>-algebras, a map</p>
<p class="center">\[
R \times S ~\to~ S,
\qquad
(r,s) ~\mapsto~ r \cdot s
\]</p>
<p>is a commutative action if and only if the following five axioms hold:</p>
<ul>
<li><p><span class="SimpleMath">\(k(r \cdot s) ~=~ (kr) \cdot s ~=~ r \cdot (ks)\)</span>,</p>
</li>
<li><p><span class="SimpleMath">\(r \cdot (s + s') ~=~ r \cdot s + r \cdot s', \qquad\)</span> (so <span class="SimpleMath">\(r \cdot 0_S = 0_S ~\forall~ r \in R\)</span>),</p>
</li>
<li><p><span class="SimpleMath">\((r + r') \cdot s ~=~ r \cdot s + r' \cdot s, \qquad\)</span> (so <span class="SimpleMath">\(0_R \cdot s = 0_S ~\forall~ s \in S\)</span>),</p>
</li>
<li><p><span class="SimpleMath">\((rr') \cdot s ~=~ r \cdot (r' \cdot s), \qquad\)</span> (so <span class="SimpleMath">\(1_R \cdot s = s ~\forall~ s \in S\)</span> when <span class="SimpleMath">\(R\)</span> has a one),</p>
</li>
</ul>
<p>for all <span class="SimpleMath">\(k \in \)</span><strong class="button">k</strong>, <span class="SimpleMath">\(r,r' \in R\), and \(s,s' \in S\)</span>.</p>
<p>Notice in particular that, for fixed <span class="SimpleMath">\(r \in R\)</span>, the map <span class="SimpleMath">\(s \mapsto r \cdot s\)</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 <spanclass="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">\(r \cdot v = (1,2,3,4,5,6) + (1,4)(2,5)(3,6) + (1,6,5,4,3,2)\)</span>, as shown in <code class="func">AlgebraActionByHomomorphism</code> (<a href="chap2_mj.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">\(\theta : B \to A\)</span> be a surjective algebra homomorphism such that <span class="SimpleMath">\(\ker\theta\)</span> is contained in the annihilator of <spanclass="SimpleMath">\(B\)</span>. Then <span class="SimpleMath">\(A\)</span> acts on <span class="SimpleMath">\(B\)</span> by <span class="SimpleMath">\(a \cdot b = pb\)</span> where <span class="SimpleMath">\(p \in (\theta^{-1}a)\)</span>. Note that this action is well defined since <span class="SimpleMath">\(\theta^{-1}a = \{ p+k ~|~ k \in \ker\theta \}\)</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">\(\theta_1 : A1 \to Q1\)</span>. The kernel of <span class="SimpleMath">\(\theta\)</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">\(\alpha : A \to C\)</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">\(R \ltimes S\)</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">\(R \ltimes S\)</span> has basis</p>
<p class="center">\[
\{(r,0_S) ~|~ r \in B_R\} ~\cup~ \{(0_R,s) ~|~ s \in B_S\}
\]</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>
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
